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 ∈ 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 3804	. . . . 5 ⊢ (A ∈ 𝑉 → ([A / x]z𝐹y ↔ ⦋A / x⦌z⦋A / x⦌𝐹⦋A / x⦌y))
2		csbconstg 2858	. . . . . 6 ⊢ (A ∈ 𝑉 → ⦋A / x⦌z = z)
3		csbconstg 2858	. . . . . 6 ⊢ (A ∈ 𝑉 → ⦋A / x⦌y = y)
4	2, 3	breq12d 3768	. . . . 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 3814	. . 3 ⊢ (A ∈ 𝑉 → {⟨y, z⟩ ∣ [A / x]z𝐹y} = {⟨y, z⟩ ∣ z⦋A / x⦌𝐹y})
7		csbopabg 3826	. . 3 ⊢ (A ∈ 𝑉 → ⦋A / x⦌{⟨y, z⟩ ∣ z𝐹y} = {⟨y, z⟩ ∣ [A / x]z𝐹y})
8		df-cnv 4296	. . . 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 2080	. 2 ⊢ (A ∈ 𝑉 → ◡⦋A / x⦌𝐹 = ⦋A / x⦌{⟨y, z⟩ ∣ z𝐹y})
11		df-cnv 4296	. . 3 ⊢ ◡𝐹 = {⟨y, z⟩ ∣ z𝐹y}
12	11	csbeq2i 2870	. 2 ⊢ ⦋A / x⦌◡𝐹 = ⦋A / x⦌{⟨y, z⟩ ∣ z𝐹y}
13	10, 12	syl6eqr 2087	1 ⊢ (A ∈ 𝑉 → ◡⦋A / x⦌𝐹 = ⦋A / x⦌◡𝐹)

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)