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Theorem brtrclfv2 40829
 Description: Two ways to indicate two elements are related by the transitive closure of a relation. (Contributed by RP, 1-Jul-2020.)
Assertion
Ref Expression
brtrclfv2 ((𝑋𝑈𝑌𝑉𝑅𝑊) → (𝑋(t+‘𝑅)𝑌𝑌 {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓}))
Distinct variable groups:   𝑅,𝑓   𝑈,𝑓   𝑓,𝑉   𝑓,𝑊   𝑓,𝑋   𝑓,𝑌

Proof of Theorem brtrclfv2
Dummy variables 𝑔 𝑟 𝑠 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-br 5036 . . . 4 (𝑋 {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}𝑌 ↔ ⟨𝑋, 𝑌⟩ ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)})
21a1i 11 . . 3 ((𝑋𝑈𝑌𝑉𝑅𝑊) → (𝑋 {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}𝑌 ↔ ⟨𝑋, 𝑌⟩ ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}))
3 trclfv 14412 . . . . 5 (𝑅𝑊 → (t+‘𝑅) = {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)})
43breqd 5046 . . . 4 (𝑅𝑊 → (𝑋(t+‘𝑅)𝑌𝑋 {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}𝑌))
543ad2ant3 1132 . . 3 ((𝑋𝑈𝑌𝑉𝑅𝑊) → (𝑋(t+‘𝑅)𝑌𝑋 {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}𝑌))
6 elimasng 5931 . . . 4 ((𝑋𝑈𝑌𝑉) → (𝑌 ∈ ( {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} “ {𝑋}) ↔ ⟨𝑋, 𝑌⟩ ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}))
763adant3 1129 . . 3 ((𝑋𝑈𝑌𝑉𝑅𝑊) → (𝑌 ∈ ( {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} “ {𝑋}) ↔ ⟨𝑋, 𝑌⟩ ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}))
82, 5, 73bitr4d 314 . 2 ((𝑋𝑈𝑌𝑉𝑅𝑊) → (𝑋(t+‘𝑅)𝑌𝑌 ∈ ( {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} “ {𝑋})))
9 intimasn 40759 . . . . 5 (𝑋𝑈 → ( {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} “ {𝑋}) = {𝑔 ∣ ∃𝑠 ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋})})
1093ad2ant1 1130 . . . 4 ((𝑋𝑈𝑌𝑉𝑅𝑊) → ( {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} “ {𝑋}) = {𝑔 ∣ ∃𝑠 ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋})})
11 simpl3 1190 . . . . . . . . . . . . . 14 (((𝑋𝑈𝑌𝑉𝑅𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → 𝑅𝑊)
12 snex 5303 . . . . . . . . . . . . . . 15 {𝑋} ∈ V
13 vex 3413 . . . . . . . . . . . . . . 15 𝑓 ∈ V
1412, 13xpex 7479 . . . . . . . . . . . . . 14 ({𝑋} × 𝑓) ∈ V
15 unexg 7475 . . . . . . . . . . . . . 14 ((𝑅𝑊 ∧ ({𝑋} × 𝑓) ∈ V) → (𝑅 ∪ ({𝑋} × 𝑓)) ∈ V)
1611, 14, 15sylancl 589 . . . . . . . . . . . . 13 (((𝑋𝑈𝑌𝑉𝑅𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → (𝑅 ∪ ({𝑋} × 𝑓)) ∈ V)
17 trclfvlb 14420 . . . . . . . . . . . . . 14 ((𝑅 ∪ ({𝑋} × 𝑓)) ∈ V → (𝑅 ∪ ({𝑋} × 𝑓)) ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))))
1817unssad 4094 . . . . . . . . . . . . 13 ((𝑅 ∪ ({𝑋} × 𝑓)) ∈ V → 𝑅 ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))))
1916, 18syl 17 . . . . . . . . . . . 12 (((𝑋𝑈𝑌𝑉𝑅𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → 𝑅 ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))))
20 trclfvcotrg 14428 . . . . . . . . . . . . 13 ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) ∘ (t+‘(𝑅 ∪ ({𝑋} × 𝑓)))) ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓)))
2120a1i 11 . . . . . . . . . . . 12 (((𝑋𝑈𝑌𝑉𝑅𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) ∘ (t+‘(𝑅 ∪ ({𝑋} × 𝑓)))) ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))))
22 simpl1 1188 . . . . . . . . . . . . . . . . 17 (((𝑋𝑈𝑌𝑉𝑅𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → 𝑋𝑈)
23 snidg 4559 . . . . . . . . . . . . . . . . 17 (𝑋𝑈𝑋 ∈ {𝑋})
2422, 23syl 17 . . . . . . . . . . . . . . . 16 (((𝑋𝑈𝑌𝑉𝑅𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → 𝑋 ∈ {𝑋})
25 inelcm 4364 . . . . . . . . . . . . . . . 16 ((𝑋 ∈ {𝑋} ∧ 𝑋 ∈ {𝑋}) → ({𝑋} ∩ {𝑋}) ≠ ∅)
2624, 24, 25syl2anc 587 . . . . . . . . . . . . . . 15 (((𝑋𝑈𝑌𝑉𝑅𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → ({𝑋} ∩ {𝑋}) ≠ ∅)
27 xpima2 6017 . . . . . . . . . . . . . . 15 (({𝑋} ∩ {𝑋}) ≠ ∅ → (({𝑋} × 𝑓) “ {𝑋}) = 𝑓)
2826, 27syl 17 . . . . . . . . . . . . . 14 (((𝑋𝑈𝑌𝑉𝑅𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → (({𝑋} × 𝑓) “ {𝑋}) = 𝑓)
2916, 17syl 17 . . . . . . . . . . . . . . . 16 (((𝑋𝑈𝑌𝑉𝑅𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → (𝑅 ∪ ({𝑋} × 𝑓)) ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))))
3029unssbd 4095 . . . . . . . . . . . . . . 15 (((𝑋𝑈𝑌𝑉𝑅𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → ({𝑋} × 𝑓) ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))))
31 imass1 5940 . . . . . . . . . . . . . . 15 (({𝑋} × 𝑓) ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))) → (({𝑋} × 𝑓) “ {𝑋}) ⊆ ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ {𝑋}))
3230, 31syl 17 . . . . . . . . . . . . . 14 (((𝑋𝑈𝑌𝑉𝑅𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → (({𝑋} × 𝑓) “ {𝑋}) ⊆ ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ {𝑋}))
3328, 32eqsstrrd 3933 . . . . . . . . . . . . 13 (((𝑋𝑈𝑌𝑉𝑅𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → 𝑓 ⊆ ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ {𝑋}))
34 imaundir 5985 . . . . . . . . . . . . . . 15 ((𝑅 ∪ ({𝑋} × 𝑓)) “ ({𝑋} ∪ 𝑓)) = ((𝑅 “ ({𝑋} ∪ 𝑓)) ∪ (({𝑋} × 𝑓) “ ({𝑋} ∪ 𝑓)))
35 simpr 488 . . . . . . . . . . . . . . . 16 (((𝑋𝑈𝑌𝑉𝑅𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓)
36 imassrn 5916 . . . . . . . . . . . . . . . . . 18 (({𝑋} × 𝑓) “ ({𝑋} ∪ 𝑓)) ⊆ ran ({𝑋} × 𝑓)
37 rnxpss 6005 . . . . . . . . . . . . . . . . . 18 ran ({𝑋} × 𝑓) ⊆ 𝑓
3836, 37sstri 3903 . . . . . . . . . . . . . . . . 17 (({𝑋} × 𝑓) “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓
3938a1i 11 . . . . . . . . . . . . . . . 16 (((𝑋𝑈𝑌𝑉𝑅𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → (({𝑋} × 𝑓) “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓)
4035, 39unssd 4093 . . . . . . . . . . . . . . 15 (((𝑋𝑈𝑌𝑉𝑅𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → ((𝑅 “ ({𝑋} ∪ 𝑓)) ∪ (({𝑋} × 𝑓) “ ({𝑋} ∪ 𝑓))) ⊆ 𝑓)
4134, 40eqsstrid 3942 . . . . . . . . . . . . . 14 (((𝑋𝑈𝑌𝑉𝑅𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → ((𝑅 ∪ ({𝑋} × 𝑓)) “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓)
42 trclimalb2 40828 . . . . . . . . . . . . . 14 (((𝑅 ∪ ({𝑋} × 𝑓)) ∈ V ∧ ((𝑅 ∪ ({𝑋} × 𝑓)) “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ {𝑋}) ⊆ 𝑓)
4316, 41, 42syl2anc 587 . . . . . . . . . . . . 13 (((𝑋𝑈𝑌𝑉𝑅𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ {𝑋}) ⊆ 𝑓)
4433, 43eqssd 3911 . . . . . . . . . . . 12 (((𝑋𝑈𝑌𝑉𝑅𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → 𝑓 = ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ {𝑋}))
45 sbcan 3747 . . . . . . . . . . . . 13 ([(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟]((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) ∧ 𝑓 = (𝑟 “ {𝑋})) ↔ ([(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟](𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) ∧ [(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟]𝑓 = (𝑟 “ {𝑋})))
46 sbcan 3747 . . . . . . . . . . . . . . 15 ([(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟](𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) ↔ ([(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟]𝑅𝑟[(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟](𝑟𝑟) ⊆ 𝑟))
47 fvex 6675 . . . . . . . . . . . . . . . . . 18 (t+‘(𝑅 ∪ ({𝑋} × 𝑓))) ∈ V
48 sbcssg 4419 . . . . . . . . . . . . . . . . . 18 ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) ∈ V → ([(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟]𝑅𝑟(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟𝑅(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟𝑟))
4947, 48ax-mp 5 . . . . . . . . . . . . . . . . 17 ([(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟]𝑅𝑟(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟𝑅(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟𝑟)
50 csbconstg 3826 . . . . . . . . . . . . . . . . . . 19 ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) ∈ V → (t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟𝑅 = 𝑅)
5147, 50ax-mp 5 . . . . . . . . . . . . . . . . . 18 (t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟𝑅 = 𝑅
5247csbvargi 4332 . . . . . . . . . . . . . . . . . 18 (t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟𝑟 = (t+‘(𝑅 ∪ ({𝑋} × 𝑓)))
5351, 52sseq12i 3924 . . . . . . . . . . . . . . . . 17 ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟𝑅(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟𝑟𝑅 ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))))
5449, 53bitri 278 . . . . . . . . . . . . . . . 16 ([(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟]𝑅𝑟𝑅 ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))))
55 sbcssg 4419 . . . . . . . . . . . . . . . . . 18 ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) ∈ V → ([(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟](𝑟𝑟) ⊆ 𝑟(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟(𝑟𝑟) ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟𝑟))
5647, 55ax-mp 5 . . . . . . . . . . . . . . . . 17 ([(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟](𝑟𝑟) ⊆ 𝑟(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟(𝑟𝑟) ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟𝑟)
57 csbcog 40751 . . . . . . . . . . . . . . . . . . . 20 ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) ∈ V → (t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟(𝑟𝑟) = ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟𝑟(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟𝑟))
5847, 57ax-mp 5 . . . . . . . . . . . . . . . . . . 19 (t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟(𝑟𝑟) = ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟𝑟(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟𝑟)
5952, 52coeq12i 5708 . . . . . . . . . . . . . . . . . . 19 ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟𝑟(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟𝑟) = ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) ∘ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))))
6058, 59eqtri 2781 . . . . . . . . . . . . . . . . . 18 (t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟(𝑟𝑟) = ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) ∘ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))))
6160, 52sseq12i 3924 . . . . . . . . . . . . . . . . 17 ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟(𝑟𝑟) ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟𝑟 ↔ ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) ∘ (t+‘(𝑅 ∪ ({𝑋} × 𝑓)))) ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))))
6256, 61bitri 278 . . . . . . . . . . . . . . . 16 ([(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟](𝑟𝑟) ⊆ 𝑟 ↔ ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) ∘ (t+‘(𝑅 ∪ ({𝑋} × 𝑓)))) ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))))
6354, 62anbi12i 629 . . . . . . . . . . . . . . 15 (([(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟]𝑅𝑟[(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟](𝑟𝑟) ⊆ 𝑟) ↔ (𝑅 ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))) ∧ ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) ∘ (t+‘(𝑅 ∪ ({𝑋} × 𝑓)))) ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓)))))
6446, 63bitri 278 . . . . . . . . . . . . . 14 ([(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟](𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) ↔ (𝑅 ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))) ∧ ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) ∘ (t+‘(𝑅 ∪ ({𝑋} × 𝑓)))) ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓)))))
65 sbceq2g 4316 . . . . . . . . . . . . . . . 16 ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) ∈ V → ([(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟]𝑓 = (𝑟 “ {𝑋}) ↔ 𝑓 = (t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟(𝑟 “ {𝑋})))
6647, 65ax-mp 5 . . . . . . . . . . . . . . 15 ([(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟]𝑓 = (𝑟 “ {𝑋}) ↔ 𝑓 = (t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟(𝑟 “ {𝑋}))
67 csbima12 5923 . . . . . . . . . . . . . . . . 17 (t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟(𝑟 “ {𝑋}) = ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟𝑟(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟{𝑋})
6852imaeq1i 5902 . . . . . . . . . . . . . . . . 17 ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟𝑟(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟{𝑋}) = ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟{𝑋})
69 csbconstg 3826 . . . . . . . . . . . . . . . . . . 19 ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) ∈ V → (t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟{𝑋} = {𝑋})
7047, 69ax-mp 5 . . . . . . . . . . . . . . . . . 18 (t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟{𝑋} = {𝑋}
7170imaeq2i 5903 . . . . . . . . . . . . . . . . 17 ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟{𝑋}) = ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ {𝑋})
7267, 68, 713eqtri 2785 . . . . . . . . . . . . . . . 16 (t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟(𝑟 “ {𝑋}) = ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ {𝑋})
7372eqeq2i 2771 . . . . . . . . . . . . . . 15 (𝑓 = (t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟(𝑟 “ {𝑋}) ↔ 𝑓 = ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ {𝑋}))
7466, 73bitri 278 . . . . . . . . . . . . . 14 ([(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟]𝑓 = (𝑟 “ {𝑋}) ↔ 𝑓 = ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ {𝑋}))
7564, 74anbi12i 629 . . . . . . . . . . . . 13 (([(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟](𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) ∧ [(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟]𝑓 = (𝑟 “ {𝑋})) ↔ ((𝑅 ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))) ∧ ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) ∘ (t+‘(𝑅 ∪ ({𝑋} × 𝑓)))) ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓)))) ∧ 𝑓 = ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ {𝑋})))
7645, 75sylbbr 239 . . . . . . . . . . . 12 (((𝑅 ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓))) ∧ ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) ∘ (t+‘(𝑅 ∪ ({𝑋} × 𝑓)))) ⊆ (t+‘(𝑅 ∪ ({𝑋} × 𝑓)))) ∧ 𝑓 = ((t+‘(𝑅 ∪ ({𝑋} × 𝑓))) “ {𝑋})) → [(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟]((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) ∧ 𝑓 = (𝑟 “ {𝑋})))
7719, 21, 44, 76syl21anc 836 . . . . . . . . . . 11 (((𝑋𝑈𝑌𝑉𝑅𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → [(t+‘(𝑅 ∪ ({𝑋} × 𝑓))) / 𝑟]((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) ∧ 𝑓 = (𝑟 “ {𝑋})))
7877spesbcd 3791 . . . . . . . . . 10 (((𝑋𝑈𝑌𝑉𝑅𝑊) ∧ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓) → ∃𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) ∧ 𝑓 = (𝑟 “ {𝑋})))
7978ex 416 . . . . . . . . 9 ((𝑋𝑈𝑌𝑉𝑅𝑊) → ((𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓 → ∃𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) ∧ 𝑓 = (𝑟 “ {𝑋}))))
80 eqeq1 2762 . . . . . . . . . . . 12 (𝑔 = 𝑓 → (𝑔 = (𝑠 “ {𝑋}) ↔ 𝑓 = (𝑠 “ {𝑋})))
8180rexbidv 3221 . . . . . . . . . . 11 (𝑔 = 𝑓 → (∃𝑠 ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋}) ↔ ∃𝑠 ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}𝑓 = (𝑠 “ {𝑋})))
82 imaeq1 5900 . . . . . . . . . . . . 13 (𝑠 = 𝑟 → (𝑠 “ {𝑋}) = (𝑟 “ {𝑋}))
8382eqeq2d 2769 . . . . . . . . . . . 12 (𝑠 = 𝑟 → (𝑓 = (𝑠 “ {𝑋}) ↔ 𝑓 = (𝑟 “ {𝑋})))
8483rexab2 3616 . . . . . . . . . . 11 (∃𝑠 ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}𝑓 = (𝑠 “ {𝑋}) ↔ ∃𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) ∧ 𝑓 = (𝑟 “ {𝑋})))
8581, 84bitrdi 290 . . . . . . . . . 10 (𝑔 = 𝑓 → (∃𝑠 ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋}) ↔ ∃𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) ∧ 𝑓 = (𝑟 “ {𝑋}))))
8613, 85elab 3590 . . . . . . . . 9 (𝑓 ∈ {𝑔 ∣ ∃𝑠 ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋})} ↔ ∃𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) ∧ 𝑓 = (𝑟 “ {𝑋})))
8779, 86syl6ibr 255 . . . . . . . 8 ((𝑋𝑈𝑌𝑉𝑅𝑊) → ((𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓𝑓 ∈ {𝑔 ∣ ∃𝑠 ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋})}))
88 intss1 4856 . . . . . . . 8 (𝑓 ∈ {𝑔 ∣ ∃𝑠 ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋})} → {𝑔 ∣ ∃𝑠 ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋})} ⊆ 𝑓)
8987, 88syl6 35 . . . . . . 7 ((𝑋𝑈𝑌𝑉𝑅𝑊) → ((𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓 {𝑔 ∣ ∃𝑠 ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋})} ⊆ 𝑓))
9089alrimiv 1928 . . . . . 6 ((𝑋𝑈𝑌𝑉𝑅𝑊) → ∀𝑓((𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓 {𝑔 ∣ ∃𝑠 ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋})} ⊆ 𝑓))
91 ssintab 4858 . . . . . 6 ( {𝑔 ∣ ∃𝑠 ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋})} ⊆ {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓} ↔ ∀𝑓((𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓 {𝑔 ∣ ∃𝑠 ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋})} ⊆ 𝑓))
9290, 91sylibr 237 . . . . 5 ((𝑋𝑈𝑌𝑉𝑅𝑊) → {𝑔 ∣ ∃𝑠 ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋})} ⊆ {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓})
93 ssintab 4858 . . . . . . 7 ( {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓} ⊆ {𝑔 ∣ ∃𝑠 ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋})} ↔ ∀𝑔(∃𝑠 ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋}) → {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓} ⊆ 𝑔))
9482eqeq2d 2769 . . . . . . . . . 10 (𝑠 = 𝑟 → (𝑔 = (𝑠 “ {𝑋}) ↔ 𝑔 = (𝑟 “ {𝑋})))
9594rexab2 3616 . . . . . . . . 9 (∃𝑠 ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋}) ↔ ∃𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) ∧ 𝑔 = (𝑟 “ {𝑋})))
96 simpr 488 . . . . . . . . . . 11 (((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) ∧ 𝑔 = (𝑟 “ {𝑋})) → 𝑔 = (𝑟 “ {𝑋}))
97 imass1 5940 . . . . . . . . . . . . . . 15 (𝑅𝑟 → (𝑅 “ {𝑋}) ⊆ (𝑟 “ {𝑋}))
9897adantr 484 . . . . . . . . . . . . . 14 ((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → (𝑅 “ {𝑋}) ⊆ (𝑟 “ {𝑋}))
99 imass1 5940 . . . . . . . . . . . . . . 15 (𝑅𝑟 → (𝑅 “ (𝑟 “ {𝑋})) ⊆ (𝑟 “ (𝑟 “ {𝑋})))
100 imaco 6085 . . . . . . . . . . . . . . . 16 ((𝑟𝑟) “ {𝑋}) = (𝑟 “ (𝑟 “ {𝑋}))
101 imass1 5940 . . . . . . . . . . . . . . . 16 ((𝑟𝑟) ⊆ 𝑟 → ((𝑟𝑟) “ {𝑋}) ⊆ (𝑟 “ {𝑋}))
102100, 101eqsstrrid 3943 . . . . . . . . . . . . . . 15 ((𝑟𝑟) ⊆ 𝑟 → (𝑟 “ (𝑟 “ {𝑋})) ⊆ (𝑟 “ {𝑋}))
10399, 102sylan9ss 3907 . . . . . . . . . . . . . 14 ((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → (𝑅 “ (𝑟 “ {𝑋})) ⊆ (𝑟 “ {𝑋}))
10498, 103jca 515 . . . . . . . . . . . . 13 ((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) → ((𝑅 “ {𝑋}) ⊆ (𝑟 “ {𝑋}) ∧ (𝑅 “ (𝑟 “ {𝑋})) ⊆ (𝑟 “ {𝑋})))
105104adantr 484 . . . . . . . . . . . 12 (((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) ∧ 𝑔 = (𝑟 “ {𝑋})) → ((𝑅 “ {𝑋}) ⊆ (𝑟 “ {𝑋}) ∧ (𝑅 “ (𝑟 “ {𝑋})) ⊆ (𝑟 “ {𝑋})))
106 vex 3413 . . . . . . . . . . . . . 14 𝑟 ∈ V
107106imaex 7631 . . . . . . . . . . . . 13 (𝑟 “ {𝑋}) ∈ V
108 imaundi 5984 . . . . . . . . . . . . . . . 16 (𝑅 “ ({𝑋} ∪ 𝑓)) = ((𝑅 “ {𝑋}) ∪ (𝑅𝑓))
109108sseq1i 3922 . . . . . . . . . . . . . . 15 ((𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓 ↔ ((𝑅 “ {𝑋}) ∪ (𝑅𝑓)) ⊆ 𝑓)
110 unss 4091 . . . . . . . . . . . . . . 15 (((𝑅 “ {𝑋}) ⊆ 𝑓 ∧ (𝑅𝑓) ⊆ 𝑓) ↔ ((𝑅 “ {𝑋}) ∪ (𝑅𝑓)) ⊆ 𝑓)
111109, 110bitr4i 281 . . . . . . . . . . . . . 14 ((𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓 ↔ ((𝑅 “ {𝑋}) ⊆ 𝑓 ∧ (𝑅𝑓) ⊆ 𝑓))
112 imaeq2 5901 . . . . . . . . . . . . . . . 16 (𝑓 = (𝑟 “ {𝑋}) → (𝑅𝑓) = (𝑅 “ (𝑟 “ {𝑋})))
113 id 22 . . . . . . . . . . . . . . . 16 (𝑓 = (𝑟 “ {𝑋}) → 𝑓 = (𝑟 “ {𝑋}))
114112, 113sseq12d 3927 . . . . . . . . . . . . . . 15 (𝑓 = (𝑟 “ {𝑋}) → ((𝑅𝑓) ⊆ 𝑓 ↔ (𝑅 “ (𝑟 “ {𝑋})) ⊆ (𝑟 “ {𝑋})))
115114cleq2lem 40709 . . . . . . . . . . . . . 14 (𝑓 = (𝑟 “ {𝑋}) → (((𝑅 “ {𝑋}) ⊆ 𝑓 ∧ (𝑅𝑓) ⊆ 𝑓) ↔ ((𝑅 “ {𝑋}) ⊆ (𝑟 “ {𝑋}) ∧ (𝑅 “ (𝑟 “ {𝑋})) ⊆ (𝑟 “ {𝑋}))))
116111, 115syl5bb 286 . . . . . . . . . . . . 13 (𝑓 = (𝑟 “ {𝑋}) → ((𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓 ↔ ((𝑅 “ {𝑋}) ⊆ (𝑟 “ {𝑋}) ∧ (𝑅 “ (𝑟 “ {𝑋})) ⊆ (𝑟 “ {𝑋}))))
117107, 116elab 3590 . . . . . . . . . . . 12 ((𝑟 “ {𝑋}) ∈ {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓} ↔ ((𝑅 “ {𝑋}) ⊆ (𝑟 “ {𝑋}) ∧ (𝑅 “ (𝑟 “ {𝑋})) ⊆ (𝑟 “ {𝑋})))
118105, 117sylibr 237 . . . . . . . . . . 11 (((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) ∧ 𝑔 = (𝑟 “ {𝑋})) → (𝑟 “ {𝑋}) ∈ {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓})
11996, 118eqeltrd 2852 . . . . . . . . . 10 (((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) ∧ 𝑔 = (𝑟 “ {𝑋})) → 𝑔 ∈ {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓})
120119exlimiv 1931 . . . . . . . . 9 (∃𝑟((𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟) ∧ 𝑔 = (𝑟 “ {𝑋})) → 𝑔 ∈ {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓})
12195, 120sylbi 220 . . . . . . . 8 (∃𝑠 ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋}) → 𝑔 ∈ {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓})
122 intss1 4856 . . . . . . . 8 (𝑔 ∈ {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓} → {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓} ⊆ 𝑔)
123121, 122syl 17 . . . . . . 7 (∃𝑠 ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋}) → {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓} ⊆ 𝑔)
12493, 123mpgbir 1801 . . . . . 6 {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓} ⊆ {𝑔 ∣ ∃𝑠 ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋})}
125124a1i 11 . . . . 5 ((𝑋𝑈𝑌𝑉𝑅𝑊) → {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓} ⊆ {𝑔 ∣ ∃𝑠 ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋})})
12692, 125eqssd 3911 . . . 4 ((𝑋𝑈𝑌𝑉𝑅𝑊) → {𝑔 ∣ ∃𝑠 ∈ {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)}𝑔 = (𝑠 “ {𝑋})} = {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓})
12710, 126eqtrd 2793 . . 3 ((𝑋𝑈𝑌𝑉𝑅𝑊) → ( {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} “ {𝑋}) = {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓})
128127eleq2d 2837 . 2 ((𝑋𝑈𝑌𝑉𝑅𝑊) → (𝑌 ∈ ( {𝑟 ∣ (𝑅𝑟 ∧ (𝑟𝑟) ⊆ 𝑟)} “ {𝑋}) ↔ 𝑌 {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓}))
1298, 128bitrd 282 1 ((𝑋𝑈𝑌𝑉𝑅𝑊) → (𝑋(t+‘𝑅)𝑌𝑌 {𝑓 ∣ (𝑅 “ ({𝑋} ∪ 𝑓)) ⊆ 𝑓}))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084  ∀wal 1536   = wceq 1538  ∃wex 1781   ∈ wcel 2111  {cab 2735   ≠ wne 2951  ∃wrex 3071  Vcvv 3409  [wsbc 3698  ⦋csb 3807   ∪ cun 3858   ∩ cin 3859   ⊆ wss 3860  ∅c0 4227  {csn 4525  ⟨cop 4531  ∩ cint 4841   class class class wbr 5035   × cxp 5525  ran crn 5528   “ cima 5530   ∘ ccom 5531  ‘cfv 6339  t+ctcl 14397 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-rep 5159  ax-sep 5172  ax-nul 5179  ax-pow 5237  ax-pr 5301  ax-un 7464  ax-cnex 10636  ax-resscn 10637  ax-1cn 10638  ax-icn 10639  ax-addcl 10640  ax-addrcl 10641  ax-mulcl 10642  ax-mulrcl 10643  ax-mulcom 10644  ax-addass 10645  ax-mulass 10646  ax-distr 10647  ax-i2m1 10648  ax-1ne0 10649  ax-1rid 10650  ax-rnegex 10651  ax-rrecex 10652  ax-cnre 10653  ax-pre-lttri 10654  ax-pre-lttrn 10655  ax-pre-ltadd 10656  ax-pre-mulgt0 10657 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-reu 3077  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-pss 3879  df-nul 4228  df-if 4424  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4802  df-int 4842  df-iun 4888  df-br 5036  df-opab 5098  df-mpt 5116  df-tr 5142  df-id 5433  df-eprel 5438  df-po 5446  df-so 5447  df-fr 5486  df-we 5488  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-pred 6130  df-ord 6176  df-on 6177  df-lim 6178  df-suc 6179  df-iota 6298  df-fun 6341  df-fn 6342  df-f 6343  df-f1 6344  df-fo 6345  df-f1o 6346  df-fv 6347  df-riota 7113  df-ov 7158  df-oprab 7159  df-mpo 7160  df-om 7585  df-2nd 7699  df-wrecs 7962  df-recs 8023  df-rdg 8061  df-er 8304  df-en 8533  df-dom 8534  df-sdom 8535  df-pnf 10720  df-mnf 10721  df-xr 10722  df-ltxr 10723  df-le 10724  df-sub 10915  df-neg 10916  df-nn 11680  df-2 11742  df-n0 11940  df-z 12026  df-uz 12288  df-seq 13424  df-trcl 14399  df-relexp 14432 This theorem is referenced by:  dffrege76  41041
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