Theorem csbcnvg 4905

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 4137	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑧𝐹𝑦 ↔ ⦋𝐴 / 𝑥⦌𝑧⦋𝐴 / 𝑥⦌𝐹⦋𝐴 / 𝑥⦌𝑦))
2		csbconstg 3138	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝑧 = 𝑧)
3		csbconstg 3138	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝑦 = 𝑦)
4	2, 3	breq12d 4095	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (⦋𝐴 / 𝑥⦌𝑧⦋𝐴 / 𝑥⦌𝐹⦋𝐴 / 𝑥⦌𝑦 ↔ 𝑧⦋𝐴 / 𝑥⦌𝐹𝑦))
5	1, 4	bitrd 188	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑧𝐹𝑦 ↔ 𝑧⦋𝐴 / 𝑥⦌𝐹𝑦))
6	5	opabbidv 4149	. . 3 ⊢ (𝐴 ∈ 𝑉 → {⟨𝑦, 𝑧⟩ ∣ [𝐴 / 𝑥]𝑧𝐹𝑦} = {⟨𝑦, 𝑧⟩ ∣ 𝑧⦋𝐴 / 𝑥⦌𝐹𝑦})
7		csbopabg 4161	. . 3 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{⟨𝑦, 𝑧⟩ ∣ 𝑧𝐹𝑦} = {⟨𝑦, 𝑧⟩ ∣ [𝐴 / 𝑥]𝑧𝐹𝑦})
8		df-cnv 4726	. . . 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 2273	. 2 ⊢ (𝐴 ∈ 𝑉 → ◡⦋𝐴 / 𝑥⦌𝐹 = ⦋𝐴 / 𝑥⦌{⟨𝑦, 𝑧⟩ ∣ 𝑧𝐹𝑦})
11		df-cnv 4726	. . 3 ⊢ ◡𝐹 = {⟨𝑦, 𝑧⟩ ∣ 𝑧𝐹𝑦}
12	11	csbeq2i 3151	. 2 ⊢ ⦋𝐴 / 𝑥⦌◡𝐹 = ⦋𝐴 / 𝑥⦌{⟨𝑦, 𝑧⟩ ∣ 𝑧𝐹𝑦}
13	10, 12	eqtr4di 2280	1 ⊢ (𝐴 ∈ 𝑉 → ◡⦋𝐴 / 𝑥⦌𝐹 = ⦋𝐴 / 𝑥⦌◡𝐹)

Syntax hints:   → wi 4   = wceq 1395   ∈ wcel 2200  [wsbc 3028  ⦋csb 3124   class class class wbr 4082  {copab 4143  ^◡ccnv 4717

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-sn 3672  df-pr 3673  df-op 3675  df-br 4083  df-opab 4145  df-cnv 4726

This theorem is referenced by: (None)