Theorem csbcnvgALT 5742

Description: Move class substitution in and out of the converse of a relation. Version of csbcnv 5741 with a sethood antecedent but depending on fewer axioms. (Contributed by Thierry Arnoux, 8-Feb-2017.) (New usage is discouraged.) (Proof modification is discouraged.)

Assertion

Ref	Expression
csbcnvgALT	⊢ (𝐴 ∈ 𝑉 → ◡⦋𝐴 / 𝑥⦌𝐹 = ⦋𝐴 / 𝑥⦌◡𝐹)

Proof of Theorem csbcnvgALT

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

Step	Hyp	Ref	Expression
1		sbcbr123 5097	. . . . 5 ⊢ ([𝐴 / 𝑥]𝑧𝐹𝑦 ↔ ⦋𝐴 / 𝑥⦌𝑧⦋𝐴 / 𝑥⦌𝐹⦋𝐴 / 𝑥⦌𝑦)
2		csbconstg 3821	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝑧 = 𝑧)
3		csbconstg 3821	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝑦 = 𝑦)
4	2, 3	breq12d 5056	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (⦋𝐴 / 𝑥⦌𝑧⦋𝐴 / 𝑥⦌𝐹⦋𝐴 / 𝑥⦌𝑦 ↔ 𝑧⦋𝐴 / 𝑥⦌𝐹𝑦))
5	1, 4	syl5bb 286	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑧𝐹𝑦 ↔ 𝑧⦋𝐴 / 𝑥⦌𝐹𝑦))
6	5	opabbidv 5109	. . 3 ⊢ (𝐴 ∈ 𝑉 → {⟨𝑦, 𝑧⟩ ∣ [𝐴 / 𝑥]𝑧𝐹𝑦} = {⟨𝑦, 𝑧⟩ ∣ 𝑧⦋𝐴 / 𝑥⦌𝐹𝑦})
7		csbopabgALT 5426	. . 3 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{⟨𝑦, 𝑧⟩ ∣ 𝑧𝐹𝑦} = {⟨𝑦, 𝑧⟩ ∣ [𝐴 / 𝑥]𝑧𝐹𝑦})
8		df-cnv 5548	. . . 4 ⊢ ◡⦋𝐴 / 𝑥⦌𝐹 = {⟨𝑦, 𝑧⟩ ∣ 𝑧⦋𝐴 / 𝑥⦌𝐹𝑦}
9	8	a1i 11	. . 3 ⊢ (𝐴 ∈ 𝑉 → ◡⦋𝐴 / 𝑥⦌𝐹 = {⟨𝑦, 𝑧⟩ ∣ 𝑧⦋𝐴 / 𝑥⦌𝐹𝑦})
10	6, 7, 9	3eqtr4rd 2785	. 2 ⊢ (𝐴 ∈ 𝑉 → ◡⦋𝐴 / 𝑥⦌𝐹 = ⦋𝐴 / 𝑥⦌{⟨𝑦, 𝑧⟩ ∣ 𝑧𝐹𝑦})
11		df-cnv 5548	. . 3 ⊢ ◡𝐹 = {⟨𝑦, 𝑧⟩ ∣ 𝑧𝐹𝑦}
12	11	csbeq2i 3810	. 2 ⊢ ⦋𝐴 / 𝑥⦌◡𝐹 = ⦋𝐴 / 𝑥⦌{⟨𝑦, 𝑧⟩ ∣ 𝑧𝐹𝑦}
13	10, 12	eqtr4di 2792	1 ⊢ (𝐴 ∈ 𝑉 → ◡⦋𝐴 / 𝑥⦌𝐹 = ⦋𝐴 / 𝑥⦌◡𝐹)

Syntax hints:   → wi 4   = wceq 1543   ∈ wcel 2110  [wsbc 3687  ⦋csb 3802   class class class wbr 5043  {copab 5105  ^◡ccnv 5539

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2706

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-clab 2713  df-cleq 2726  df-clel 2812  df-nfc 2882  df-rab 3063  df-v 3403  df-sbc 3688  df-csb 3803  df-dif 3860  df-un 3862  df-nul 4228  df-if 4430  df-sn 4532  df-pr 4534  df-op 4538  df-br 5044  df-opab 5106  df-cnv 5548

This theorem is referenced by: (None)