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Theorem isuspgrim0 47809
Description: An isomorphism of simple pseudographs is a bijection between their vertices which induces a bijection between their edges. (Contributed by AV, 21-Apr-2025.)
Hypotheses
Ref Expression
isusgrim.v 𝑉 = (Vtx‘𝐺)
isusgrim.w 𝑊 = (Vtx‘𝐻)
isusgrim.e 𝐸 = (Edg‘𝐺)
isusgrim.d 𝐷 = (Edg‘𝐻)
Assertion
Ref Expression
isuspgrim0 ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹𝑋) → (𝐹 ∈ (𝐺 GraphIso 𝐻) ↔ (𝐹:𝑉1-1-onto𝑊 ∧ (𝑒𝐸 ↦ (𝐹𝑒)):𝐸1-1-onto𝐷)))
Distinct variable groups:   𝐷,𝑒   𝑒,𝐸   𝑒,𝐹   𝑒,𝐺   𝑒,𝐻   𝑒,𝑉   𝑒,𝑊   𝑒,𝑋

Proof of Theorem isuspgrim0
Dummy variables 𝑑 𝑖 𝑥 𝑗 𝑘 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isusgrim.v . . 3 𝑉 = (Vtx‘𝐺)
2 isusgrim.w . . 3 𝑊 = (Vtx‘𝐻)
3 eqid 2734 . . 3 (iEdg‘𝐺) = (iEdg‘𝐺)
4 eqid 2734 . . 3 (iEdg‘𝐻) = (iEdg‘𝐻)
51, 2, 3, 4isgrim 47805 . 2 ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹𝑋) → (𝐹 ∈ (𝐺 GraphIso 𝐻) ↔ (𝐹:𝑉1-1-onto𝑊 ∧ ∃𝑗(𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖))))))
6 isusgrim.e . . . . . . . . . . . . . . 15 𝐸 = (Edg‘𝐺)
76eleq2i 2830 . . . . . . . . . . . . . 14 (𝑒𝐸𝑒 ∈ (Edg‘𝐺))
8 uspgruhgr 29215 . . . . . . . . . . . . . . 15 (𝐺 ∈ USPGraph → 𝐺 ∈ UHGraph)
93uhgredgiedgb 29157 . . . . . . . . . . . . . . 15 (𝐺 ∈ UHGraph → (𝑒 ∈ (Edg‘𝐺) ↔ ∃𝑘 ∈ dom (iEdg‘𝐺)𝑒 = ((iEdg‘𝐺)‘𝑘)))
108, 9syl 17 . . . . . . . . . . . . . 14 (𝐺 ∈ USPGraph → (𝑒 ∈ (Edg‘𝐺) ↔ ∃𝑘 ∈ dom (iEdg‘𝐺)𝑒 = ((iEdg‘𝐺)‘𝑘)))
117, 10bitrid 283 . . . . . . . . . . . . 13 (𝐺 ∈ USPGraph → (𝑒𝐸 ↔ ∃𝑘 ∈ dom (iEdg‘𝐺)𝑒 = ((iEdg‘𝐺)‘𝑘)))
12113ad2ant1 1132 . . . . . . . . . . . 12 ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹𝑋) → (𝑒𝐸 ↔ ∃𝑘 ∈ dom (iEdg‘𝐺)𝑒 = ((iEdg‘𝐺)‘𝑘)))
1312ad2antrr 726 . . . . . . . . . . 11 ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹𝑋) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) → (𝑒𝐸 ↔ ∃𝑘 ∈ dom (iEdg‘𝐺)𝑒 = ((iEdg‘𝐺)‘𝑘)))
1413biimpa 476 . . . . . . . . . 10 (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹𝑋) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ 𝑒𝐸) → ∃𝑘 ∈ dom (iEdg‘𝐺)𝑒 = ((iEdg‘𝐺)‘𝑘))
15 2fveq3 6911 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 = 𝑘 → ((iEdg‘𝐻)‘(𝑗𝑖)) = ((iEdg‘𝐻)‘(𝑗𝑘)))
16 fveq2 6906 . . . . . . . . . . . . . . . . . . . . . 22 (𝑖 = 𝑘 → ((iEdg‘𝐺)‘𝑖) = ((iEdg‘𝐺)‘𝑘))
1716imaeq2d 6079 . . . . . . . . . . . . . . . . . . . . 21 (𝑖 = 𝑘 → (𝐹 “ ((iEdg‘𝐺)‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑘)))
1815, 17eqeq12d 2750 . . . . . . . . . . . . . . . . . . . 20 (𝑖 = 𝑘 → (((iEdg‘𝐻)‘(𝑗𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)) ↔ ((iEdg‘𝐻)‘(𝑗𝑘)) = (𝐹 “ ((iEdg‘𝐺)‘𝑘))))
1918rspcv 3617 . . . . . . . . . . . . . . . . . . 19 (𝑘 ∈ dom (iEdg‘𝐺) → (∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)) → ((iEdg‘𝐻)‘(𝑗𝑘)) = (𝐹 “ ((iEdg‘𝐺)‘𝑘))))
2019adantl 481 . . . . . . . . . . . . . . . . . 18 (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹𝑋) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ 𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻)) ∧ 𝑘 ∈ dom (iEdg‘𝐺)) → (∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)) → ((iEdg‘𝐻)‘(𝑗𝑘)) = (𝐹 “ ((iEdg‘𝐺)‘𝑘))))
21 uspgruhgr 29215 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐻 ∈ USPGraph → 𝐻 ∈ UHGraph)
224uhgrfun 29097 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝐻 ∈ UHGraph → Fun (iEdg‘𝐻))
2321, 22syl 17 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐻 ∈ USPGraph → Fun (iEdg‘𝐻))
24233ad2ant2 1133 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹𝑋) → Fun (iEdg‘𝐻))
2524ad3antrrr 730 . . . . . . . . . . . . . . . . . . . . 21 (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹𝑋) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ 𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻)) ∧ 𝑘 ∈ dom (iEdg‘𝐺)) → Fun (iEdg‘𝐻))
26 f1of 6848 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) → 𝑗:dom (iEdg‘𝐺)⟶dom (iEdg‘𝐻))
2726adantl 481 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹𝑋) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ 𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻)) → 𝑗:dom (iEdg‘𝐺)⟶dom (iEdg‘𝐻))
2827ffvelcdmda 7103 . . . . . . . . . . . . . . . . . . . . 21 (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹𝑋) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ 𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻)) ∧ 𝑘 ∈ dom (iEdg‘𝐺)) → (𝑗𝑘) ∈ dom (iEdg‘𝐻))
294iedgedg 29081 . . . . . . . . . . . . . . . . . . . . 21 ((Fun (iEdg‘𝐻) ∧ (𝑗𝑘) ∈ dom (iEdg‘𝐻)) → ((iEdg‘𝐻)‘(𝑗𝑘)) ∈ (Edg‘𝐻))
3025, 28, 29syl2anc 584 . . . . . . . . . . . . . . . . . . . 20 (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹𝑋) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ 𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻)) ∧ 𝑘 ∈ dom (iEdg‘𝐺)) → ((iEdg‘𝐻)‘(𝑗𝑘)) ∈ (Edg‘𝐻))
31 isusgrim.d . . . . . . . . . . . . . . . . . . . . 21 𝐷 = (Edg‘𝐻)
3231eleq2i 2830 . . . . . . . . . . . . . . . . . . . 20 (((iEdg‘𝐻)‘(𝑗𝑘)) ∈ 𝐷 ↔ ((iEdg‘𝐻)‘(𝑗𝑘)) ∈ (Edg‘𝐻))
3330, 32sylibr 234 . . . . . . . . . . . . . . . . . . 19 (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹𝑋) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ 𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻)) ∧ 𝑘 ∈ dom (iEdg‘𝐺)) → ((iEdg‘𝐻)‘(𝑗𝑘)) ∈ 𝐷)
34 eleq1 2826 . . . . . . . . . . . . . . . . . . 19 (((iEdg‘𝐻)‘(𝑗𝑘)) = (𝐹 “ ((iEdg‘𝐺)‘𝑘)) → (((iEdg‘𝐻)‘(𝑗𝑘)) ∈ 𝐷 ↔ (𝐹 “ ((iEdg‘𝐺)‘𝑘)) ∈ 𝐷))
3533, 34syl5ibcom 245 . . . . . . . . . . . . . . . . . 18 (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹𝑋) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ 𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻)) ∧ 𝑘 ∈ dom (iEdg‘𝐺)) → (((iEdg‘𝐻)‘(𝑗𝑘)) = (𝐹 “ ((iEdg‘𝐺)‘𝑘)) → (𝐹 “ ((iEdg‘𝐺)‘𝑘)) ∈ 𝐷))
3620, 35syld 47 . . . . . . . . . . . . . . . . 17 (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹𝑋) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ 𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻)) ∧ 𝑘 ∈ dom (iEdg‘𝐺)) → (∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)) → (𝐹 “ ((iEdg‘𝐺)‘𝑘)) ∈ 𝐷))
3736ex 412 . . . . . . . . . . . . . . . 16 ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹𝑋) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ 𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻)) → (𝑘 ∈ dom (iEdg‘𝐺) → (∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)) → (𝐹 “ ((iEdg‘𝐺)‘𝑘)) ∈ 𝐷)))
3837com23 86 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹𝑋) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ 𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻)) → (∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)) → (𝑘 ∈ dom (iEdg‘𝐺) → (𝐹 “ ((iEdg‘𝐺)‘𝑘)) ∈ 𝐷)))
3938impr 454 . . . . . . . . . . . . . 14 ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹𝑋) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) → (𝑘 ∈ dom (iEdg‘𝐺) → (𝐹 “ ((iEdg‘𝐺)‘𝑘)) ∈ 𝐷))
4039adantr 480 . . . . . . . . . . . . 13 (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹𝑋) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ 𝑒𝐸) → (𝑘 ∈ dom (iEdg‘𝐺) → (𝐹 “ ((iEdg‘𝐺)‘𝑘)) ∈ 𝐷))
4140imp 406 . . . . . . . . . . . 12 ((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹𝑋) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ 𝑒𝐸) ∧ 𝑘 ∈ dom (iEdg‘𝐺)) → (𝐹 “ ((iEdg‘𝐺)‘𝑘)) ∈ 𝐷)
42 imaeq2 6075 . . . . . . . . . . . . 13 (𝑒 = ((iEdg‘𝐺)‘𝑘) → (𝐹𝑒) = (𝐹 “ ((iEdg‘𝐺)‘𝑘)))
4342eleq1d 2823 . . . . . . . . . . . 12 (𝑒 = ((iEdg‘𝐺)‘𝑘) → ((𝐹𝑒) ∈ 𝐷 ↔ (𝐹 “ ((iEdg‘𝐺)‘𝑘)) ∈ 𝐷))
4441, 43syl5ibrcom 247 . . . . . . . . . . 11 ((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹𝑋) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ 𝑒𝐸) ∧ 𝑘 ∈ dom (iEdg‘𝐺)) → (𝑒 = ((iEdg‘𝐺)‘𝑘) → (𝐹𝑒) ∈ 𝐷))
4544rexlimdva 3152 . . . . . . . . . 10 (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹𝑋) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ 𝑒𝐸) → (∃𝑘 ∈ dom (iEdg‘𝐺)𝑒 = ((iEdg‘𝐺)‘𝑘) → (𝐹𝑒) ∈ 𝐷))
4614, 45mpd 15 . . . . . . . . 9 (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹𝑋) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ 𝑒𝐸) → (𝐹𝑒) ∈ 𝐷)
4746ralrimiva 3143 . . . . . . . 8 ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹𝑋) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) → ∀𝑒𝐸 (𝐹𝑒) ∈ 𝐷)
4831eleq2i 2830 . . . . . . . . . . . . 13 (𝑑𝐷𝑑 ∈ (Edg‘𝐻))
494uhgredgiedgb 29157 . . . . . . . . . . . . . 14 (𝐻 ∈ UHGraph → (𝑑 ∈ (Edg‘𝐻) ↔ ∃𝑘 ∈ dom (iEdg‘𝐻)𝑑 = ((iEdg‘𝐻)‘𝑘)))
5021, 49syl 17 . . . . . . . . . . . . 13 (𝐻 ∈ USPGraph → (𝑑 ∈ (Edg‘𝐻) ↔ ∃𝑘 ∈ dom (iEdg‘𝐻)𝑑 = ((iEdg‘𝐻)‘𝑘)))
5148, 50bitrid 283 . . . . . . . . . . . 12 (𝐻 ∈ USPGraph → (𝑑𝐷 ↔ ∃𝑘 ∈ dom (iEdg‘𝐻)𝑑 = ((iEdg‘𝐻)‘𝑘)))
52513ad2ant2 1133 . . . . . . . . . . 11 ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹𝑋) → (𝑑𝐷 ↔ ∃𝑘 ∈ dom (iEdg‘𝐻)𝑑 = ((iEdg‘𝐻)‘𝑘)))
5352ad2antrr 726 . . . . . . . . . 10 ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹𝑋) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) → (𝑑𝐷 ↔ ∃𝑘 ∈ dom (iEdg‘𝐻)𝑑 = ((iEdg‘𝐻)‘𝑘)))
54 simprl 771 . . . . . . . . . . . . 13 ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹𝑋) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) → 𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻))
55 f1ocnvdm 7304 . . . . . . . . . . . . 13 ((𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ 𝑘 ∈ dom (iEdg‘𝐻)) → (𝑗𝑘) ∈ dom (iEdg‘𝐺))
5654, 55sylan 580 . . . . . . . . . . . 12 (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹𝑋) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ 𝑘 ∈ dom (iEdg‘𝐻)) → (𝑗𝑘) ∈ dom (iEdg‘𝐺))
57 2fveq3 6911 . . . . . . . . . . . . . . . . . 18 (𝑖 = (𝑗𝑘) → ((iEdg‘𝐻)‘(𝑗𝑖)) = ((iEdg‘𝐻)‘(𝑗‘(𝑗𝑘))))
58 fveq2 6906 . . . . . . . . . . . . . . . . . . 19 (𝑖 = (𝑗𝑘) → ((iEdg‘𝐺)‘𝑖) = ((iEdg‘𝐺)‘(𝑗𝑘)))
5958imaeq2d 6079 . . . . . . . . . . . . . . . . . 18 (𝑖 = (𝑗𝑘) → (𝐹 “ ((iEdg‘𝐺)‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘(𝑗𝑘))))
6057, 59eqeq12d 2750 . . . . . . . . . . . . . . . . 17 (𝑖 = (𝑗𝑘) → (((iEdg‘𝐻)‘(𝑗𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)) ↔ ((iEdg‘𝐻)‘(𝑗‘(𝑗𝑘))) = (𝐹 “ ((iEdg‘𝐺)‘(𝑗𝑘)))))
6160rspccv 3618 . . . . . . . . . . . . . . . 16 (∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)) → ((𝑗𝑘) ∈ dom (iEdg‘𝐺) → ((iEdg‘𝐻)‘(𝑗‘(𝑗𝑘))) = (𝐹 “ ((iEdg‘𝐺)‘(𝑗𝑘)))))
6261adantl 481 . . . . . . . . . . . . . . 15 ((𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖))) → ((𝑗𝑘) ∈ dom (iEdg‘𝐺) → ((iEdg‘𝐻)‘(𝑗‘(𝑗𝑘))) = (𝐹 “ ((iEdg‘𝐺)‘(𝑗𝑘)))))
6362adantl 481 . . . . . . . . . . . . . 14 ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹𝑋) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) → ((𝑗𝑘) ∈ dom (iEdg‘𝐺) → ((iEdg‘𝐻)‘(𝑗‘(𝑗𝑘))) = (𝐹 “ ((iEdg‘𝐺)‘(𝑗𝑘)))))
6463adantr 480 . . . . . . . . . . . . 13 (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹𝑋) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ 𝑘 ∈ dom (iEdg‘𝐻)) → ((𝑗𝑘) ∈ dom (iEdg‘𝐺) → ((iEdg‘𝐻)‘(𝑗‘(𝑗𝑘))) = (𝐹 “ ((iEdg‘𝐺)‘(𝑗𝑘)))))
65 f1ocnvfv2 7296 . . . . . . . . . . . . . . . 16 ((𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ 𝑘 ∈ dom (iEdg‘𝐻)) → (𝑗‘(𝑗𝑘)) = 𝑘)
6654, 65sylan 580 . . . . . . . . . . . . . . 15 (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹𝑋) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ 𝑘 ∈ dom (iEdg‘𝐻)) → (𝑗‘(𝑗𝑘)) = 𝑘)
6766fveqeq2d 6914 . . . . . . . . . . . . . 14 (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹𝑋) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ 𝑘 ∈ dom (iEdg‘𝐻)) → (((iEdg‘𝐻)‘(𝑗‘(𝑗𝑘))) = (𝐹 “ ((iEdg‘𝐺)‘(𝑗𝑘))) ↔ ((iEdg‘𝐻)‘𝑘) = (𝐹 “ ((iEdg‘𝐺)‘(𝑗𝑘)))))
68 eqeq2 2746 . . . . . . . . . . . . . . . . 17 (((iEdg‘𝐻)‘𝑘) = (𝐹 “ ((iEdg‘𝐺)‘(𝑗𝑘))) → (𝑑 = ((iEdg‘𝐻)‘𝑘) ↔ 𝑑 = (𝐹 “ ((iEdg‘𝐺)‘(𝑗𝑘)))))
6968adantl 481 . . . . . . . . . . . . . . . 16 ((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹𝑋) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ 𝑘 ∈ dom (iEdg‘𝐻)) ∧ ((iEdg‘𝐻)‘𝑘) = (𝐹 “ ((iEdg‘𝐺)‘(𝑗𝑘)))) → (𝑑 = ((iEdg‘𝐻)‘𝑘) ↔ 𝑑 = (𝐹 “ ((iEdg‘𝐺)‘(𝑗𝑘)))))
70 simpll1 1211 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹𝑋) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) → 𝐺 ∈ USPGraph)
716, 3uspgriedgedg 29207 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐺 ∈ USPGraph ∧ (𝑗𝑘) ∈ dom (iEdg‘𝐺)) → ∃!𝑒𝐸 𝑒 = ((iEdg‘𝐺)‘(𝑗𝑘)))
7270, 56, 71syl2an2r 685 . . . . . . . . . . . . . . . . . . . . 21 (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹𝑋) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ 𝑘 ∈ dom (iEdg‘𝐻)) → ∃!𝑒𝐸 𝑒 = ((iEdg‘𝐺)‘(𝑗𝑘)))
73 eqcom 2741 . . . . . . . . . . . . . . . . . . . . . 22 (((iEdg‘𝐺)‘(𝑗𝑘)) = 𝑒𝑒 = ((iEdg‘𝐺)‘(𝑗𝑘)))
7473reubii 3386 . . . . . . . . . . . . . . . . . . . . 21 (∃!𝑒𝐸 ((iEdg‘𝐺)‘(𝑗𝑘)) = 𝑒 ↔ ∃!𝑒𝐸 𝑒 = ((iEdg‘𝐺)‘(𝑗𝑘)))
7572, 74sylibr 234 . . . . . . . . . . . . . . . . . . . 20 (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹𝑋) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ 𝑘 ∈ dom (iEdg‘𝐻)) → ∃!𝑒𝐸 ((iEdg‘𝐺)‘(𝑗𝑘)) = 𝑒)
76 f1of1 6847 . . . . . . . . . . . . . . . . . . . . . . 23 (𝐹:𝑉1-1-onto𝑊𝐹:𝑉1-1𝑊)
7776ad4antlr 733 . . . . . . . . . . . . . . . . . . . . . 22 ((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹𝑋) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ 𝑘 ∈ dom (iEdg‘𝐻)) ∧ 𝑒𝐸) → 𝐹:𝑉1-1𝑊)
78 uspgrupgr 29209 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph)
79783ad2ant1 1132 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹𝑋) → 𝐺 ∈ UPGraph)
8079ad3antrrr 730 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹𝑋) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ 𝑘 ∈ dom (iEdg‘𝐻)) → 𝐺 ∈ UPGraph)
8180, 56jca 511 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹𝑋) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ 𝑘 ∈ dom (iEdg‘𝐻)) → (𝐺 ∈ UPGraph ∧ (𝑗𝑘) ∈ dom (iEdg‘𝐺)))
8281adantr 480 . . . . . . . . . . . . . . . . . . . . . . 23 ((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹𝑋) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ 𝑘 ∈ dom (iEdg‘𝐻)) ∧ 𝑒𝐸) → (𝐺 ∈ UPGraph ∧ (𝑗𝑘) ∈ dom (iEdg‘𝐺)))
831, 3upgrss 29119 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐺 ∈ UPGraph ∧ (𝑗𝑘) ∈ dom (iEdg‘𝐺)) → ((iEdg‘𝐺)‘(𝑗𝑘)) ⊆ 𝑉)
8482, 83syl 17 . . . . . . . . . . . . . . . . . . . . . 22 ((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹𝑋) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ 𝑘 ∈ dom (iEdg‘𝐻)) ∧ 𝑒𝐸) → ((iEdg‘𝐺)‘(𝑗𝑘)) ⊆ 𝑉)
857biimpi 216 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑒𝐸𝑒 ∈ (Edg‘𝐺))
86 edgupgr 29165 . . . . . . . . . . . . . . . . . . . . . . . . . 26 ((𝐺 ∈ UPGraph ∧ 𝑒 ∈ (Edg‘𝐺)) → (𝑒 ∈ 𝒫 (Vtx‘𝐺) ∧ 𝑒 ≠ ∅ ∧ (♯‘𝑒) ≤ 2))
8780, 85, 86syl2an 596 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹𝑋) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ 𝑘 ∈ dom (iEdg‘𝐻)) ∧ 𝑒𝐸) → (𝑒 ∈ 𝒫 (Vtx‘𝐺) ∧ 𝑒 ≠ ∅ ∧ (♯‘𝑒) ≤ 2))
8887simp1d 1141 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹𝑋) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ 𝑘 ∈ dom (iEdg‘𝐻)) ∧ 𝑒𝐸) → 𝑒 ∈ 𝒫 (Vtx‘𝐺))
8988elpwid 4613 . . . . . . . . . . . . . . . . . . . . . . 23 ((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹𝑋) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ 𝑘 ∈ dom (iEdg‘𝐻)) ∧ 𝑒𝐸) → 𝑒 ⊆ (Vtx‘𝐺))
9089, 1sseqtrrdi 4046 . . . . . . . . . . . . . . . . . . . . . 22 ((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹𝑋) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ 𝑘 ∈ dom (iEdg‘𝐻)) ∧ 𝑒𝐸) → 𝑒𝑉)
91 f1imaeq 7284 . . . . . . . . . . . . . . . . . . . . . 22 ((𝐹:𝑉1-1𝑊 ∧ (((iEdg‘𝐺)‘(𝑗𝑘)) ⊆ 𝑉𝑒𝑉)) → ((𝐹 “ ((iEdg‘𝐺)‘(𝑗𝑘))) = (𝐹𝑒) ↔ ((iEdg‘𝐺)‘(𝑗𝑘)) = 𝑒))
9277, 84, 90, 91syl12anc 837 . . . . . . . . . . . . . . . . . . . . 21 ((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹𝑋) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ 𝑘 ∈ dom (iEdg‘𝐻)) ∧ 𝑒𝐸) → ((𝐹 “ ((iEdg‘𝐺)‘(𝑗𝑘))) = (𝐹𝑒) ↔ ((iEdg‘𝐺)‘(𝑗𝑘)) = 𝑒))
9392reubidva 3393 . . . . . . . . . . . . . . . . . . . 20 (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹𝑋) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ 𝑘 ∈ dom (iEdg‘𝐻)) → (∃!𝑒𝐸 (𝐹 “ ((iEdg‘𝐺)‘(𝑗𝑘))) = (𝐹𝑒) ↔ ∃!𝑒𝐸 ((iEdg‘𝐺)‘(𝑗𝑘)) = 𝑒))
9475, 93mpbird 257 . . . . . . . . . . . . . . . . . . 19 (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹𝑋) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ 𝑘 ∈ dom (iEdg‘𝐻)) → ∃!𝑒𝐸 (𝐹 “ ((iEdg‘𝐺)‘(𝑗𝑘))) = (𝐹𝑒))
9594ad2antrr 726 . . . . . . . . . . . . . . . . . 18 (((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹𝑋) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ 𝑘 ∈ dom (iEdg‘𝐻)) ∧ ((iEdg‘𝐻)‘𝑘) = (𝐹 “ ((iEdg‘𝐺)‘(𝑗𝑘)))) ∧ 𝑑 = (𝐹 “ ((iEdg‘𝐺)‘(𝑗𝑘)))) → ∃!𝑒𝐸 (𝐹 “ ((iEdg‘𝐺)‘(𝑗𝑘))) = (𝐹𝑒))
96 eqeq1 2738 . . . . . . . . . . . . . . . . . . . 20 (𝑑 = (𝐹 “ ((iEdg‘𝐺)‘(𝑗𝑘))) → (𝑑 = (𝐹𝑒) ↔ (𝐹 “ ((iEdg‘𝐺)‘(𝑗𝑘))) = (𝐹𝑒)))
9796reubidv 3395 . . . . . . . . . . . . . . . . . . 19 (𝑑 = (𝐹 “ ((iEdg‘𝐺)‘(𝑗𝑘))) → (∃!𝑒𝐸 𝑑 = (𝐹𝑒) ↔ ∃!𝑒𝐸 (𝐹 “ ((iEdg‘𝐺)‘(𝑗𝑘))) = (𝐹𝑒)))
9897adantl 481 . . . . . . . . . . . . . . . . . 18 (((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹𝑋) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ 𝑘 ∈ dom (iEdg‘𝐻)) ∧ ((iEdg‘𝐻)‘𝑘) = (𝐹 “ ((iEdg‘𝐺)‘(𝑗𝑘)))) ∧ 𝑑 = (𝐹 “ ((iEdg‘𝐺)‘(𝑗𝑘)))) → (∃!𝑒𝐸 𝑑 = (𝐹𝑒) ↔ ∃!𝑒𝐸 (𝐹 “ ((iEdg‘𝐺)‘(𝑗𝑘))) = (𝐹𝑒)))
9995, 98mpbird 257 . . . . . . . . . . . . . . . . 17 (((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹𝑋) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ 𝑘 ∈ dom (iEdg‘𝐻)) ∧ ((iEdg‘𝐻)‘𝑘) = (𝐹 “ ((iEdg‘𝐺)‘(𝑗𝑘)))) ∧ 𝑑 = (𝐹 “ ((iEdg‘𝐺)‘(𝑗𝑘)))) → ∃!𝑒𝐸 𝑑 = (𝐹𝑒))
10099ex 412 . . . . . . . . . . . . . . . 16 ((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹𝑋) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ 𝑘 ∈ dom (iEdg‘𝐻)) ∧ ((iEdg‘𝐻)‘𝑘) = (𝐹 “ ((iEdg‘𝐺)‘(𝑗𝑘)))) → (𝑑 = (𝐹 “ ((iEdg‘𝐺)‘(𝑗𝑘))) → ∃!𝑒𝐸 𝑑 = (𝐹𝑒)))
10169, 100sylbid 240 . . . . . . . . . . . . . . 15 ((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹𝑋) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ 𝑘 ∈ dom (iEdg‘𝐻)) ∧ ((iEdg‘𝐻)‘𝑘) = (𝐹 “ ((iEdg‘𝐺)‘(𝑗𝑘)))) → (𝑑 = ((iEdg‘𝐻)‘𝑘) → ∃!𝑒𝐸 𝑑 = (𝐹𝑒)))
102101ex 412 . . . . . . . . . . . . . 14 (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹𝑋) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ 𝑘 ∈ dom (iEdg‘𝐻)) → (((iEdg‘𝐻)‘𝑘) = (𝐹 “ ((iEdg‘𝐺)‘(𝑗𝑘))) → (𝑑 = ((iEdg‘𝐻)‘𝑘) → ∃!𝑒𝐸 𝑑 = (𝐹𝑒))))
10367, 102sylbid 240 . . . . . . . . . . . . 13 (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹𝑋) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ 𝑘 ∈ dom (iEdg‘𝐻)) → (((iEdg‘𝐻)‘(𝑗‘(𝑗𝑘))) = (𝐹 “ ((iEdg‘𝐺)‘(𝑗𝑘))) → (𝑑 = ((iEdg‘𝐻)‘𝑘) → ∃!𝑒𝐸 𝑑 = (𝐹𝑒))))
10464, 103syld 47 . . . . . . . . . . . 12 (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹𝑋) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ 𝑘 ∈ dom (iEdg‘𝐻)) → ((𝑗𝑘) ∈ dom (iEdg‘𝐺) → (𝑑 = ((iEdg‘𝐻)‘𝑘) → ∃!𝑒𝐸 𝑑 = (𝐹𝑒))))
10556, 104mpd 15 . . . . . . . . . . 11 (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹𝑋) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ∧ 𝑘 ∈ dom (iEdg‘𝐻)) → (𝑑 = ((iEdg‘𝐻)‘𝑘) → ∃!𝑒𝐸 𝑑 = (𝐹𝑒)))
106105rexlimdva 3152 . . . . . . . . . 10 ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹𝑋) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) → (∃𝑘 ∈ dom (iEdg‘𝐻)𝑑 = ((iEdg‘𝐻)‘𝑘) → ∃!𝑒𝐸 𝑑 = (𝐹𝑒)))
10753, 106sylbid 240 . . . . . . . . 9 ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹𝑋) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) → (𝑑𝐷 → ∃!𝑒𝐸 𝑑 = (𝐹𝑒)))
108107ralrimiv 3142 . . . . . . . 8 ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹𝑋) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) → ∀𝑑𝐷 ∃!𝑒𝐸 𝑑 = (𝐹𝑒))
109 imaeq2 6075 . . . . . . . . . 10 (𝑥 = 𝑒 → (𝐹𝑥) = (𝐹𝑒))
110109cbvmptv 5260 . . . . . . . . 9 (𝑥𝐸 ↦ (𝐹𝑥)) = (𝑒𝐸 ↦ (𝐹𝑒))
111110f1ompt 7130 . . . . . . . 8 ((𝑥𝐸 ↦ (𝐹𝑥)):𝐸1-1-onto𝐷 ↔ (∀𝑒𝐸 (𝐹𝑒) ∈ 𝐷 ∧ ∀𝑑𝐷 ∃!𝑒𝐸 𝑑 = (𝐹𝑒)))
11247, 108, 111sylanbrc 583 . . . . . . 7 ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹𝑋) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) → (𝑥𝐸 ↦ (𝐹𝑥)):𝐸1-1-onto𝐷)
113112ex 412 . . . . . 6 (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹𝑋) ∧ 𝐹:𝑉1-1-onto𝑊) → ((𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖))) → (𝑥𝐸 ↦ (𝐹𝑥)):𝐸1-1-onto𝐷))
114113exlimdv 1930 . . . . 5 (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹𝑋) ∧ 𝐹:𝑉1-1-onto𝑊) → (∃𝑗(𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖))) → (𝑥𝐸 ↦ (𝐹𝑥)):𝐸1-1-onto𝐷))
115 fvex 6919 . . . . . . . . . 10 (iEdg‘𝐺) ∈ V
116115dmex 7931 . . . . . . . . 9 dom (iEdg‘𝐺) ∈ V
117116mptex 7242 . . . . . . . 8 (𝑒 ∈ dom (iEdg‘𝐺) ↦ ((iEdg‘𝐻)‘((𝑥𝐸 ↦ (𝐹𝑥))‘((iEdg‘𝐺)‘𝑒)))) ∈ V
118117a1i 11 . . . . . . 7 ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹𝑋) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ (𝑥𝐸 ↦ (𝐹𝑥)):𝐸1-1-onto𝐷) → (𝑒 ∈ dom (iEdg‘𝐺) ↦ ((iEdg‘𝐻)‘((𝑥𝐸 ↦ (𝐹𝑥))‘((iEdg‘𝐺)‘𝑒)))) ∈ V)
119 eqid 2734 . . . . . . . 8 (𝑒 ∈ dom (iEdg‘𝐺) ↦ ((iEdg‘𝐻)‘((𝑥𝐸 ↦ (𝐹𝑥))‘((iEdg‘𝐺)‘𝑒)))) = (𝑒 ∈ dom (iEdg‘𝐺) ↦ ((iEdg‘𝐻)‘((𝑥𝐸 ↦ (𝐹𝑥))‘((iEdg‘𝐺)‘𝑒))))
1201, 2, 6, 31, 3, 4, 110, 119isuspgrim0lem 47808 . . . . . . 7 ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹𝑋) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ (𝑥𝐸 ↦ (𝐹𝑥)):𝐸1-1-onto𝐷) → ((𝑒 ∈ dom (iEdg‘𝐺) ↦ ((iEdg‘𝐻)‘((𝑥𝐸 ↦ (𝐹𝑥))‘((iEdg‘𝐺)‘𝑒)))):dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘((𝑒 ∈ dom (iEdg‘𝐺) ↦ ((iEdg‘𝐻)‘((𝑥𝐸 ↦ (𝐹𝑥))‘((iEdg‘𝐺)‘𝑒))))‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖))))
121 f1oeq1 6836 . . . . . . . 8 (𝑗 = (𝑒 ∈ dom (iEdg‘𝐺) ↦ ((iEdg‘𝐻)‘((𝑥𝐸 ↦ (𝐹𝑥))‘((iEdg‘𝐺)‘𝑒)))) → (𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ↔ (𝑒 ∈ dom (iEdg‘𝐺) ↦ ((iEdg‘𝐻)‘((𝑥𝐸 ↦ (𝐹𝑥))‘((iEdg‘𝐺)‘𝑒)))):dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻)))
122 fveq1 6905 . . . . . . . . . 10 (𝑗 = (𝑒 ∈ dom (iEdg‘𝐺) ↦ ((iEdg‘𝐻)‘((𝑥𝐸 ↦ (𝐹𝑥))‘((iEdg‘𝐺)‘𝑒)))) → (𝑗𝑖) = ((𝑒 ∈ dom (iEdg‘𝐺) ↦ ((iEdg‘𝐻)‘((𝑥𝐸 ↦ (𝐹𝑥))‘((iEdg‘𝐺)‘𝑒))))‘𝑖))
123122fveqeq2d 6914 . . . . . . . . 9 (𝑗 = (𝑒 ∈ dom (iEdg‘𝐺) ↦ ((iEdg‘𝐻)‘((𝑥𝐸 ↦ (𝐹𝑥))‘((iEdg‘𝐺)‘𝑒)))) → (((iEdg‘𝐻)‘(𝑗𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)) ↔ ((iEdg‘𝐻)‘((𝑒 ∈ dom (iEdg‘𝐺) ↦ ((iEdg‘𝐻)‘((𝑥𝐸 ↦ (𝐹𝑥))‘((iEdg‘𝐺)‘𝑒))))‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖))))
124123ralbidv 3175 . . . . . . . 8 (𝑗 = (𝑒 ∈ dom (iEdg‘𝐺) ↦ ((iEdg‘𝐻)‘((𝑥𝐸 ↦ (𝐹𝑥))‘((iEdg‘𝐺)‘𝑒)))) → (∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)) ↔ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘((𝑒 ∈ dom (iEdg‘𝐺) ↦ ((iEdg‘𝐻)‘((𝑥𝐸 ↦ (𝐹𝑥))‘((iEdg‘𝐺)‘𝑒))))‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖))))
125121, 124anbi12d 632 . . . . . . 7 (𝑗 = (𝑒 ∈ dom (iEdg‘𝐺) ↦ ((iEdg‘𝐻)‘((𝑥𝐸 ↦ (𝐹𝑥))‘((iEdg‘𝐺)‘𝑒)))) → ((𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖))) ↔ ((𝑒 ∈ dom (iEdg‘𝐺) ↦ ((iEdg‘𝐻)‘((𝑥𝐸 ↦ (𝐹𝑥))‘((iEdg‘𝐺)‘𝑒)))):dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘((𝑒 ∈ dom (iEdg‘𝐺) ↦ ((iEdg‘𝐻)‘((𝑥𝐸 ↦ (𝐹𝑥))‘((iEdg‘𝐺)‘𝑒))))‘𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))))
126118, 120, 125spcedv 3597 . . . . . 6 ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹𝑋) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ (𝑥𝐸 ↦ (𝐹𝑥)):𝐸1-1-onto𝐷) → ∃𝑗(𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖))))
127126ex 412 . . . . 5 (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹𝑋) ∧ 𝐹:𝑉1-1-onto𝑊) → ((𝑥𝐸 ↦ (𝐹𝑥)):𝐸1-1-onto𝐷 → ∃𝑗(𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))))
128114, 127impbid 212 . . . 4 (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹𝑋) ∧ 𝐹:𝑉1-1-onto𝑊) → (∃𝑗(𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖))) ↔ (𝑥𝐸 ↦ (𝐹𝑥)):𝐸1-1-onto𝐷))
129 f1oeq1 6836 . . . . 5 ((𝑥𝐸 ↦ (𝐹𝑥)) = (𝑒𝐸 ↦ (𝐹𝑒)) → ((𝑥𝐸 ↦ (𝐹𝑥)):𝐸1-1-onto𝐷 ↔ (𝑒𝐸 ↦ (𝐹𝑒)):𝐸1-1-onto𝐷))
130110, 129mp1i 13 . . . 4 (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹𝑋) ∧ 𝐹:𝑉1-1-onto𝑊) → ((𝑥𝐸 ↦ (𝐹𝑥)):𝐸1-1-onto𝐷 ↔ (𝑒𝐸 ↦ (𝐹𝑒)):𝐸1-1-onto𝐷))
131128, 130bitrd 279 . . 3 (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹𝑋) ∧ 𝐹:𝑉1-1-onto𝑊) → (∃𝑗(𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖))) ↔ (𝑒𝐸 ↦ (𝐹𝑒)):𝐸1-1-onto𝐷))
132131pm5.32da 579 . 2 ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹𝑋) → ((𝐹:𝑉1-1-onto𝑊 ∧ ∃𝑗(𝑗:dom (iEdg‘𝐺)–1-1-onto→dom (iEdg‘𝐻) ∧ ∀𝑖 ∈ dom (iEdg‘𝐺)((iEdg‘𝐻)‘(𝑗𝑖)) = (𝐹 “ ((iEdg‘𝐺)‘𝑖)))) ↔ (𝐹:𝑉1-1-onto𝑊 ∧ (𝑒𝐸 ↦ (𝐹𝑒)):𝐸1-1-onto𝐷)))
1335, 132bitrd 279 1 ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph ∧ 𝐹𝑋) → (𝐹 ∈ (𝐺 GraphIso 𝐻) ↔ (𝐹:𝑉1-1-onto𝑊 ∧ (𝑒𝐸 ↦ (𝐹𝑒)):𝐸1-1-onto𝐷)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1536  wex 1775  wcel 2105  wne 2937  wral 3058  wrex 3067  ∃!wreu 3375  Vcvv 3477  wss 3962  c0 4338  𝒫 cpw 4604   class class class wbr 5147  cmpt 5230  ccnv 5687  dom cdm 5688  cima 5691  Fun wfun 6556  wf 6558  1-1wf1 6559  1-1-ontowf1o 6561  cfv 6562  (class class class)co 7430  cle 11293  2c2 12318  chash 14365  Vtxcvtx 29027  iEdgciedg 29028  Edgcedg 29078  UHGraphcuhgr 29087  UPGraphcupgr 29111  USPGraphcuspgr 29179   GraphIso cgrim 47798
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-rep 5284  ax-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rmo 3377  df-reu 3378  df-rab 3433  df-v 3479  df-sbc 3791  df-csb 3908  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-ima 5701  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-f1 6567  df-fo 6568  df-f1o 6569  df-fv 6570  df-ov 7433  df-oprab 7434  df-mpo 7435  df-map 8866  df-edg 29079  df-uhgr 29089  df-upgr 29113  df-uspgr 29181  df-grim 47801
This theorem is referenced by:  uspgrimprop  47810  isuspgrim  47812
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