Theorem csbrng 5198

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

Assertion

Ref	Expression
csbrng	⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌ran 𝐵 = ran ⦋𝐴 / 𝑥⦌𝐵)

Proof of Theorem csbrng

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

Step	Hyp	Ref	Expression
1		csbabg 3189	. . 3 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{𝑦 ∣ ∃𝑤⟨𝑤, 𝑦⟩ ∈ 𝐵} = {𝑦 ∣ [𝐴 / 𝑥]∃𝑤⟨𝑤, 𝑦⟩ ∈ 𝐵})
2		sbcexg 3086	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]∃𝑤⟨𝑤, 𝑦⟩ ∈ 𝐵 ↔ ∃𝑤[𝐴 / 𝑥]⟨𝑤, 𝑦⟩ ∈ 𝐵))
3		sbcel2g 3148	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]⟨𝑤, 𝑦⟩ ∈ 𝐵 ↔ ⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵))
4	3	exbidv 1873	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (∃𝑤[𝐴 / 𝑥]⟨𝑤, 𝑦⟩ ∈ 𝐵 ↔ ∃𝑤⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵))
5	2, 4	bitrd 188	. . . 4 ⊢ (𝐴 ∈ 𝑉 → ([𝐴 / 𝑥]∃𝑤⟨𝑤, 𝑦⟩ ∈ 𝐵 ↔ ∃𝑤⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵))
6	5	abbidv 2349	. . 3 ⊢ (𝐴 ∈ 𝑉 → {𝑦 ∣ [𝐴 / 𝑥]∃𝑤⟨𝑤, 𝑦⟩ ∈ 𝐵} = {𝑦 ∣ ∃𝑤⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵})
7	1, 6	eqtrd 2264	. 2 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌{𝑦 ∣ ∃𝑤⟨𝑤, 𝑦⟩ ∈ 𝐵} = {𝑦 ∣ ∃𝑤⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵})
8		dfrn3 4919	. . 3 ⊢ ran 𝐵 = {𝑦 ∣ ∃𝑤⟨𝑤, 𝑦⟩ ∈ 𝐵}
9	8	csbeq2i 3154	. 2 ⊢ ⦋𝐴 / 𝑥⦌ran 𝐵 = ⦋𝐴 / 𝑥⦌{𝑦 ∣ ∃𝑤⟨𝑤, 𝑦⟩ ∈ 𝐵}
10		dfrn3 4919	. 2 ⊢ ran ⦋𝐴 / 𝑥⦌𝐵 = {𝑦 ∣ ∃𝑤⟨𝑤, 𝑦⟩ ∈ ⦋𝐴 / 𝑥⦌𝐵}
11	7, 9, 10	3eqtr4g 2289	1 ⊢ (𝐴 ∈ 𝑉 → ⦋𝐴 / 𝑥⦌ran 𝐵 = ran ⦋𝐴 / 𝑥⦌𝐵)

Syntax hints:   → wi 4   = wceq 1397  ∃wex 1540   ∈ wcel 2202  {cab 2217  [wsbc 3031  ⦋csb 3127  ⟨cop 3672  ran crn 4726

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-sbc 3032  df-csb 3128  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-opab 4151  df-cnv 4733  df-dm 4735  df-rn 4736

This theorem is referenced by:  sbcfg  5481