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Theorem csbcnvg 4723
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 3982 . . . . 5 (𝐴𝑉 → ([𝐴 / 𝑥]𝑧𝐹𝑦𝐴 / 𝑥𝑧𝐴 / 𝑥𝐹𝐴 / 𝑥𝑦))
2 csbconstg 3016 . . . . . 6 (𝐴𝑉𝐴 / 𝑥𝑧 = 𝑧)
3 csbconstg 3016 . . . . . 6 (𝐴𝑉𝐴 / 𝑥𝑦 = 𝑦)
42, 3breq12d 3942 . . . . 5 (𝐴𝑉 → (𝐴 / 𝑥𝑧𝐴 / 𝑥𝐹𝐴 / 𝑥𝑦𝑧𝐴 / 𝑥𝐹𝑦))
51, 4bitrd 187 . . . 4 (𝐴𝑉 → ([𝐴 / 𝑥]𝑧𝐹𝑦𝑧𝐴 / 𝑥𝐹𝑦))
65opabbidv 3994 . . 3 (𝐴𝑉 → {⟨𝑦, 𝑧⟩ ∣ [𝐴 / 𝑥]𝑧𝐹𝑦} = {⟨𝑦, 𝑧⟩ ∣ 𝑧𝐴 / 𝑥𝐹𝑦})
7 csbopabg 4006 . . 3 (𝐴𝑉𝐴 / 𝑥{⟨𝑦, 𝑧⟩ ∣ 𝑧𝐹𝑦} = {⟨𝑦, 𝑧⟩ ∣ [𝐴 / 𝑥]𝑧𝐹𝑦})
8 df-cnv 4547 . . . 4 𝐴 / 𝑥𝐹 = {⟨𝑦, 𝑧⟩ ∣ 𝑧𝐴 / 𝑥𝐹𝑦}
98a1i 9 . . 3 (𝐴𝑉𝐴 / 𝑥𝐹 = {⟨𝑦, 𝑧⟩ ∣ 𝑧𝐴 / 𝑥𝐹𝑦})
106, 7, 93eqtr4rd 2183 . 2 (𝐴𝑉𝐴 / 𝑥𝐹 = 𝐴 / 𝑥{⟨𝑦, 𝑧⟩ ∣ 𝑧𝐹𝑦})
11 df-cnv 4547 . . 3 𝐹 = {⟨𝑦, 𝑧⟩ ∣ 𝑧𝐹𝑦}
1211csbeq2i 3029 . 2 𝐴 / 𝑥𝐹 = 𝐴 / 𝑥{⟨𝑦, 𝑧⟩ ∣ 𝑧𝐹𝑦}
1310, 12eqtr4di 2190 1 (𝐴𝑉𝐴 / 𝑥𝐹 = 𝐴 / 𝑥𝐹)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1331  wcel 1480  [wsbc 2909  csb 3003   class class class wbr 3929  {copab 3988  ccnv 4538
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-sbc 2910  df-csb 3004  df-un 3075  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-opab 3990  df-cnv 4547
This theorem is referenced by: (None)
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