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 ∈ 𝑉 → ◡⦋A / x⦌𝐹 = ⦋A / x⦌◡𝐹)

Proof of Theorem csbcnvg

Dummy variables y z are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		sbcbrg 3776	. . . . 5 ⊢ (A ∈ 𝑉 → ([A / x]z𝐹y ↔ ⦋A / x⦌z⦋A / x⦌𝐹⦋A / x⦌y))
2		csbconstg 2832	. . . . . 6 ⊢ (A ∈ 𝑉 → ⦋A / x⦌z = z)
3		csbconstg 2832	. . . . . 6 ⊢ (A ∈ 𝑉 → ⦋A / x⦌y = y)
4	2, 3	breq12d 3740	. . . . 5 ⊢ (A ∈ 𝑉 → (⦋A / x⦌z⦋A / x⦌𝐹⦋A / x⦌y ↔ z⦋A / x⦌𝐹y))
5	1, 4	bitrd 177	. . . 4 ⊢ (A ∈ 𝑉 → ([A / x]z𝐹y ↔ z⦋A / x⦌𝐹y))
6	5	opabbidv 3786	. . 3 ⊢ (A ∈ 𝑉 → {⟨y, z⟩ ∣ [A / x]z𝐹y} = {⟨y, z⟩ ∣ z⦋A / x⦌𝐹y})
7		csbopabg 3798	. . 3 ⊢ (A ∈ 𝑉 → ⦋A / x⦌{⟨y, z⟩ ∣ z𝐹y} = {⟨y, z⟩ ∣ [A / x]z𝐹y})
8		df-cnv 4268	. . . 4 ⊢ ◡⦋A / x⦌𝐹 = {⟨y, z⟩ ∣ z⦋A / x⦌𝐹y}
9	8	a1i 9	. . 3 ⊢ (A ∈ 𝑉 → ◡⦋A / x⦌𝐹 = {⟨y, z⟩ ∣ z⦋A / x⦌𝐹y})
10	6, 7, 9	3eqtr4rd 2056	. 2 ⊢ (A ∈ 𝑉 → ◡⦋A / x⦌𝐹 = ⦋A / x⦌{⟨y, z⟩ ∣ z𝐹y})
11		df-cnv 4268	. . 3 ⊢ ◡𝐹 = {⟨y, z⟩ ∣ z𝐹y}
12	11	csbeq2i 2844	. 2 ⊢ ⦋A / x⦌◡𝐹 = ⦋A / x⦌{⟨y, z⟩ ∣ z𝐹y}
13	10, 12	syl6eqr 2063	1 ⊢ (A ∈ 𝑉 → ◡⦋A / x⦌𝐹 = ⦋A / x⦌◡𝐹)

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)