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Theorem fcoinvbr 30273
Description: Binary relation for the equivalence relation from fcoinver 30272. (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 5069 . . . 4 (𝑋 𝑌𝑋(𝐹𝐹)𝑌)
3 brcog 5736 . . . 4 ((𝑋𝐴𝑌𝐴) → (𝑋(𝐹𝐹)𝑌 ↔ ∃𝑧(𝑋𝐹𝑧𝑧𝐹𝑌)))
42, 3syl5bb 284 . . 3 ((𝑋𝐴𝑌𝐴) → (𝑋 𝑌 ↔ ∃𝑧(𝑋𝐹𝑧𝑧𝐹𝑌)))
543adant1 1124 . 2 ((𝐹 Fn 𝐴𝑋𝐴𝑌𝐴) → (𝑋 𝑌 ↔ ∃𝑧(𝑋𝐹𝑧𝑧𝐹𝑌)))
6 fvex 6680 . . . . 5 (𝐹𝑋) ∈ V
76eqvinc 3646 . . . 4 ((𝐹𝑋) = (𝐹𝑌) ↔ ∃𝑧(𝑧 = (𝐹𝑋) ∧ 𝑧 = (𝐹𝑌)))
8 eqcom 2833 . . . . . 6 (𝑧 = (𝐹𝑋) ↔ (𝐹𝑋) = 𝑧)
9 eqcom 2833 . . . . . 6 (𝑧 = (𝐹𝑌) ↔ (𝐹𝑌) = 𝑧)
108, 9anbi12i 626 . . . . 5 ((𝑧 = (𝐹𝑋) ∧ 𝑧 = (𝐹𝑌)) ↔ ((𝐹𝑋) = 𝑧 ∧ (𝐹𝑌) = 𝑧))
1110exbii 1841 . . . 4 (∃𝑧(𝑧 = (𝐹𝑋) ∧ 𝑧 = (𝐹𝑌)) ↔ ∃𝑧((𝐹𝑋) = 𝑧 ∧ (𝐹𝑌) = 𝑧))
127, 11bitri 276 . . 3 ((𝐹𝑋) = (𝐹𝑌) ↔ ∃𝑧((𝐹𝑋) = 𝑧 ∧ (𝐹𝑌) = 𝑧))
13 fnbrfvb 6715 . . . . . . 7 ((𝐹 Fn 𝐴𝑋𝐴) → ((𝐹𝑋) = 𝑧𝑋𝐹𝑧))
14133adant3 1126 . . . . . 6 ((𝐹 Fn 𝐴𝑋𝐴𝑌𝐴) → ((𝐹𝑋) = 𝑧𝑋𝐹𝑧))
15 fnbrfvb 6715 . . . . . . 7 ((𝐹 Fn 𝐴𝑌𝐴) → ((𝐹𝑌) = 𝑧𝑌𝐹𝑧))
16153adant2 1125 . . . . . 6 ((𝐹 Fn 𝐴𝑋𝐴𝑌𝐴) → ((𝐹𝑌) = 𝑧𝑌𝐹𝑧))
1714, 16anbi12d 630 . . . . 5 ((𝐹 Fn 𝐴𝑋𝐴𝑌𝐴) → (((𝐹𝑋) = 𝑧 ∧ (𝐹𝑌) = 𝑧) ↔ (𝑋𝐹𝑧𝑌𝐹𝑧)))
18 vex 3503 . . . . . . . 8 𝑧 ∈ V
19 brcnvg 5749 . . . . . . . 8 ((𝑧 ∈ V ∧ 𝑌𝐴) → (𝑧𝐹𝑌𝑌𝐹𝑧))
2018, 19mpan 686 . . . . . . 7 (𝑌𝐴 → (𝑧𝐹𝑌𝑌𝐹𝑧))
21203ad2ant3 1129 . . . . . 6 ((𝐹 Fn 𝐴𝑋𝐴𝑌𝐴) → (𝑧𝐹𝑌𝑌𝐹𝑧))
2221anbi2d 628 . . . . 5 ((𝐹 Fn 𝐴𝑋𝐴𝑌𝐴) → ((𝑋𝐹𝑧𝑧𝐹𝑌) ↔ (𝑋𝐹𝑧𝑌𝐹𝑧)))
2317, 22bitr4d 283 . . . 4 ((𝐹 Fn 𝐴𝑋𝐴𝑌𝐴) → (((𝐹𝑋) = 𝑧 ∧ (𝐹𝑌) = 𝑧) ↔ (𝑋𝐹𝑧𝑧𝐹𝑌)))
2423exbidv 1915 . . 3 ((𝐹 Fn 𝐴𝑋𝐴𝑌𝐴) → (∃𝑧((𝐹𝑋) = 𝑧 ∧ (𝐹𝑌) = 𝑧) ↔ ∃𝑧(𝑋𝐹𝑧𝑧𝐹𝑌)))
2512, 24syl5bb 284 . 2 ((𝐹 Fn 𝐴𝑋𝐴𝑌𝐴) → ((𝐹𝑋) = (𝐹𝑌) ↔ ∃𝑧(𝑋𝐹𝑧𝑧𝐹𝑌)))
265, 25bitr4d 283 1 ((𝐹 Fn 𝐴𝑋𝐴𝑌𝐴) → (𝑋 𝑌 ↔ (𝐹𝑋) = (𝐹𝑌)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  w3a 1081   = wceq 1530  wex 1773  wcel 2107  Vcvv 3500   class class class wbr 5063  ccnv 5553  ccom 5558   Fn wfn 6347  cfv 6352
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pr 5326
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-sbc 3777  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-iota 6312  df-fun 6354  df-fn 6355  df-fv 6360
This theorem is referenced by:  qtophaus  30986
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