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Theorem fcoinvbr 30947
Description: Binary relation for the equivalence relation from fcoinver 30946. (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 5080 . . . 4 (𝑋 𝑌𝑋(𝐹𝐹)𝑌)
3 brcog 5775 . . . 4 ((𝑋𝐴𝑌𝐴) → (𝑋(𝐹𝐹)𝑌 ↔ ∃𝑧(𝑋𝐹𝑧𝑧𝐹𝑌)))
42, 3syl5bb 283 . . 3 ((𝑋𝐴𝑌𝐴) → (𝑋 𝑌 ↔ ∃𝑧(𝑋𝐹𝑧𝑧𝐹𝑌)))
543adant1 1129 . 2 ((𝐹 Fn 𝐴𝑋𝐴𝑌𝐴) → (𝑋 𝑌 ↔ ∃𝑧(𝑋𝐹𝑧𝑧𝐹𝑌)))
6 fvex 6787 . . . . 5 (𝐹𝑋) ∈ V
76eqvinc 3579 . . . 4 ((𝐹𝑋) = (𝐹𝑌) ↔ ∃𝑧(𝑧 = (𝐹𝑋) ∧ 𝑧 = (𝐹𝑌)))
8 eqcom 2745 . . . . . 6 (𝑧 = (𝐹𝑋) ↔ (𝐹𝑋) = 𝑧)
9 eqcom 2745 . . . . . 6 (𝑧 = (𝐹𝑌) ↔ (𝐹𝑌) = 𝑧)
108, 9anbi12i 627 . . . . 5 ((𝑧 = (𝐹𝑋) ∧ 𝑧 = (𝐹𝑌)) ↔ ((𝐹𝑋) = 𝑧 ∧ (𝐹𝑌) = 𝑧))
1110exbii 1850 . . . 4 (∃𝑧(𝑧 = (𝐹𝑋) ∧ 𝑧 = (𝐹𝑌)) ↔ ∃𝑧((𝐹𝑋) = 𝑧 ∧ (𝐹𝑌) = 𝑧))
127, 11bitri 274 . . 3 ((𝐹𝑋) = (𝐹𝑌) ↔ ∃𝑧((𝐹𝑋) = 𝑧 ∧ (𝐹𝑌) = 𝑧))
13 fnbrfvb 6822 . . . . . . 7 ((𝐹 Fn 𝐴𝑋𝐴) → ((𝐹𝑋) = 𝑧𝑋𝐹𝑧))
14133adant3 1131 . . . . . 6 ((𝐹 Fn 𝐴𝑋𝐴𝑌𝐴) → ((𝐹𝑋) = 𝑧𝑋𝐹𝑧))
15 fnbrfvb 6822 . . . . . . 7 ((𝐹 Fn 𝐴𝑌𝐴) → ((𝐹𝑌) = 𝑧𝑌𝐹𝑧))
16153adant2 1130 . . . . . 6 ((𝐹 Fn 𝐴𝑋𝐴𝑌𝐴) → ((𝐹𝑌) = 𝑧𝑌𝐹𝑧))
1714, 16anbi12d 631 . . . . 5 ((𝐹 Fn 𝐴𝑋𝐴𝑌𝐴) → (((𝐹𝑋) = 𝑧 ∧ (𝐹𝑌) = 𝑧) ↔ (𝑋𝐹𝑧𝑌𝐹𝑧)))
18 vex 3436 . . . . . . . 8 𝑧 ∈ V
19 brcnvg 5788 . . . . . . . 8 ((𝑧 ∈ V ∧ 𝑌𝐴) → (𝑧𝐹𝑌𝑌𝐹𝑧))
2018, 19mpan 687 . . . . . . 7 (𝑌𝐴 → (𝑧𝐹𝑌𝑌𝐹𝑧))
21203ad2ant3 1134 . . . . . 6 ((𝐹 Fn 𝐴𝑋𝐴𝑌𝐴) → (𝑧𝐹𝑌𝑌𝐹𝑧))
2221anbi2d 629 . . . . 5 ((𝐹 Fn 𝐴𝑋𝐴𝑌𝐴) → ((𝑋𝐹𝑧𝑧𝐹𝑌) ↔ (𝑋𝐹𝑧𝑌𝐹𝑧)))
2317, 22bitr4d 281 . . . 4 ((𝐹 Fn 𝐴𝑋𝐴𝑌𝐴) → (((𝐹𝑋) = 𝑧 ∧ (𝐹𝑌) = 𝑧) ↔ (𝑋𝐹𝑧𝑧𝐹𝑌)))
2423exbidv 1924 . . 3 ((𝐹 Fn 𝐴𝑋𝐴𝑌𝐴) → (∃𝑧((𝐹𝑋) = 𝑧 ∧ (𝐹𝑌) = 𝑧) ↔ ∃𝑧(𝑋𝐹𝑧𝑧𝐹𝑌)))
2512, 24syl5bb 283 . 2 ((𝐹 Fn 𝐴𝑋𝐴𝑌𝐴) → ((𝐹𝑋) = (𝐹𝑌) ↔ ∃𝑧(𝑋𝐹𝑧𝑧𝐹𝑌)))
265, 25bitr4d 281 1 ((𝐹 Fn 𝐴𝑋𝐴𝑌𝐴) → (𝑋 𝑌 ↔ (𝐹𝑋) = (𝐹𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1086   = wceq 1539  wex 1782  wcel 2106  Vcvv 3432   class class class wbr 5074  ccnv 5588  ccom 5593   Fn wfn 6428  cfv 6433
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-iota 6391  df-fun 6435  df-fn 6436  df-fv 6441
This theorem is referenced by:  qtophaus  31786
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