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Theorem csbcnvg 4519
 Description: Move class substitution in and out of the converse of a function. (Contributed by Thierry Arnoux, 8-Feb-2017.)
Assertion
Ref Expression
csbcnvg (𝐴𝑉𝐴 / 𝑥𝐹 = 𝐴 / 𝑥𝐹)

Proof of Theorem csbcnvg
Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sbcbrg 3813 . . . . 5 (𝐴𝑉 → ([𝐴 / 𝑥]𝑧𝐹𝑦𝐴 / 𝑥𝑧𝐴 / 𝑥𝐹𝐴 / 𝑥𝑦))
2 csbconstg 2864 . . . . . 6 (𝐴𝑉𝐴 / 𝑥𝑧 = 𝑧)
3 csbconstg 2864 . . . . . 6 (𝐴𝑉𝐴 / 𝑥𝑦 = 𝑦)
42, 3breq12d 3777 . . . . 5 (𝐴𝑉 → (𝐴 / 𝑥𝑧𝐴 / 𝑥𝐹𝐴 / 𝑥𝑦𝑧𝐴 / 𝑥𝐹𝑦))
51, 4bitrd 177 . . . 4 (𝐴𝑉 → ([𝐴 / 𝑥]𝑧𝐹𝑦𝑧𝐴 / 𝑥𝐹𝑦))
65opabbidv 3823 . . 3 (𝐴𝑉 → {⟨𝑦, 𝑧⟩ ∣ [𝐴 / 𝑥]𝑧𝐹𝑦} = {⟨𝑦, 𝑧⟩ ∣ 𝑧𝐴 / 𝑥𝐹𝑦})
7 csbopabg 3835 . . 3 (𝐴𝑉𝐴 / 𝑥{⟨𝑦, 𝑧⟩ ∣ 𝑧𝐹𝑦} = {⟨𝑦, 𝑧⟩ ∣ [𝐴 / 𝑥]𝑧𝐹𝑦})
8 df-cnv 4353 . . . 4 𝐴 / 𝑥𝐹 = {⟨𝑦, 𝑧⟩ ∣ 𝑧𝐴 / 𝑥𝐹𝑦}
98a1i 9 . . 3 (𝐴𝑉𝐴 / 𝑥𝐹 = {⟨𝑦, 𝑧⟩ ∣ 𝑧𝐴 / 𝑥𝐹𝑦})
106, 7, 93eqtr4rd 2083 . 2 (𝐴𝑉𝐴 / 𝑥𝐹 = 𝐴 / 𝑥{⟨𝑦, 𝑧⟩ ∣ 𝑧𝐹𝑦})
11 df-cnv 4353 . . 3 𝐹 = {⟨𝑦, 𝑧⟩ ∣ 𝑧𝐹𝑦}
1211csbeq2i 2876 . 2 𝐴 / 𝑥𝐹 = 𝐴 / 𝑥{⟨𝑦, 𝑧⟩ ∣ 𝑧𝐹𝑦}
1310, 12syl6eqr 2090 1 (𝐴𝑉𝐴 / 𝑥𝐹 = 𝐴 / 𝑥𝐹)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1243   ∈ wcel 1393  [wsbc 2764  ⦋csb 2852   class class class wbr 3764  {copab 3817  ◡ccnv 4344 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-sbc 2765  df-csb 2853  df-un 2922  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-cnv 4353 This theorem is referenced by: (None)
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