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Theorem fcoinvbr 32625
Description: Binary relation for the equivalence relation from fcoinver 32624. (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 5154 . . . 4 (𝑋 𝑌𝑋(𝐹𝐹)𝑌)
3 brcog 5880 . . . 4 ((𝑋𝐴𝑌𝐴) → (𝑋(𝐹𝐹)𝑌 ↔ ∃𝑧(𝑋𝐹𝑧𝑧𝐹𝑌)))
42, 3bitrid 283 . . 3 ((𝑋𝐴𝑌𝐴) → (𝑋 𝑌 ↔ ∃𝑧(𝑋𝐹𝑧𝑧𝐹𝑌)))
543adant1 1129 . 2 ((𝐹 Fn 𝐴𝑋𝐴𝑌𝐴) → (𝑋 𝑌 ↔ ∃𝑧(𝑋𝐹𝑧𝑧𝐹𝑌)))
6 fvex 6920 . . . . 5 (𝐹𝑋) ∈ V
76eqvinc 3649 . . . 4 ((𝐹𝑋) = (𝐹𝑌) ↔ ∃𝑧(𝑧 = (𝐹𝑋) ∧ 𝑧 = (𝐹𝑌)))
8 eqcom 2742 . . . . . 6 (𝑧 = (𝐹𝑋) ↔ (𝐹𝑋) = 𝑧)
9 eqcom 2742 . . . . . 6 (𝑧 = (𝐹𝑌) ↔ (𝐹𝑌) = 𝑧)
108, 9anbi12i 628 . . . . 5 ((𝑧 = (𝐹𝑋) ∧ 𝑧 = (𝐹𝑌)) ↔ ((𝐹𝑋) = 𝑧 ∧ (𝐹𝑌) = 𝑧))
1110exbii 1845 . . . 4 (∃𝑧(𝑧 = (𝐹𝑋) ∧ 𝑧 = (𝐹𝑌)) ↔ ∃𝑧((𝐹𝑋) = 𝑧 ∧ (𝐹𝑌) = 𝑧))
127, 11bitri 275 . . 3 ((𝐹𝑋) = (𝐹𝑌) ↔ ∃𝑧((𝐹𝑋) = 𝑧 ∧ (𝐹𝑌) = 𝑧))
13 fnbrfvb 6960 . . . . . . 7 ((𝐹 Fn 𝐴𝑋𝐴) → ((𝐹𝑋) = 𝑧𝑋𝐹𝑧))
14133adant3 1131 . . . . . 6 ((𝐹 Fn 𝐴𝑋𝐴𝑌𝐴) → ((𝐹𝑋) = 𝑧𝑋𝐹𝑧))
15 fnbrfvb 6960 . . . . . . 7 ((𝐹 Fn 𝐴𝑌𝐴) → ((𝐹𝑌) = 𝑧𝑌𝐹𝑧))
16153adant2 1130 . . . . . 6 ((𝐹 Fn 𝐴𝑋𝐴𝑌𝐴) → ((𝐹𝑌) = 𝑧𝑌𝐹𝑧))
1714, 16anbi12d 632 . . . . 5 ((𝐹 Fn 𝐴𝑋𝐴𝑌𝐴) → (((𝐹𝑋) = 𝑧 ∧ (𝐹𝑌) = 𝑧) ↔ (𝑋𝐹𝑧𝑌𝐹𝑧)))
18 vex 3482 . . . . . . . 8 𝑧 ∈ V
19 brcnvg 5893 . . . . . . . 8 ((𝑧 ∈ V ∧ 𝑌𝐴) → (𝑧𝐹𝑌𝑌𝐹𝑧))
2018, 19mpan 690 . . . . . . 7 (𝑌𝐴 → (𝑧𝐹𝑌𝑌𝐹𝑧))
21203ad2ant3 1134 . . . . . 6 ((𝐹 Fn 𝐴𝑋𝐴𝑌𝐴) → (𝑧𝐹𝑌𝑌𝐹𝑧))
2221anbi2d 630 . . . . 5 ((𝐹 Fn 𝐴𝑋𝐴𝑌𝐴) → ((𝑋𝐹𝑧𝑧𝐹𝑌) ↔ (𝑋𝐹𝑧𝑌𝐹𝑧)))
2317, 22bitr4d 282 . . . 4 ((𝐹 Fn 𝐴𝑋𝐴𝑌𝐴) → (((𝐹𝑋) = 𝑧 ∧ (𝐹𝑌) = 𝑧) ↔ (𝑋𝐹𝑧𝑧𝐹𝑌)))
2423exbidv 1919 . . 3 ((𝐹 Fn 𝐴𝑋𝐴𝑌𝐴) → (∃𝑧((𝐹𝑋) = 𝑧 ∧ (𝐹𝑌) = 𝑧) ↔ ∃𝑧(𝑋𝐹𝑧𝑧𝐹𝑌)))
2512, 24bitrid 283 . 2 ((𝐹 Fn 𝐴𝑋𝐴𝑌𝐴) → ((𝐹𝑋) = (𝐹𝑌) ↔ ∃𝑧(𝑋𝐹𝑧𝑧𝐹𝑌)))
265, 25bitr4d 282 1 ((𝐹 Fn 𝐴𝑋𝐴𝑌𝐴) → (𝑋 𝑌 ↔ (𝐹𝑋) = (𝐹𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  w3a 1086   = wceq 1537  wex 1776  wcel 2106  Vcvv 3478   class class class wbr 5148  ccnv 5688  ccom 5693   Fn wfn 6558  cfv 6563
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-iota 6516  df-fun 6565  df-fn 6566  df-fv 6571
This theorem is referenced by:  qtophaus  33797
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