Theorem csbcnvgALT 5830

Description: Move class substitution in and out of the converse of a relation. Version of csbcnv 5829 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 5149	. . . . 5 ⊢ ([𝐴 / 𝑥]𝑧𝐹𝑦 ↔ ⦋𝐴 / 𝑥⦌𝑧⦋𝐴 / 𝑥⦌𝐹⦋𝐴 / 𝑥⦌𝑦)
2		csbconstg 3865	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝑧 = 𝑧)
3		csbconstg 3865	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌𝑦 = 𝑦)
4	2, 3	breq12d 5108	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (⦋𝐴 / 𝑥⦌𝑧⦋𝐴 / 𝑥⦌𝐹⦋𝐴 / 𝑥⦌𝑦 ↔ 𝑧⦋𝐴 / 𝑥⦌𝐹𝑦))
5	1, 4	bitrid 283	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]𝑧𝐹𝑦 ↔ 𝑧⦋𝐴 / 𝑥⦌𝐹𝑦))
6	5	opabbidv 5161	. . 3 ⊢ (𝐴 ∈ 𝑉 → {⟨𝑦, 𝑧⟩ ∣ [𝐴 / 𝑥]𝑧𝐹𝑦} = {⟨𝑦, 𝑧⟩ ∣ 𝑧⦋𝐴 / 𝑥⦌𝐹𝑦})
7		csbopabgALT 5501	. . 3 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{⟨𝑦, 𝑧⟩ ∣ 𝑧𝐹𝑦} = {⟨𝑦, 𝑧⟩ ∣ [𝐴 / 𝑥]𝑧𝐹𝑦})
8		df-cnv 5629	. . . 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 2779	. 2 ⊢ (𝐴 ∈ 𝑉 → ◡⦋𝐴 / 𝑥⦌𝐹 = ⦋𝐴 / 𝑥⦌{⟨𝑦, 𝑧⟩ ∣ 𝑧𝐹𝑦})
11		df-cnv 5629	. . 3 ⊢ ◡𝐹 = {⟨𝑦, 𝑧⟩ ∣ 𝑧𝐹𝑦}
12	11	csbeq2i 3854	. 2 ⊢ ⦋𝐴 / 𝑥⦌◡𝐹 = ⦋𝐴 / 𝑥⦌{⟨𝑦, 𝑧⟩ ∣ 𝑧𝐹𝑦}
13	10, 12	eqtr4di 2786	1 ⊢ (𝐴 ∈ 𝑉 → ◡⦋𝐴 / 𝑥⦌𝐹 = ⦋𝐴 / 𝑥⦌◡𝐹)

Syntax hints:   → wi 4   = wceq 1541   ∈ wcel 2113  [wsbc 3737  ⦋csb 3846   class class class wbr 5095  {copab 5157  ^◡ccnv 5620

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-ss 3915  df-nul 4283  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-br 5096  df-opab 5158  df-cnv 5629

This theorem is referenced by: (None)