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Theorem fcoinvbr 30534
Description: Binary relation for the equivalence relation from fcoinver 30533. (Contributed by Thierry Arnoux, 3-Jan-2020.)
Hypothesis
Ref Expression
fcoinvbr.e = (𝐹𝐹)
Assertion
Ref Expression
fcoinvbr ((𝐹 Fn 𝐴𝑋𝐴𝑌𝐴) → (𝑋 𝑌 ↔ (𝐹𝑋) = (𝐹𝑌)))

Proof of Theorem fcoinvbr
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 fcoinvbr.e . . . . 5 = (𝐹𝐹)
21breqi 5046 . . . 4 (𝑋 𝑌𝑋(𝐹𝐹)𝑌)
3 brcog 5719 . . . 4 ((𝑋𝐴𝑌𝐴) → (𝑋(𝐹𝐹)𝑌 ↔ ∃𝑧(𝑋𝐹𝑧𝑧𝐹𝑌)))
42, 3syl5bb 286 . . 3 ((𝑋𝐴𝑌𝐴) → (𝑋 𝑌 ↔ ∃𝑧(𝑋𝐹𝑧𝑧𝐹𝑌)))
543adant1 1131 . 2 ((𝐹 Fn 𝐴𝑋𝐴𝑌𝐴) → (𝑋 𝑌 ↔ ∃𝑧(𝑋𝐹𝑧𝑧𝐹𝑌)))
6 fvex 6700 . . . . 5 (𝐹𝑋) ∈ V
76eqvinc 3548 . . . 4 ((𝐹𝑋) = (𝐹𝑌) ↔ ∃𝑧(𝑧 = (𝐹𝑋) ∧ 𝑧 = (𝐹𝑌)))
8 eqcom 2746 . . . . . 6 (𝑧 = (𝐹𝑋) ↔ (𝐹𝑋) = 𝑧)
9 eqcom 2746 . . . . . 6 (𝑧 = (𝐹𝑌) ↔ (𝐹𝑌) = 𝑧)
108, 9anbi12i 630 . . . . 5 ((𝑧 = (𝐹𝑋) ∧ 𝑧 = (𝐹𝑌)) ↔ ((𝐹𝑋) = 𝑧 ∧ (𝐹𝑌) = 𝑧))
1110exbii 1854 . . . 4 (∃𝑧(𝑧 = (𝐹𝑋) ∧ 𝑧 = (𝐹𝑌)) ↔ ∃𝑧((𝐹𝑋) = 𝑧 ∧ (𝐹𝑌) = 𝑧))
127, 11bitri 278 . . 3 ((𝐹𝑋) = (𝐹𝑌) ↔ ∃𝑧((𝐹𝑋) = 𝑧 ∧ (𝐹𝑌) = 𝑧))
13 fnbrfvb 6735 . . . . . . 7 ((𝐹 Fn 𝐴𝑋𝐴) → ((𝐹𝑋) = 𝑧𝑋𝐹𝑧))
14133adant3 1133 . . . . . 6 ((𝐹 Fn 𝐴𝑋𝐴𝑌𝐴) → ((𝐹𝑋) = 𝑧𝑋𝐹𝑧))
15 fnbrfvb 6735 . . . . . . 7 ((𝐹 Fn 𝐴𝑌𝐴) → ((𝐹𝑌) = 𝑧𝑌𝐹𝑧))
16153adant2 1132 . . . . . 6 ((𝐹 Fn 𝐴𝑋𝐴𝑌𝐴) → ((𝐹𝑌) = 𝑧𝑌𝐹𝑧))
1714, 16anbi12d 634 . . . . 5 ((𝐹 Fn 𝐴𝑋𝐴𝑌𝐴) → (((𝐹𝑋) = 𝑧 ∧ (𝐹𝑌) = 𝑧) ↔ (𝑋𝐹𝑧𝑌𝐹𝑧)))
18 vex 3404 . . . . . . . 8 𝑧 ∈ V
19 brcnvg 5732 . . . . . . . 8 ((𝑧 ∈ V ∧ 𝑌𝐴) → (𝑧𝐹𝑌𝑌𝐹𝑧))
2018, 19mpan 690 . . . . . . 7 (𝑌𝐴 → (𝑧𝐹𝑌𝑌𝐹𝑧))
21203ad2ant3 1136 . . . . . 6 ((𝐹 Fn 𝐴𝑋𝐴𝑌𝐴) → (𝑧𝐹𝑌𝑌𝐹𝑧))
2221anbi2d 632 . . . . 5 ((𝐹 Fn 𝐴𝑋𝐴𝑌𝐴) → ((𝑋𝐹𝑧𝑧𝐹𝑌) ↔ (𝑋𝐹𝑧𝑌𝐹𝑧)))
2317, 22bitr4d 285 . . . 4 ((𝐹 Fn 𝐴𝑋𝐴𝑌𝐴) → (((𝐹𝑋) = 𝑧 ∧ (𝐹𝑌) = 𝑧) ↔ (𝑋𝐹𝑧𝑧𝐹𝑌)))
2423exbidv 1928 . . 3 ((𝐹 Fn 𝐴𝑋𝐴𝑌𝐴) → (∃𝑧((𝐹𝑋) = 𝑧 ∧ (𝐹𝑌) = 𝑧) ↔ ∃𝑧(𝑋𝐹𝑧𝑧𝐹𝑌)))
2512, 24syl5bb 286 . 2 ((𝐹 Fn 𝐴𝑋𝐴𝑌𝐴) → ((𝐹𝑋) = (𝐹𝑌) ↔ ∃𝑧(𝑋𝐹𝑧𝑧𝐹𝑌)))
265, 25bitr4d 285 1 ((𝐹 Fn 𝐴𝑋𝐴𝑌𝐴) → (𝑋 𝑌 ↔ (𝐹𝑋) = (𝐹𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  w3a 1088   = wceq 1542  wex 1786  wcel 2114  Vcvv 3400   class class class wbr 5040  ccnv 5534  ccom 5539   Fn wfn 6345  cfv 6350
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-sep 5177  ax-nul 5184  ax-pr 5306
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ral 3059  df-rex 3060  df-v 3402  df-sbc 3686  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-nul 4222  df-if 4425  df-sn 4527  df-pr 4529  df-op 4533  df-uni 4807  df-br 5041  df-opab 5103  df-id 5439  df-xp 5541  df-rel 5542  df-cnv 5543  df-co 5544  df-dm 5545  df-iota 6308  df-fun 6352  df-fn 6353  df-fv 6358
This theorem is referenced by:  qtophaus  31371
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