Theorem csbcnv 5872

Description: Move class substitution in and out of the converse of a relation. (Contributed by Thierry Arnoux, 8-Feb-2017.) (Revised by NM, 23-Aug-2018.) Remove dependency on ax-sep 5257 and ax-pr 5404. (Revised by Eric Schmidt, 4-Jun-2026.)

Assertion

Ref	Expression
csbcnv	⊢ ◡⦋𝐴 / 𝑥⦌𝐹 = ⦋𝐴 / 𝑥⦌◡𝐹

Proof of Theorem csbcnv

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

Step	Hyp	Ref	Expression
1		df-cnv 5669	. . . . 5 ⊢ ◡⦋𝐴 / 𝑥⦌𝐹 = {⟨𝑦, 𝑧⟩ ∣ 𝑧⦋𝐴 / 𝑥⦌𝐹𝑦}
2		sbcbr 5166	. . . . . 6 ⊢ ([𝐴 / 𝑥]𝑧𝐹𝑦 ↔ 𝑧⦋𝐴 / 𝑥⦌𝐹𝑦)
3	2	opabbii 5178	. . . . 5 ⊢ {⟨𝑦, 𝑧⟩ ∣ [𝐴 / 𝑥]𝑧𝐹𝑦} = {⟨𝑦, 𝑧⟩ ∣ 𝑧⦋𝐴 / 𝑥⦌𝐹𝑦}
4	1, 3	eqtr4i 2789	. . . 4 ⊢ ◡⦋𝐴 / 𝑥⦌𝐹 = {⟨𝑦, 𝑧⟩ ∣ [𝐴 / 𝑥]𝑧𝐹𝑦}
5		csbopabw 5541	. . . 4 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌{⟨𝑦, 𝑧⟩ ∣ 𝑧𝐹𝑦} = {⟨𝑦, 𝑧⟩ ∣ [𝐴 / 𝑥]𝑧𝐹𝑦})
6	4, 5	eqtr4id 2817	. . 3 ⊢ (𝐴 ∈ V → ◡⦋𝐴 / 𝑥⦌𝐹 = ⦋𝐴 / 𝑥⦌{⟨𝑦, 𝑧⟩ ∣ 𝑧𝐹𝑦})
7		df-cnv 5669	. . . 4 ⊢ ◡𝐹 = {⟨𝑦, 𝑧⟩ ∣ 𝑧𝐹𝑦}
8	7	csbeq2i 3861	. . 3 ⊢ ⦋𝐴 / 𝑥⦌◡𝐹 = ⦋𝐴 / 𝑥⦌{⟨𝑦, 𝑧⟩ ∣ 𝑧𝐹𝑦}
9	6, 8	eqtr4di 2816	. 2 ⊢ (𝐴 ∈ V → ◡⦋𝐴 / 𝑥⦌𝐹 = ⦋𝐴 / 𝑥⦌◡𝐹)
10		cnv0 5869	. . 3 ⊢ ◡∅ = ∅
11		csbprc 4374	. . . 4 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐹 = ∅)
12	11	cnveqd 5861	. . 3 ⊢ (¬ 𝐴 ∈ V → ◡⦋𝐴 / 𝑥⦌𝐹 = ◡∅)
13		csbprc 4374	. . 3 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌◡𝐹 = ∅)
14	10, 12, 13	3eqtr4a 2824	. 2 ⊢ (¬ 𝐴 ∈ V → ◡⦋𝐴 / 𝑥⦌𝐹 = ⦋𝐴 / 𝑥⦌◡𝐹)
15	9, 14	pm2.61i 184	1 ⊢ ◡⦋𝐴 / 𝑥⦌𝐹 = ⦋𝐴 / 𝑥⦌◡𝐹

This proof depends on syntax axioms:  ¬ wn 3   = wceq 1570   ∈ wcel 2143  Vcvv 3455  [wsbc 3744  ⦋csb 3853  ∅c0 4286   class class class wbr 5109  {copab 5173  ^◡ccnv 5660

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-rab 3417  df-v 3457  df-sbc 3745  df-csb 3854  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-br 5110  df-opab 5174  df-cnv 5669

This theorem is used by:  csbpredg  6308  esum2dlem  34491