Theorem csbrn 6150

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 6028	. . 3 ⊢ ⦋𝐴 / 𝑥⦌(𝐵 “ V) = (⦋𝐴 / 𝑥⦌𝐵 “ ⦋𝐴 / 𝑥⦌V)
2		csbconstg 3869	. . . . 5 ⊢ (𝐴 ∈ V → ⦋𝐴 / 𝑥⦌V = V)
3	2	imaeq2d 6009	. . . 4 ⊢ (𝐴 ∈ V → (⦋𝐴 / 𝑥⦌𝐵 “ ⦋𝐴 / 𝑥⦌V) = (⦋𝐴 / 𝑥⦌𝐵 “ V))
4		0ima 6027	. . . . . 6 ⊢ (∅ “ V) = ∅
5	4	eqcomi 2740	. . . . 5 ⊢ ∅ = (∅ “ V)
6		csbprc 4359	. . . . . . 7 ⊢ (¬ 𝐴 ∈ V → ⦋𝐴 / 𝑥⦌𝐵 = ∅)
7	6	imaeq1d 6008	. . . . . 6 ⊢ (¬ 𝐴 ∈ V → (⦋𝐴 / 𝑥⦌𝐵 “ ⦋𝐴 / 𝑥⦌V) = (∅ “ ⦋𝐴 / 𝑥⦌V))
8		0ima 6027	. . . . . 6 ⊢ (∅ “ ⦋𝐴 / 𝑥⦌V) = ∅
9	7, 8	eqtrdi 2782	. . . . 5 ⊢ (¬ 𝐴 ∈ V → (⦋𝐴 / 𝑥⦌𝐵 “ ⦋𝐴 / 𝑥⦌V) = ∅)
10	6	imaeq1d 6008	. . . . 5 ⊢ (¬ 𝐴 ∈ V → (⦋𝐴 / 𝑥⦌𝐵 “ V) = (∅ “ V))
11	5, 9, 10	3eqtr4a 2792	. . . 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 2754	. 2 ⊢ ⦋𝐴 / 𝑥⦌(𝐵 “ V) = (⦋𝐴 / 𝑥⦌𝐵 “ V)
14		dfrn4 6149	. . 3 ⊢ ran 𝐵 = (𝐵 “ V)
15	14	csbeq2i 3858	. 2 ⊢ ⦋𝐴 / 𝑥⦌ran 𝐵 = ⦋𝐴 / 𝑥⦌(𝐵 “ V)
16		dfrn4 6149	. 2 ⊢ ran ⦋𝐴 / 𝑥⦌𝐵 = (⦋𝐴 / 𝑥⦌𝐵 “ V)
17	13, 15, 16	3eqtr4i 2764	1 ⊢ ⦋𝐴 / 𝑥⦌ran 𝐵 = ran ⦋𝐴 / 𝑥⦌𝐵

Syntax hints:  ¬ wn 3   = wceq 1541   ∈ wcel 2111  Vcvv 3436  ⦋csb 3850  ∅c0 4283  ran crn 5617   “ cima 5619

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 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5234  ax-nul 5244  ax-pr 5370

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 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4476  df-sn 4577  df-pr 4579  df-op 4583  df-br 5092  df-opab 5154  df-xp 5622  df-rel 5623  df-cnv 5624  df-dm 5626  df-rn 5627  df-res 5628  df-ima 5629

This theorem is referenced by:  sbcfg  6649  csbima12gALTVD  44928