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Theorem csbcnvgALT 5339
Description: Move class substitution in and out of the converse of a function. Version of csbcnv 5338 with a sethood antecedent but depending on fewer axioms. (Contributed by Thierry Arnoux, 8-Feb-2017.) (New usage is discouraged.) (Proof modification is discouraged.)
Assertion
Ref Expression
csbcnvgALT (𝐴𝑉𝐴 / 𝑥𝐹 = 𝐴 / 𝑥𝐹)

Proof of Theorem csbcnvgALT
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sbcbr123 4739 . . . . 5 ([𝐴 / 𝑥]𝑧𝐹𝑦𝐴 / 𝑥𝑧𝐴 / 𝑥𝐹𝐴 / 𝑥𝑦)
2 csbconstg 3579 . . . . . 6 (𝐴𝑉𝐴 / 𝑥𝑧 = 𝑧)
3 csbconstg 3579 . . . . . 6 (𝐴𝑉𝐴 / 𝑥𝑦 = 𝑦)
42, 3breq12d 4698 . . . . 5 (𝐴𝑉 → (𝐴 / 𝑥𝑧𝐴 / 𝑥𝐹𝐴 / 𝑥𝑦𝑧𝐴 / 𝑥𝐹𝑦))
51, 4syl5bb 272 . . . 4 (𝐴𝑉 → ([𝐴 / 𝑥]𝑧𝐹𝑦𝑧𝐴 / 𝑥𝐹𝑦))
65opabbidv 4749 . . 3 (𝐴𝑉 → {⟨𝑦, 𝑧⟩ ∣ [𝐴 / 𝑥]𝑧𝐹𝑦} = {⟨𝑦, 𝑧⟩ ∣ 𝑧𝐴 / 𝑥𝐹𝑦})
7 csbopabgALT 5038 . . 3 (𝐴𝑉𝐴 / 𝑥{⟨𝑦, 𝑧⟩ ∣ 𝑧𝐹𝑦} = {⟨𝑦, 𝑧⟩ ∣ [𝐴 / 𝑥]𝑧𝐹𝑦})
8 df-cnv 5151 . . . 4 𝐴 / 𝑥𝐹 = {⟨𝑦, 𝑧⟩ ∣ 𝑧𝐴 / 𝑥𝐹𝑦}
98a1i 11 . . 3 (𝐴𝑉𝐴 / 𝑥𝐹 = {⟨𝑦, 𝑧⟩ ∣ 𝑧𝐴 / 𝑥𝐹𝑦})
106, 7, 93eqtr4rd 2696 . 2 (𝐴𝑉𝐴 / 𝑥𝐹 = 𝐴 / 𝑥{⟨𝑦, 𝑧⟩ ∣ 𝑧𝐹𝑦})
11 df-cnv 5151 . . 3 𝐹 = {⟨𝑦, 𝑧⟩ ∣ 𝑧𝐹𝑦}
1211csbeq2i 4026 . 2 𝐴 / 𝑥𝐹 = 𝐴 / 𝑥{⟨𝑦, 𝑧⟩ ∣ 𝑧𝐹𝑦}
1310, 12syl6eqr 2703 1 (𝐴𝑉𝐴 / 𝑥𝐹 = 𝐴 / 𝑥𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523  wcel 2030  [wsbc 3468  csb 3566   class class class wbr 4685  {copab 4745  ccnv 5142
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-fal 1529  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-cnv 5151
This theorem is referenced by: (None)
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