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Theorem isuspgrimlem 47811
Description: Lemma for isuspgrim 47812. (Contributed by AV, 27-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
isuspgrimlem ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ ∀𝑥𝑉𝑦𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹𝑥), (𝐹𝑦)} ∈ 𝐷)) → (𝑒𝐸 ↦ (𝐹𝑒)):𝐸1-1-onto𝐷)
Distinct variable groups:   𝑥,𝐸   𝑥,𝐹   𝐷,𝑒   𝑒,𝐸,𝑥   𝑒,𝐹   𝑒,𝐺   𝑒,𝐻   𝑒,𝑉   𝑒,𝑊   𝑥,𝐷,𝑦   𝑦,𝐸,𝑒   𝑦,𝐹   𝑥,𝐺,𝑦   𝑥,𝐻,𝑦   𝑥,𝑉,𝑦   𝑥,𝑊,𝑦

Proof of Theorem isuspgrimlem
Dummy variables 𝑑 𝑎 𝑏 𝑚 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 uspgrupgr 29209 . . . . . . . . 9 (𝐺 ∈ USPGraph → 𝐺 ∈ UPGraph)
21adantr 480 . . . . . . . 8 ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) → 𝐺 ∈ UPGraph)
32adantr 480 . . . . . . 7 (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉1-1-onto𝑊) → 𝐺 ∈ UPGraph)
43adantr 480 . . . . . 6 ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ ∀𝑥𝑉𝑦𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹𝑥), (𝐹𝑦)} ∈ 𝐷)) → 𝐺 ∈ UPGraph)
5 isusgrim.v . . . . . . 7 𝑉 = (Vtx‘𝐺)
6 isusgrim.e . . . . . . 7 𝐸 = (Edg‘𝐺)
75, 6upgredg 29168 . . . . . 6 ((𝐺 ∈ UPGraph ∧ 𝑒𝐸) → ∃𝑎𝑉𝑏𝑉 𝑒 = {𝑎, 𝑏})
84, 7sylan 580 . . . . 5 (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ ∀𝑥𝑉𝑦𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹𝑥), (𝐹𝑦)} ∈ 𝐷)) ∧ 𝑒𝐸) → ∃𝑎𝑉𝑏𝑉 𝑒 = {𝑎, 𝑏})
9 preq12 4739 . . . . . . . . . . . . . . . . . . . 20 ((𝑥 = 𝑎𝑦 = 𝑏) → {𝑥, 𝑦} = {𝑎, 𝑏})
109eleq1d 2823 . . . . . . . . . . . . . . . . . . 19 ((𝑥 = 𝑎𝑦 = 𝑏) → ({𝑥, 𝑦} ∈ 𝐸 ↔ {𝑎, 𝑏} ∈ 𝐸))
11 fveq2 6906 . . . . . . . . . . . . . . . . . . . . . 22 (𝑥 = 𝑎 → (𝐹𝑥) = (𝐹𝑎))
1211adantr 480 . . . . . . . . . . . . . . . . . . . . 21 ((𝑥 = 𝑎𝑦 = 𝑏) → (𝐹𝑥) = (𝐹𝑎))
13 fveq2 6906 . . . . . . . . . . . . . . . . . . . . . 22 (𝑦 = 𝑏 → (𝐹𝑦) = (𝐹𝑏))
1413adantl 481 . . . . . . . . . . . . . . . . . . . . 21 ((𝑥 = 𝑎𝑦 = 𝑏) → (𝐹𝑦) = (𝐹𝑏))
1512, 14preq12d 4745 . . . . . . . . . . . . . . . . . . . 20 ((𝑥 = 𝑎𝑦 = 𝑏) → {(𝐹𝑥), (𝐹𝑦)} = {(𝐹𝑎), (𝐹𝑏)})
1615eleq1d 2823 . . . . . . . . . . . . . . . . . . 19 ((𝑥 = 𝑎𝑦 = 𝑏) → ({(𝐹𝑥), (𝐹𝑦)} ∈ 𝐷 ↔ {(𝐹𝑎), (𝐹𝑏)} ∈ 𝐷))
1710, 16bibi12d 345 . . . . . . . . . . . . . . . . . 18 ((𝑥 = 𝑎𝑦 = 𝑏) → (({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹𝑥), (𝐹𝑦)} ∈ 𝐷) ↔ ({𝑎, 𝑏} ∈ 𝐸 ↔ {(𝐹𝑎), (𝐹𝑏)} ∈ 𝐷)))
1817rspc2gv 3631 . . . . . . . . . . . . . . . . 17 ((𝑎𝑉𝑏𝑉) → (∀𝑥𝑉𝑦𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹𝑥), (𝐹𝑦)} ∈ 𝐷) → ({𝑎, 𝑏} ∈ 𝐸 ↔ {(𝐹𝑎), (𝐹𝑏)} ∈ 𝐷)))
1918com12 32 . . . . . . . . . . . . . . . 16 (∀𝑥𝑉𝑦𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹𝑥), (𝐹𝑦)} ∈ 𝐷) → ((𝑎𝑉𝑏𝑉) → ({𝑎, 𝑏} ∈ 𝐸 ↔ {(𝐹𝑎), (𝐹𝑏)} ∈ 𝐷)))
2019adantl 481 . . . . . . . . . . . . . . 15 ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ ∀𝑥𝑉𝑦𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹𝑥), (𝐹𝑦)} ∈ 𝐷)) → ((𝑎𝑉𝑏𝑉) → ({𝑎, 𝑏} ∈ 𝐸 ↔ {(𝐹𝑎), (𝐹𝑏)} ∈ 𝐷)))
2120imp 406 . . . . . . . . . . . . . 14 (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ ∀𝑥𝑉𝑦𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹𝑥), (𝐹𝑦)} ∈ 𝐷)) ∧ (𝑎𝑉𝑏𝑉)) → ({𝑎, 𝑏} ∈ 𝐸 ↔ {(𝐹𝑎), (𝐹𝑏)} ∈ 𝐷))
22 f1ofn 6849 . . . . . . . . . . . . . . . . . 18 (𝐹:𝑉1-1-onto𝑊𝐹 Fn 𝑉)
2322ad3antlr 731 . . . . . . . . . . . . . . . . 17 (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ ∀𝑥𝑉𝑦𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹𝑥), (𝐹𝑦)} ∈ 𝐷)) ∧ (𝑎𝑉𝑏𝑉)) → 𝐹 Fn 𝑉)
24 simprl 771 . . . . . . . . . . . . . . . . 17 (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ ∀𝑥𝑉𝑦𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹𝑥), (𝐹𝑦)} ∈ 𝐷)) ∧ (𝑎𝑉𝑏𝑉)) → 𝑎𝑉)
25 simpr 484 . . . . . . . . . . . . . . . . . 18 ((𝑎𝑉𝑏𝑉) → 𝑏𝑉)
2625adantl 481 . . . . . . . . . . . . . . . . 17 (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ ∀𝑥𝑉𝑦𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹𝑥), (𝐹𝑦)} ∈ 𝐷)) ∧ (𝑎𝑉𝑏𝑉)) → 𝑏𝑉)
27 fnimapr 6991 . . . . . . . . . . . . . . . . 17 ((𝐹 Fn 𝑉𝑎𝑉𝑏𝑉) → (𝐹 “ {𝑎, 𝑏}) = {(𝐹𝑎), (𝐹𝑏)})
2823, 24, 26, 27syl3anc 1370 . . . . . . . . . . . . . . . 16 (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ ∀𝑥𝑉𝑦𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹𝑥), (𝐹𝑦)} ∈ 𝐷)) ∧ (𝑎𝑉𝑏𝑉)) → (𝐹 “ {𝑎, 𝑏}) = {(𝐹𝑎), (𝐹𝑏)})
2928eqcomd 2740 . . . . . . . . . . . . . . 15 (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ ∀𝑥𝑉𝑦𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹𝑥), (𝐹𝑦)} ∈ 𝐷)) ∧ (𝑎𝑉𝑏𝑉)) → {(𝐹𝑎), (𝐹𝑏)} = (𝐹 “ {𝑎, 𝑏}))
3029eleq1d 2823 . . . . . . . . . . . . . 14 (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ ∀𝑥𝑉𝑦𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹𝑥), (𝐹𝑦)} ∈ 𝐷)) ∧ (𝑎𝑉𝑏𝑉)) → ({(𝐹𝑎), (𝐹𝑏)} ∈ 𝐷 ↔ (𝐹 “ {𝑎, 𝑏}) ∈ 𝐷))
3121, 30bitrd 279 . . . . . . . . . . . . 13 (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ ∀𝑥𝑉𝑦𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹𝑥), (𝐹𝑦)} ∈ 𝐷)) ∧ (𝑎𝑉𝑏𝑉)) → ({𝑎, 𝑏} ∈ 𝐸 ↔ (𝐹 “ {𝑎, 𝑏}) ∈ 𝐷))
3231adantr 480 . . . . . . . . . . . 12 ((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ ∀𝑥𝑉𝑦𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹𝑥), (𝐹𝑦)} ∈ 𝐷)) ∧ (𝑎𝑉𝑏𝑉)) ∧ 𝑒 = {𝑎, 𝑏}) → ({𝑎, 𝑏} ∈ 𝐸 ↔ (𝐹 “ {𝑎, 𝑏}) ∈ 𝐷))
3332biimpd 229 . . . . . . . . . . 11 ((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ ∀𝑥𝑉𝑦𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹𝑥), (𝐹𝑦)} ∈ 𝐷)) ∧ (𝑎𝑉𝑏𝑉)) ∧ 𝑒 = {𝑎, 𝑏}) → ({𝑎, 𝑏} ∈ 𝐸 → (𝐹 “ {𝑎, 𝑏}) ∈ 𝐷))
34 eleq1 2826 . . . . . . . . . . . . 13 (𝑒 = {𝑎, 𝑏} → (𝑒𝐸 ↔ {𝑎, 𝑏} ∈ 𝐸))
35 imaeq2 6075 . . . . . . . . . . . . . 14 (𝑒 = {𝑎, 𝑏} → (𝐹𝑒) = (𝐹 “ {𝑎, 𝑏}))
3635eleq1d 2823 . . . . . . . . . . . . 13 (𝑒 = {𝑎, 𝑏} → ((𝐹𝑒) ∈ 𝐷 ↔ (𝐹 “ {𝑎, 𝑏}) ∈ 𝐷))
3734, 36imbi12d 344 . . . . . . . . . . . 12 (𝑒 = {𝑎, 𝑏} → ((𝑒𝐸 → (𝐹𝑒) ∈ 𝐷) ↔ ({𝑎, 𝑏} ∈ 𝐸 → (𝐹 “ {𝑎, 𝑏}) ∈ 𝐷)))
3837adantl 481 . . . . . . . . . . 11 ((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ ∀𝑥𝑉𝑦𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹𝑥), (𝐹𝑦)} ∈ 𝐷)) ∧ (𝑎𝑉𝑏𝑉)) ∧ 𝑒 = {𝑎, 𝑏}) → ((𝑒𝐸 → (𝐹𝑒) ∈ 𝐷) ↔ ({𝑎, 𝑏} ∈ 𝐸 → (𝐹 “ {𝑎, 𝑏}) ∈ 𝐷)))
3933, 38mpbird 257 . . . . . . . . . 10 ((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ ∀𝑥𝑉𝑦𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹𝑥), (𝐹𝑦)} ∈ 𝐷)) ∧ (𝑎𝑉𝑏𝑉)) ∧ 𝑒 = {𝑎, 𝑏}) → (𝑒𝐸 → (𝐹𝑒) ∈ 𝐷))
4039exp31 419 . . . . . . . . 9 ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ ∀𝑥𝑉𝑦𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹𝑥), (𝐹𝑦)} ∈ 𝐷)) → ((𝑎𝑉𝑏𝑉) → (𝑒 = {𝑎, 𝑏} → (𝑒𝐸 → (𝐹𝑒) ∈ 𝐷))))
4140com23 86 . . . . . . . 8 ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ ∀𝑥𝑉𝑦𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹𝑥), (𝐹𝑦)} ∈ 𝐷)) → (𝑒 = {𝑎, 𝑏} → ((𝑎𝑉𝑏𝑉) → (𝑒𝐸 → (𝐹𝑒) ∈ 𝐷))))
4241com24 95 . . . . . . 7 ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ ∀𝑥𝑉𝑦𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹𝑥), (𝐹𝑦)} ∈ 𝐷)) → (𝑒𝐸 → ((𝑎𝑉𝑏𝑉) → (𝑒 = {𝑎, 𝑏} → (𝐹𝑒) ∈ 𝐷))))
4342imp 406 . . . . . 6 (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ ∀𝑥𝑉𝑦𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹𝑥), (𝐹𝑦)} ∈ 𝐷)) ∧ 𝑒𝐸) → ((𝑎𝑉𝑏𝑉) → (𝑒 = {𝑎, 𝑏} → (𝐹𝑒) ∈ 𝐷)))
4443rexlimdvv 3209 . . . . 5 (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ ∀𝑥𝑉𝑦𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹𝑥), (𝐹𝑦)} ∈ 𝐷)) ∧ 𝑒𝐸) → (∃𝑎𝑉𝑏𝑉 𝑒 = {𝑎, 𝑏} → (𝐹𝑒) ∈ 𝐷))
458, 44mpd 15 . . . 4 (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ ∀𝑥𝑉𝑦𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹𝑥), (𝐹𝑦)} ∈ 𝐷)) ∧ 𝑒𝐸) → (𝐹𝑒) ∈ 𝐷)
4645ex 412 . . 3 ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ ∀𝑥𝑉𝑦𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹𝑥), (𝐹𝑦)} ∈ 𝐷)) → (𝑒𝐸 → (𝐹𝑒) ∈ 𝐷))
4746ralrimiv 3142 . 2 ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ ∀𝑥𝑉𝑦𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹𝑥), (𝐹𝑦)} ∈ 𝐷)) → ∀𝑒𝐸 (𝐹𝑒) ∈ 𝐷)
48 uspgrupgr 29209 . . . . . 6 (𝐻 ∈ USPGraph → 𝐻 ∈ UPGraph)
4948ad3antlr 731 . . . . 5 ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ ∀𝑥𝑉𝑦𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹𝑥), (𝐹𝑦)} ∈ 𝐷)) → 𝐻 ∈ UPGraph)
50 isusgrim.w . . . . . 6 𝑊 = (Vtx‘𝐻)
51 isusgrim.d . . . . . 6 𝐷 = (Edg‘𝐻)
5250, 51upgredg 29168 . . . . 5 ((𝐻 ∈ UPGraph ∧ 𝑑𝐷) → ∃𝑎𝑊𝑏𝑊 𝑑 = {𝑎, 𝑏})
5349, 52sylan 580 . . . 4 (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ ∀𝑥𝑉𝑦𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹𝑥), (𝐹𝑦)} ∈ 𝐷)) ∧ 𝑑𝐷) → ∃𝑎𝑊𝑏𝑊 𝑑 = {𝑎, 𝑏})
54 f1ofo 6855 . . . . . . . . . . . 12 (𝐹:𝑉1-1-onto𝑊𝐹:𝑉onto𝑊)
55 foelrn 7126 . . . . . . . . . . . . . 14 ((𝐹:𝑉onto𝑊𝑎𝑊) → ∃𝑚𝑉 𝑎 = (𝐹𝑚))
5655ex 412 . . . . . . . . . . . . 13 (𝐹:𝑉onto𝑊 → (𝑎𝑊 → ∃𝑚𝑉 𝑎 = (𝐹𝑚)))
57 foelrn 7126 . . . . . . . . . . . . . 14 ((𝐹:𝑉onto𝑊𝑏𝑊) → ∃𝑛𝑉 𝑏 = (𝐹𝑛))
5857ex 412 . . . . . . . . . . . . 13 (𝐹:𝑉onto𝑊 → (𝑏𝑊 → ∃𝑛𝑉 𝑏 = (𝐹𝑛)))
5956, 58anim12d 609 . . . . . . . . . . . 12 (𝐹:𝑉onto𝑊 → ((𝑎𝑊𝑏𝑊) → (∃𝑚𝑉 𝑎 = (𝐹𝑚) ∧ ∃𝑛𝑉 𝑏 = (𝐹𝑛))))
6054, 59syl 17 . . . . . . . . . . 11 (𝐹:𝑉1-1-onto𝑊 → ((𝑎𝑊𝑏𝑊) → (∃𝑚𝑉 𝑎 = (𝐹𝑚) ∧ ∃𝑛𝑉 𝑏 = (𝐹𝑛))))
6160adantl 481 . . . . . . . . . 10 (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉1-1-onto𝑊) → ((𝑎𝑊𝑏𝑊) → (∃𝑚𝑉 𝑎 = (𝐹𝑚) ∧ ∃𝑛𝑉 𝑏 = (𝐹𝑛))))
6261adantr 480 . . . . . . . . 9 ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ ∀𝑥𝑉𝑦𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹𝑥), (𝐹𝑦)} ∈ 𝐷)) → ((𝑎𝑊𝑏𝑊) → (∃𝑚𝑉 𝑎 = (𝐹𝑚) ∧ ∃𝑛𝑉 𝑏 = (𝐹𝑛))))
6362imp 406 . . . . . . . 8 (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ ∀𝑥𝑉𝑦𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹𝑥), (𝐹𝑦)} ∈ 𝐷)) ∧ (𝑎𝑊𝑏𝑊)) → (∃𝑚𝑉 𝑎 = (𝐹𝑚) ∧ ∃𝑛𝑉 𝑏 = (𝐹𝑛)))
64 preq12 4739 . . . . . . . . . . . . . . . . 17 ((𝑎 = (𝐹𝑚) ∧ 𝑏 = (𝐹𝑛)) → {𝑎, 𝑏} = {(𝐹𝑚), (𝐹𝑛)})
6564eqeq2d 2745 . . . . . . . . . . . . . . . 16 ((𝑎 = (𝐹𝑚) ∧ 𝑏 = (𝐹𝑛)) → (𝑑 = {𝑎, 𝑏} ↔ 𝑑 = {(𝐹𝑚), (𝐹𝑛)}))
6665ancoms 458 . . . . . . . . . . . . . . 15 ((𝑏 = (𝐹𝑛) ∧ 𝑎 = (𝐹𝑚)) → (𝑑 = {𝑎, 𝑏} ↔ 𝑑 = {(𝐹𝑚), (𝐹𝑛)}))
6766adantl 481 . . . . . . . . . . . . . 14 ((((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ ∀𝑥𝑉𝑦𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹𝑥), (𝐹𝑦)} ∈ 𝐷)) ∧ (𝑎𝑊𝑏𝑊)) ∧ 𝑚𝑉) ∧ 𝑛𝑉) ∧ (𝑏 = (𝐹𝑛) ∧ 𝑎 = (𝐹𝑚))) → (𝑑 = {𝑎, 𝑏} ↔ 𝑑 = {(𝐹𝑚), (𝐹𝑛)}))
68 preq12 4739 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑥 = 𝑚𝑦 = 𝑛) → {𝑥, 𝑦} = {𝑚, 𝑛})
6968eleq1d 2823 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑥 = 𝑚𝑦 = 𝑛) → ({𝑥, 𝑦} ∈ 𝐸 ↔ {𝑚, 𝑛} ∈ 𝐸))
70 fveq2 6906 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑥 = 𝑚 → (𝐹𝑥) = (𝐹𝑚))
7170adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑥 = 𝑚𝑦 = 𝑛) → (𝐹𝑥) = (𝐹𝑚))
72 fveq2 6906 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (𝑦 = 𝑛 → (𝐹𝑦) = (𝐹𝑛))
7372adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((𝑥 = 𝑚𝑦 = 𝑛) → (𝐹𝑦) = (𝐹𝑛))
7471, 73preq12d 4745 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝑥 = 𝑚𝑦 = 𝑛) → {(𝐹𝑥), (𝐹𝑦)} = {(𝐹𝑚), (𝐹𝑛)})
7574eleq1d 2823 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑥 = 𝑚𝑦 = 𝑛) → ({(𝐹𝑥), (𝐹𝑦)} ∈ 𝐷 ↔ {(𝐹𝑚), (𝐹𝑛)} ∈ 𝐷))
7669, 75bibi12d 345 . . . . . . . . . . . . . . . . . . . . . 22 ((𝑥 = 𝑚𝑦 = 𝑛) → (({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹𝑥), (𝐹𝑦)} ∈ 𝐷) ↔ ({𝑚, 𝑛} ∈ 𝐸 ↔ {(𝐹𝑚), (𝐹𝑛)} ∈ 𝐷)))
7776rspc2gv 3631 . . . . . . . . . . . . . . . . . . . . 21 ((𝑚𝑉𝑛𝑉) → (∀𝑥𝑉𝑦𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹𝑥), (𝐹𝑦)} ∈ 𝐷) → ({𝑚, 𝑛} ∈ 𝐸 ↔ {(𝐹𝑚), (𝐹𝑛)} ∈ 𝐷)))
7877adantl 481 . . . . . . . . . . . . . . . . . . . 20 ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ (𝑚𝑉𝑛𝑉)) → (∀𝑥𝑉𝑦𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹𝑥), (𝐹𝑦)} ∈ 𝐷) → ({𝑚, 𝑛} ∈ 𝐸 ↔ {(𝐹𝑚), (𝐹𝑛)} ∈ 𝐷)))
7922adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉1-1-onto𝑊) → 𝐹 Fn 𝑉)
8079anim1i 615 . . . . . . . . . . . . . . . . . . . . . . . 24 ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ (𝑚𝑉𝑛𝑉)) → (𝐹 Fn 𝑉 ∧ (𝑚𝑉𝑛𝑉)))
81 3anass 1094 . . . . . . . . . . . . . . . . . . . . . . . 24 ((𝐹 Fn 𝑉𝑚𝑉𝑛𝑉) ↔ (𝐹 Fn 𝑉 ∧ (𝑚𝑉𝑛𝑉)))
8280, 81sylibr 234 . . . . . . . . . . . . . . . . . . . . . . 23 ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ (𝑚𝑉𝑛𝑉)) → (𝐹 Fn 𝑉𝑚𝑉𝑛𝑉))
83 fnimapr 6991 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝐹 Fn 𝑉𝑚𝑉𝑛𝑉) → (𝐹 “ {𝑚, 𝑛}) = {(𝐹𝑚), (𝐹𝑛)})
8482, 83syl 17 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ (𝑚𝑉𝑛𝑉)) → (𝐹 “ {𝑚, 𝑛}) = {(𝐹𝑚), (𝐹𝑛)})
8584eqcomd 2740 . . . . . . . . . . . . . . . . . . . . 21 ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ (𝑚𝑉𝑛𝑉)) → {(𝐹𝑚), (𝐹𝑛)} = (𝐹 “ {𝑚, 𝑛}))
86 simpr 484 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ (𝑚𝑉𝑛𝑉)) ∧ ({𝑚, 𝑛} ∈ 𝐸 ↔ (𝐹 “ {𝑚, 𝑛}) ∈ 𝐷)) → ({𝑚, 𝑛} ∈ 𝐸 ↔ (𝐹 “ {𝑚, 𝑛}) ∈ 𝐷))
87 simpr 484 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ (𝑚𝑉𝑛𝑉)) ∧ {𝑚, 𝑛} ∈ 𝐸) → {𝑚, 𝑛} ∈ 𝐸)
88 reueq 3745 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ({𝑚, 𝑛} ∈ 𝐸 ↔ ∃!𝑒𝐸 𝑒 = {𝑚, 𝑛})
8987, 88sylib 218 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ (𝑚𝑉𝑛𝑉)) ∧ {𝑚, 𝑛} ∈ 𝐸) → ∃!𝑒𝐸 𝑒 = {𝑚, 𝑛})
90 eqcom 2741 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ({𝑚, 𝑛} = 𝑒𝑒 = {𝑚, 𝑛})
9190reubii 3386 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (∃!𝑒𝐸 {𝑚, 𝑛} = 𝑒 ↔ ∃!𝑒𝐸 𝑒 = {𝑚, 𝑛})
9289, 91sylibr 234 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ (𝑚𝑉𝑛𝑉)) ∧ {𝑚, 𝑛} ∈ 𝐸) → ∃!𝑒𝐸 {𝑚, 𝑛} = 𝑒)
93 f1of1 6847 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝐹:𝑉1-1-onto𝑊𝐹:𝑉1-1𝑊)
9493adantl 481 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 (((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉1-1-onto𝑊) → 𝐹:𝑉1-1𝑊)
9594ad5ant12 756 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ (𝑚𝑉𝑛𝑉)) ∧ {𝑚, 𝑛} ∈ 𝐸) ∧ 𝑒𝐸) → 𝐹:𝑉1-1𝑊)
96 prssi 4825 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((𝑚𝑉𝑛𝑉) → {𝑚, 𝑛} ⊆ 𝑉)
9796ad3antlr 731 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ (𝑚𝑉𝑛𝑉)) ∧ {𝑚, 𝑛} ∈ 𝐸) ∧ 𝑒𝐸) → {𝑚, 𝑛} ⊆ 𝑉)
98 uspgruhgr 29215 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 (𝐺 ∈ USPGraph → 𝐺 ∈ UHGraph)
9998adantr 480 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) → 𝐺 ∈ UHGraph)
10099ad5ant12 756 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ (𝑚𝑉𝑛𝑉)) ∧ {𝑚, 𝑛} ∈ 𝐸) → 𝐺 ∈ UHGraph)
1016eleq2i 2830 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 (𝑒𝐸𝑒 ∈ (Edg‘𝐺))
102101biimpi 216 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 (𝑒𝐸𝑒 ∈ (Edg‘𝐺))
103 edguhgr 29160 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ((𝐺 ∈ UHGraph ∧ 𝑒 ∈ (Edg‘𝐺)) → 𝑒 ∈ 𝒫 (Vtx‘𝐺))
1045pweqi 4620 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 𝒫 𝑉 = 𝒫 (Vtx‘𝐺)
105103, 104eleqtrrdi 2849 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ((𝐺 ∈ UHGraph ∧ 𝑒 ∈ (Edg‘𝐺)) → 𝑒 ∈ 𝒫 𝑉)
106100, 102, 105syl2an 596 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ (𝑚𝑉𝑛𝑉)) ∧ {𝑚, 𝑛} ∈ 𝐸) ∧ 𝑒𝐸) → 𝑒 ∈ 𝒫 𝑉)
107106elpwid 4613 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ (𝑚𝑉𝑛𝑉)) ∧ {𝑚, 𝑛} ∈ 𝐸) ∧ 𝑒𝐸) → 𝑒𝑉)
108 f1imaeq 7284 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝐹:𝑉1-1𝑊 ∧ ({𝑚, 𝑛} ⊆ 𝑉𝑒𝑉)) → ((𝐹 “ {𝑚, 𝑛}) = (𝐹𝑒) ↔ {𝑚, 𝑛} = 𝑒))
10995, 97, 107, 108syl12anc 837 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ (𝑚𝑉𝑛𝑉)) ∧ {𝑚, 𝑛} ∈ 𝐸) ∧ 𝑒𝐸) → ((𝐹 “ {𝑚, 𝑛}) = (𝐹𝑒) ↔ {𝑚, 𝑛} = 𝑒))
110109reubidva 3393 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ (𝑚𝑉𝑛𝑉)) ∧ {𝑚, 𝑛} ∈ 𝐸) → (∃!𝑒𝐸 (𝐹 “ {𝑚, 𝑛}) = (𝐹𝑒) ↔ ∃!𝑒𝐸 {𝑚, 𝑛} = 𝑒))
11192, 110mpbird 257 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ (𝑚𝑉𝑛𝑉)) ∧ {𝑚, 𝑛} ∈ 𝐸) → ∃!𝑒𝐸 (𝐹 “ {𝑚, 𝑛}) = (𝐹𝑒))
112111ex 412 . . . . . . . . . . . . . . . . . . . . . . . . 25 ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ (𝑚𝑉𝑛𝑉)) → ({𝑚, 𝑛} ∈ 𝐸 → ∃!𝑒𝐸 (𝐹 “ {𝑚, 𝑛}) = (𝐹𝑒)))
113112adantr 480 . . . . . . . . . . . . . . . . . . . . . . . 24 (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ (𝑚𝑉𝑛𝑉)) ∧ ({𝑚, 𝑛} ∈ 𝐸 ↔ (𝐹 “ {𝑚, 𝑛}) ∈ 𝐷)) → ({𝑚, 𝑛} ∈ 𝐸 → ∃!𝑒𝐸 (𝐹 “ {𝑚, 𝑛}) = (𝐹𝑒)))
11486, 113sylbird 260 . . . . . . . . . . . . . . . . . . . . . . 23 (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ (𝑚𝑉𝑛𝑉)) ∧ ({𝑚, 𝑛} ∈ 𝐸 ↔ (𝐹 “ {𝑚, 𝑛}) ∈ 𝐷)) → ((𝐹 “ {𝑚, 𝑛}) ∈ 𝐷 → ∃!𝑒𝐸 (𝐹 “ {𝑚, 𝑛}) = (𝐹𝑒)))
115114ex 412 . . . . . . . . . . . . . . . . . . . . . 22 ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ (𝑚𝑉𝑛𝑉)) → (({𝑚, 𝑛} ∈ 𝐸 ↔ (𝐹 “ {𝑚, 𝑛}) ∈ 𝐷) → ((𝐹 “ {𝑚, 𝑛}) ∈ 𝐷 → ∃!𝑒𝐸 (𝐹 “ {𝑚, 𝑛}) = (𝐹𝑒))))
116 eleq1 2826 . . . . . . . . . . . . . . . . . . . . . . . 24 ({(𝐹𝑚), (𝐹𝑛)} = (𝐹 “ {𝑚, 𝑛}) → ({(𝐹𝑚), (𝐹𝑛)} ∈ 𝐷 ↔ (𝐹 “ {𝑚, 𝑛}) ∈ 𝐷))
117116bibi2d 342 . . . . . . . . . . . . . . . . . . . . . . 23 ({(𝐹𝑚), (𝐹𝑛)} = (𝐹 “ {𝑚, 𝑛}) → (({𝑚, 𝑛} ∈ 𝐸 ↔ {(𝐹𝑚), (𝐹𝑛)} ∈ 𝐷) ↔ ({𝑚, 𝑛} ∈ 𝐸 ↔ (𝐹 “ {𝑚, 𝑛}) ∈ 𝐷)))
118 eqeq1 2738 . . . . . . . . . . . . . . . . . . . . . . . . 25 ({(𝐹𝑚), (𝐹𝑛)} = (𝐹 “ {𝑚, 𝑛}) → ({(𝐹𝑚), (𝐹𝑛)} = (𝐹𝑒) ↔ (𝐹 “ {𝑚, 𝑛}) = (𝐹𝑒)))
119118reubidv 3395 . . . . . . . . . . . . . . . . . . . . . . . 24 ({(𝐹𝑚), (𝐹𝑛)} = (𝐹 “ {𝑚, 𝑛}) → (∃!𝑒𝐸 {(𝐹𝑚), (𝐹𝑛)} = (𝐹𝑒) ↔ ∃!𝑒𝐸 (𝐹 “ {𝑚, 𝑛}) = (𝐹𝑒)))
120116, 119imbi12d 344 . . . . . . . . . . . . . . . . . . . . . . 23 ({(𝐹𝑚), (𝐹𝑛)} = (𝐹 “ {𝑚, 𝑛}) → (({(𝐹𝑚), (𝐹𝑛)} ∈ 𝐷 → ∃!𝑒𝐸 {(𝐹𝑚), (𝐹𝑛)} = (𝐹𝑒)) ↔ ((𝐹 “ {𝑚, 𝑛}) ∈ 𝐷 → ∃!𝑒𝐸 (𝐹 “ {𝑚, 𝑛}) = (𝐹𝑒))))
121117, 120imbi12d 344 . . . . . . . . . . . . . . . . . . . . . 22 ({(𝐹𝑚), (𝐹𝑛)} = (𝐹 “ {𝑚, 𝑛}) → ((({𝑚, 𝑛} ∈ 𝐸 ↔ {(𝐹𝑚), (𝐹𝑛)} ∈ 𝐷) → ({(𝐹𝑚), (𝐹𝑛)} ∈ 𝐷 → ∃!𝑒𝐸 {(𝐹𝑚), (𝐹𝑛)} = (𝐹𝑒))) ↔ (({𝑚, 𝑛} ∈ 𝐸 ↔ (𝐹 “ {𝑚, 𝑛}) ∈ 𝐷) → ((𝐹 “ {𝑚, 𝑛}) ∈ 𝐷 → ∃!𝑒𝐸 (𝐹 “ {𝑚, 𝑛}) = (𝐹𝑒)))))
122115, 121syl5ibrcom 247 . . . . . . . . . . . . . . . . . . . . 21 ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ (𝑚𝑉𝑛𝑉)) → ({(𝐹𝑚), (𝐹𝑛)} = (𝐹 “ {𝑚, 𝑛}) → (({𝑚, 𝑛} ∈ 𝐸 ↔ {(𝐹𝑚), (𝐹𝑛)} ∈ 𝐷) → ({(𝐹𝑚), (𝐹𝑛)} ∈ 𝐷 → ∃!𝑒𝐸 {(𝐹𝑚), (𝐹𝑛)} = (𝐹𝑒)))))
12385, 122mpd 15 . . . . . . . . . . . . . . . . . . . 20 ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ (𝑚𝑉𝑛𝑉)) → (({𝑚, 𝑛} ∈ 𝐸 ↔ {(𝐹𝑚), (𝐹𝑛)} ∈ 𝐷) → ({(𝐹𝑚), (𝐹𝑛)} ∈ 𝐷 → ∃!𝑒𝐸 {(𝐹𝑚), (𝐹𝑛)} = (𝐹𝑒))))
12478, 123syld 47 . . . . . . . . . . . . . . . . . . 19 ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ (𝑚𝑉𝑛𝑉)) → (∀𝑥𝑉𝑦𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹𝑥), (𝐹𝑦)} ∈ 𝐷) → ({(𝐹𝑚), (𝐹𝑛)} ∈ 𝐷 → ∃!𝑒𝐸 {(𝐹𝑚), (𝐹𝑛)} = (𝐹𝑒))))
125124impancom 451 . . . . . . . . . . . . . . . . . 18 ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ ∀𝑥𝑉𝑦𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹𝑥), (𝐹𝑦)} ∈ 𝐷)) → ((𝑚𝑉𝑛𝑉) → ({(𝐹𝑚), (𝐹𝑛)} ∈ 𝐷 → ∃!𝑒𝐸 {(𝐹𝑚), (𝐹𝑛)} = (𝐹𝑒))))
126125adantr 480 . . . . . . . . . . . . . . . . 17 (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ ∀𝑥𝑉𝑦𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹𝑥), (𝐹𝑦)} ∈ 𝐷)) ∧ (𝑎𝑊𝑏𝑊)) → ((𝑚𝑉𝑛𝑉) → ({(𝐹𝑚), (𝐹𝑛)} ∈ 𝐷 → ∃!𝑒𝐸 {(𝐹𝑚), (𝐹𝑛)} = (𝐹𝑒))))
127126impl 455 . . . . . . . . . . . . . . . 16 (((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ ∀𝑥𝑉𝑦𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹𝑥), (𝐹𝑦)} ∈ 𝐷)) ∧ (𝑎𝑊𝑏𝑊)) ∧ 𝑚𝑉) ∧ 𝑛𝑉) → ({(𝐹𝑚), (𝐹𝑛)} ∈ 𝐷 → ∃!𝑒𝐸 {(𝐹𝑚), (𝐹𝑛)} = (𝐹𝑒)))
128 eleq1 2826 . . . . . . . . . . . . . . . . 17 (𝑑 = {(𝐹𝑚), (𝐹𝑛)} → (𝑑𝐷 ↔ {(𝐹𝑚), (𝐹𝑛)} ∈ 𝐷))
129 eqeq1 2738 . . . . . . . . . . . . . . . . . 18 (𝑑 = {(𝐹𝑚), (𝐹𝑛)} → (𝑑 = (𝐹𝑒) ↔ {(𝐹𝑚), (𝐹𝑛)} = (𝐹𝑒)))
130129reubidv 3395 . . . . . . . . . . . . . . . . 17 (𝑑 = {(𝐹𝑚), (𝐹𝑛)} → (∃!𝑒𝐸 𝑑 = (𝐹𝑒) ↔ ∃!𝑒𝐸 {(𝐹𝑚), (𝐹𝑛)} = (𝐹𝑒)))
131128, 130imbi12d 344 . . . . . . . . . . . . . . . 16 (𝑑 = {(𝐹𝑚), (𝐹𝑛)} → ((𝑑𝐷 → ∃!𝑒𝐸 𝑑 = (𝐹𝑒)) ↔ ({(𝐹𝑚), (𝐹𝑛)} ∈ 𝐷 → ∃!𝑒𝐸 {(𝐹𝑚), (𝐹𝑛)} = (𝐹𝑒))))
132127, 131syl5ibrcom 247 . . . . . . . . . . . . . . 15 (((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ ∀𝑥𝑉𝑦𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹𝑥), (𝐹𝑦)} ∈ 𝐷)) ∧ (𝑎𝑊𝑏𝑊)) ∧ 𝑚𝑉) ∧ 𝑛𝑉) → (𝑑 = {(𝐹𝑚), (𝐹𝑛)} → (𝑑𝐷 → ∃!𝑒𝐸 𝑑 = (𝐹𝑒))))
133132adantr 480 . . . . . . . . . . . . . 14 ((((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ ∀𝑥𝑉𝑦𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹𝑥), (𝐹𝑦)} ∈ 𝐷)) ∧ (𝑎𝑊𝑏𝑊)) ∧ 𝑚𝑉) ∧ 𝑛𝑉) ∧ (𝑏 = (𝐹𝑛) ∧ 𝑎 = (𝐹𝑚))) → (𝑑 = {(𝐹𝑚), (𝐹𝑛)} → (𝑑𝐷 → ∃!𝑒𝐸 𝑑 = (𝐹𝑒))))
13467, 133sylbid 240 . . . . . . . . . . . . 13 ((((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ ∀𝑥𝑉𝑦𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹𝑥), (𝐹𝑦)} ∈ 𝐷)) ∧ (𝑎𝑊𝑏𝑊)) ∧ 𝑚𝑉) ∧ 𝑛𝑉) ∧ (𝑏 = (𝐹𝑛) ∧ 𝑎 = (𝐹𝑚))) → (𝑑 = {𝑎, 𝑏} → (𝑑𝐷 → ∃!𝑒𝐸 𝑑 = (𝐹𝑒))))
135134exp32 420 . . . . . . . . . . . 12 (((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ ∀𝑥𝑉𝑦𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹𝑥), (𝐹𝑦)} ∈ 𝐷)) ∧ (𝑎𝑊𝑏𝑊)) ∧ 𝑚𝑉) ∧ 𝑛𝑉) → (𝑏 = (𝐹𝑛) → (𝑎 = (𝐹𝑚) → (𝑑 = {𝑎, 𝑏} → (𝑑𝐷 → ∃!𝑒𝐸 𝑑 = (𝐹𝑒))))))
136135rexlimdva 3152 . . . . . . . . . . 11 ((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ ∀𝑥𝑉𝑦𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹𝑥), (𝐹𝑦)} ∈ 𝐷)) ∧ (𝑎𝑊𝑏𝑊)) ∧ 𝑚𝑉) → (∃𝑛𝑉 𝑏 = (𝐹𝑛) → (𝑎 = (𝐹𝑚) → (𝑑 = {𝑎, 𝑏} → (𝑑𝐷 → ∃!𝑒𝐸 𝑑 = (𝐹𝑒))))))
137136com23 86 . . . . . . . . . 10 ((((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ ∀𝑥𝑉𝑦𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹𝑥), (𝐹𝑦)} ∈ 𝐷)) ∧ (𝑎𝑊𝑏𝑊)) ∧ 𝑚𝑉) → (𝑎 = (𝐹𝑚) → (∃𝑛𝑉 𝑏 = (𝐹𝑛) → (𝑑 = {𝑎, 𝑏} → (𝑑𝐷 → ∃!𝑒𝐸 𝑑 = (𝐹𝑒))))))
138137rexlimdva 3152 . . . . . . . . 9 (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ ∀𝑥𝑉𝑦𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹𝑥), (𝐹𝑦)} ∈ 𝐷)) ∧ (𝑎𝑊𝑏𝑊)) → (∃𝑚𝑉 𝑎 = (𝐹𝑚) → (∃𝑛𝑉 𝑏 = (𝐹𝑛) → (𝑑 = {𝑎, 𝑏} → (𝑑𝐷 → ∃!𝑒𝐸 𝑑 = (𝐹𝑒))))))
139138impd 410 . . . . . . . 8 (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ ∀𝑥𝑉𝑦𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹𝑥), (𝐹𝑦)} ∈ 𝐷)) ∧ (𝑎𝑊𝑏𝑊)) → ((∃𝑚𝑉 𝑎 = (𝐹𝑚) ∧ ∃𝑛𝑉 𝑏 = (𝐹𝑛)) → (𝑑 = {𝑎, 𝑏} → (𝑑𝐷 → ∃!𝑒𝐸 𝑑 = (𝐹𝑒)))))
14063, 139mpd 15 . . . . . . 7 (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ ∀𝑥𝑉𝑦𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹𝑥), (𝐹𝑦)} ∈ 𝐷)) ∧ (𝑎𝑊𝑏𝑊)) → (𝑑 = {𝑎, 𝑏} → (𝑑𝐷 → ∃!𝑒𝐸 𝑑 = (𝐹𝑒))))
141140com23 86 . . . . . 6 (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ ∀𝑥𝑉𝑦𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹𝑥), (𝐹𝑦)} ∈ 𝐷)) ∧ (𝑎𝑊𝑏𝑊)) → (𝑑𝐷 → (𝑑 = {𝑎, 𝑏} → ∃!𝑒𝐸 𝑑 = (𝐹𝑒))))
142141impancom 451 . . . . 5 (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ ∀𝑥𝑉𝑦𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹𝑥), (𝐹𝑦)} ∈ 𝐷)) ∧ 𝑑𝐷) → ((𝑎𝑊𝑏𝑊) → (𝑑 = {𝑎, 𝑏} → ∃!𝑒𝐸 𝑑 = (𝐹𝑒))))
143142rexlimdvv 3209 . . . 4 (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ ∀𝑥𝑉𝑦𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹𝑥), (𝐹𝑦)} ∈ 𝐷)) ∧ 𝑑𝐷) → (∃𝑎𝑊𝑏𝑊 𝑑 = {𝑎, 𝑏} → ∃!𝑒𝐸 𝑑 = (𝐹𝑒)))
14453, 143mpd 15 . . 3 (((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ ∀𝑥𝑉𝑦𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹𝑥), (𝐹𝑦)} ∈ 𝐷)) ∧ 𝑑𝐷) → ∃!𝑒𝐸 𝑑 = (𝐹𝑒))
145144ralrimiva 3143 . 2 ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ ∀𝑥𝑉𝑦𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹𝑥), (𝐹𝑦)} ∈ 𝐷)) → ∀𝑑𝐷 ∃!𝑒𝐸 𝑑 = (𝐹𝑒))
146 eqid 2734 . . 3 (𝑒𝐸 ↦ (𝐹𝑒)) = (𝑒𝐸 ↦ (𝐹𝑒))
147146f1ompt 7130 . 2 ((𝑒𝐸 ↦ (𝐹𝑒)):𝐸1-1-onto𝐷 ↔ (∀𝑒𝐸 (𝐹𝑒) ∈ 𝐷 ∧ ∀𝑑𝐷 ∃!𝑒𝐸 𝑑 = (𝐹𝑒)))
14847, 145, 147sylanbrc 583 1 ((((𝐺 ∈ USPGraph ∧ 𝐻 ∈ USPGraph) ∧ 𝐹:𝑉1-1-onto𝑊) ∧ ∀𝑥𝑉𝑦𝑉 ({𝑥, 𝑦} ∈ 𝐸 ↔ {(𝐹𝑥), (𝐹𝑦)} ∈ 𝐷)) → (𝑒𝐸 ↦ (𝐹𝑒)):𝐸1-1-onto𝐷)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1536  wcel 2105  wral 3058  wrex 3067  ∃!wreu 3375  wss 3962  𝒫 cpw 4604  {cpr 4632  cmpt 5230  cima 5691   Fn wfn 6557  1-1wf1 6559  ontowfo 6560  1-1-ontowf1o 6561  cfv 6562  Vtxcvtx 29027  Edgcedg 29078  UHGraphcuhgr 29087  UPGraphcupgr 29111  USPGraphcuspgr 29179
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-sep 5301  ax-nul 5311  ax-pow 5370  ax-pr 5437  ax-un 7753  ax-cnex 11208  ax-resscn 11209  ax-1cn 11210  ax-icn 11211  ax-addcl 11212  ax-addrcl 11213  ax-mulcl 11214  ax-mulrcl 11215  ax-mulcom 11216  ax-addass 11217  ax-mulass 11218  ax-distr 11219  ax-i2m1 11220  ax-1ne0 11221  ax-1rid 11222  ax-rnegex 11223  ax-rrecex 11224  ax-cnre 11225  ax-pre-lttri 11226  ax-pre-lttrn 11227  ax-pre-ltadd 11228  ax-pre-mulgt0 11229
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  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-nel 3044  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-pss 3982  df-nul 4339  df-if 4531  df-pw 4606  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-int 4951  df-iun 4997  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5582  df-eprel 5588  df-po 5596  df-so 5597  df-fr 5640  df-we 5642  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-pred 6322  df-ord 6388  df-on 6389  df-lim 6390  df-suc 6391  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-riota 7387  df-ov 7433  df-oprab 7434  df-mpo 7435  df-om 7887  df-1st 8012  df-2nd 8013  df-frecs 8304  df-wrecs 8335  df-recs 8409  df-rdg 8448  df-1o 8504  df-2o 8505  df-oadd 8508  df-er 8743  df-en 8984  df-dom 8985  df-sdom 8986  df-fin 8987  df-dju 9938  df-card 9976  df-pnf 11294  df-mnf 11295  df-xr 11296  df-ltxr 11297  df-le 11298  df-sub 11491  df-neg 11492  df-nn 12264  df-2 12326  df-n0 12524  df-xnn0 12597  df-z 12611  df-uz 12876  df-fz 13544  df-hash 14366  df-edg 29079  df-uhgr 29089  df-upgr 29113  df-uspgr 29181
This theorem is referenced by:  isuspgrim  47812
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