Theorem csbrn 6199

Description: Distribute proper substitution through the range of a class. (Contributed by Alan Sare, 10-Nov-2012.)

Assertion

Ref	Expression
csbrn	⊢ ⦋𝐴 / 𝑥⦌ran 𝐵 = ran ⦋𝐴 / 𝑥⦌𝐵

Proof of Theorem csbrn

Step	Hyp	Ref	Expression
1		csbima12 6075	. . 3 ⊢ ⦋𝐴 / 𝑥⦌(𝐵 “ V) = (⦋𝐴 / 𝑥⦌𝐵 “ ⦋𝐴 / 𝑥⦌V)
2		csbconstg 3911	. . . . 5 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌V = V)
3	2	imaeq2d 6057	. . . 4 ⊢ (𝐴 ∈ V → (⦋𝐴 / 𝑥⦌𝐵 “ ⦋𝐴 / 𝑥⦌V) = (⦋𝐴 / 𝑥⦌𝐵 “ V))
4		0ima 6074	. . . . . 6 ⊢ (∅ “ V) = ∅
5	4	eqcomi 2742	. . . . 5 ⊢ ∅ = (∅ “ V)
6		csbprc 4405	. . . . . . 7 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 = ∅)
7	6	imaeq1d 6056	. . . . . 6 ⊢ (¬ 𝐴 ∈ V → (⦋𝐴 / 𝑥⦌𝐵 “ ⦋𝐴 / 𝑥⦌V) = (∅ “ ⦋𝐴 / 𝑥⦌V))
8		0ima 6074	. . . . . 6 ⊢ (∅ “ ⦋𝐴 / 𝑥⦌V) = ∅
9	7, 8	eqtrdi 2789	. . . . 5 ⊢ (¬ 𝐴 ∈ V → (⦋𝐴 / 𝑥⦌𝐵 “ ⦋𝐴 / 𝑥⦌V) = ∅)
10	6	imaeq1d 6056	. . . . 5 ⊢ (¬ 𝐴 ∈ V → (⦋𝐴 / 𝑥⦌𝐵 “ V) = (∅ “ V))
11	5, 9, 10	3eqtr4a 2799	. . . 4 ⊢ (¬ 𝐴 ∈ V → (⦋𝐴 / 𝑥⦌𝐵 “ ⦋𝐴 / 𝑥⦌V) = (⦋𝐴 / 𝑥⦌𝐵 “ V))
12	3, 11	pm2.61i 182	. . 3 ⊢ (⦋𝐴 / 𝑥⦌𝐵 “ ⦋𝐴 / 𝑥⦌V) = (⦋𝐴 / 𝑥⦌𝐵 “ V)
13	1, 12	eqtri 2761	. 2 ⊢ ⦋𝐴 / 𝑥⦌(𝐵 “ V) = (⦋𝐴 / 𝑥⦌𝐵 “ V)
14		dfrn4 6198	. . 3 ⊢ ran 𝐵 = (𝐵 “ V)
15	14	csbeq2i 3900	. 2 ⊢ ⦋𝐴 / 𝑥⦌ran 𝐵 = ⦋𝐴 / 𝑥⦌(𝐵 “ V)
16		dfrn4 6198	. 2 ⊢ ran ⦋𝐴 / 𝑥⦌𝐵 = (⦋𝐴 / 𝑥⦌𝐵 “ V)
17	13, 15, 16	3eqtr4i 2771	1 ⊢ ⦋𝐴 / 𝑥⦌ran 𝐵 = ran ⦋𝐴 / 𝑥⦌𝐵

Syntax hints:  ¬ wn 3   = wceq 1542   ∈ wcel 2107  Vcvv 3475  ⦋csb 3892  ∅c0 4321  ran crn 5676   “ cima 5678

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-xp 5681  df-rel 5682  df-cnv 5683  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688

This theorem is referenced by:  sbcfg  6712  csbima12gALTVD  43591