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Theorem fcoinvbr 32618
Description: Binary relation for the equivalence relation from fcoinver 32617. (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 5149 . . . 4 (𝑋 𝑌𝑋(𝐹𝐹)𝑌)
3 brcog 5877 . . . 4 ((𝑋𝐴𝑌𝐴) → (𝑋(𝐹𝐹)𝑌 ↔ ∃𝑧(𝑋𝐹𝑧𝑧𝐹𝑌)))
42, 3bitrid 283 . . 3 ((𝑋𝐴𝑌𝐴) → (𝑋 𝑌 ↔ ∃𝑧(𝑋𝐹𝑧𝑧𝐹𝑌)))
543adant1 1131 . 2 ((𝐹 Fn 𝐴𝑋𝐴𝑌𝐴) → (𝑋 𝑌 ↔ ∃𝑧(𝑋𝐹𝑧𝑧𝐹𝑌)))
6 fvex 6919 . . . . 5 (𝐹𝑋) ∈ V
76eqvinc 3649 . . . 4 ((𝐹𝑋) = (𝐹𝑌) ↔ ∃𝑧(𝑧 = (𝐹𝑋) ∧ 𝑧 = (𝐹𝑌)))
8 eqcom 2744 . . . . . 6 (𝑧 = (𝐹𝑋) ↔ (𝐹𝑋) = 𝑧)
9 eqcom 2744 . . . . . 6 (𝑧 = (𝐹𝑌) ↔ (𝐹𝑌) = 𝑧)
108, 9anbi12i 628 . . . . 5 ((𝑧 = (𝐹𝑋) ∧ 𝑧 = (𝐹𝑌)) ↔ ((𝐹𝑋) = 𝑧 ∧ (𝐹𝑌) = 𝑧))
1110exbii 1848 . . . 4 (∃𝑧(𝑧 = (𝐹𝑋) ∧ 𝑧 = (𝐹𝑌)) ↔ ∃𝑧((𝐹𝑋) = 𝑧 ∧ (𝐹𝑌) = 𝑧))
127, 11bitri 275 . . 3 ((𝐹𝑋) = (𝐹𝑌) ↔ ∃𝑧((𝐹𝑋) = 𝑧 ∧ (𝐹𝑌) = 𝑧))
13 fnbrfvb 6959 . . . . . . 7 ((𝐹 Fn 𝐴𝑋𝐴) → ((𝐹𝑋) = 𝑧𝑋𝐹𝑧))
14133adant3 1133 . . . . . 6 ((𝐹 Fn 𝐴𝑋𝐴𝑌𝐴) → ((𝐹𝑋) = 𝑧𝑋𝐹𝑧))
15 fnbrfvb 6959 . . . . . . 7 ((𝐹 Fn 𝐴𝑌𝐴) → ((𝐹𝑌) = 𝑧𝑌𝐹𝑧))
16153adant2 1132 . . . . . 6 ((𝐹 Fn 𝐴𝑋𝐴𝑌𝐴) → ((𝐹𝑌) = 𝑧𝑌𝐹𝑧))
1714, 16anbi12d 632 . . . . 5 ((𝐹 Fn 𝐴𝑋𝐴𝑌𝐴) → (((𝐹𝑋) = 𝑧 ∧ (𝐹𝑌) = 𝑧) ↔ (𝑋𝐹𝑧𝑌𝐹𝑧)))
18 vex 3484 . . . . . . . 8 𝑧 ∈ V
19 brcnvg 5890 . . . . . . . 8 ((𝑧 ∈ V ∧ 𝑌𝐴) → (𝑧𝐹𝑌𝑌𝐹𝑧))
2018, 19mpan 690 . . . . . . 7 (𝑌𝐴 → (𝑧𝐹𝑌𝑌𝐹𝑧))
21203ad2ant3 1136 . . . . . 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 1921 . . 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 1087   = wceq 1540  wex 1779  wcel 2108  Vcvv 3480   class class class wbr 5143  ccnv 5684  ccom 5689   Fn wfn 6556  cfv 6561
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-iota 6514  df-fun 6563  df-fn 6564  df-fv 6569
This theorem is referenced by:  qtophaus  33835
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