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Theorem csbcnvg 4462
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 3804 . . . . 5 (A 𝑉 → ([A / x]z𝐹yA / xzA / x𝐹A / xy))
2 csbconstg 2858 . . . . . 6 (A 𝑉A / xz = z)
3 csbconstg 2858 . . . . . 6 (A 𝑉A / xy = y)
42, 3breq12d 3768 . . . . 5 (A 𝑉 → (A / xzA / x𝐹A / xyzA / x𝐹y))
51, 4bitrd 177 . . . 4 (A 𝑉 → ([A / x]z𝐹yzA / x𝐹y))
65opabbidv 3814 . . 3 (A 𝑉 → {⟨y, z⟩ ∣ [A / x]z𝐹y} = {⟨y, z⟩ ∣ zA / x𝐹y})
7 csbopabg 3826 . . 3 (A 𝑉A / x{⟨y, z⟩ ∣ z𝐹y} = {⟨y, z⟩ ∣ [A / x]z𝐹y})
8 df-cnv 4296 . . . 4 A / x𝐹 = {⟨y, z⟩ ∣ zA / x𝐹y}
98a1i 9 . . 3 (A 𝑉A / x𝐹 = {⟨y, z⟩ ∣ zA / x𝐹y})
106, 7, 93eqtr4rd 2080 . 2 (A 𝑉A / x𝐹 = A / x{⟨y, z⟩ ∣ z𝐹y})
11 df-cnv 4296 . . 3 𝐹 = {⟨y, z⟩ ∣ z𝐹y}
1211csbeq2i 2870 . 2 A / x𝐹 = A / x{⟨y, z⟩ ∣ z𝐹y}
1310, 12syl6eqr 2087 1 (A 𝑉A / x𝐹 = A / x𝐹)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1242   wcel 1390  [wsbc 2758  csb 2846   class class class wbr 3755  {copab 3808  ccnv 4287
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-cnv 4296
This theorem is referenced by: (None)
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