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Theorem csbcnvg 4434
 Description: Move class substitution in and out of the converse of a function. (Contributed by Thierry Arnoux, 8-Feb-2017.)
Assertion
Ref Expression
csbcnvg (A 𝑉A / x𝐹 = A / x𝐹)

Proof of Theorem csbcnvg
Dummy variables y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sbcbrg 3776 . . . . 5 (A 𝑉 → ([A / x]z𝐹yA / xzA / x𝐹A / xy))
2 csbconstg 2832 . . . . . 6 (A 𝑉A / xz = z)
3 csbconstg 2832 . . . . . 6 (A 𝑉A / xy = y)
42, 3breq12d 3740 . . . . 5 (A 𝑉 → (A / xzA / x𝐹A / xyzA / x𝐹y))
51, 4bitrd 177 . . . 4 (A 𝑉 → ([A / x]z𝐹yzA / x𝐹y))
65opabbidv 3786 . . 3 (A 𝑉 → {⟨y, z⟩ ∣ [A / x]z𝐹y} = {⟨y, z⟩ ∣ zA / x𝐹y})
7 csbopabg 3798 . . 3 (A 𝑉A / x{⟨y, z⟩ ∣ z𝐹y} = {⟨y, z⟩ ∣ [A / x]z𝐹y})
8 df-cnv 4268 . . . 4 A / x𝐹 = {⟨y, z⟩ ∣ zA / x𝐹y}
98a1i 9 . . 3 (A 𝑉A / x𝐹 = {⟨y, z⟩ ∣ zA / x𝐹y})
106, 7, 93eqtr4rd 2056 . 2 (A 𝑉A / x𝐹 = A / x{⟨y, z⟩ ∣ z𝐹y})
11 df-cnv 4268 . . 3 𝐹 = {⟨y, z⟩ ∣ z𝐹y}
1211csbeq2i 2844 . 2 A / x𝐹 = A / x{⟨y, z⟩ ∣ z𝐹y}
1310, 12syl6eqr 2063 1 (A 𝑉A / x𝐹 = A / x𝐹)
 Colors of variables: wff set class Syntax hints:   → wi 4   = wceq 1223   ∈ wcel 1366  [wsbc 2732  ⦋csb 2820   class class class wbr 3727  {copab 3780  ◡ccnv 4259 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 614  ax-5 1309  ax-7 1310  ax-gen 1311  ax-ie1 1355  ax-ie2 1356  ax-8 1368  ax-10 1369  ax-11 1370  ax-i12 1371  ax-bnd 1372  ax-4 1373  ax-17 1392  ax-i9 1396  ax-ial 1400  ax-i5r 1401  ax-ext 1995 This theorem depends on definitions:  df-bi 110  df-3an 869  df-tru 1226  df-nf 1323  df-sb 1619  df-clab 2000  df-cleq 2006  df-clel 2009  df-nfc 2140  df-v 2528  df-sbc 2733  df-csb 2821  df-un 2890  df-sn 3345  df-pr 3346  df-op 3348  df-br 3728  df-opab 3782  df-cnv 4268 This theorem is referenced by: (None)
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