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Theorem csbrng 5072
Description: Distribute proper substitution through the range of a class. (Contributed by Alan Sare, 10-Nov-2012.)
Assertion
Ref Expression
csbrng (𝐴𝑉𝐴 / 𝑥ran 𝐵 = ran 𝐴 / 𝑥𝐵)

Proof of Theorem csbrng
Dummy variables 𝑤 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 csbabg 3110 . . 3 (𝐴𝑉𝐴 / 𝑥{𝑦 ∣ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐵} = {𝑦[𝐴 / 𝑥]𝑤𝑤, 𝑦⟩ ∈ 𝐵})
2 sbcexg 3009 . . . . 5 (𝐴𝑉 → ([𝐴 / 𝑥]𝑤𝑤, 𝑦⟩ ∈ 𝐵 ↔ ∃𝑤[𝐴 / 𝑥]𝑤, 𝑦⟩ ∈ 𝐵))
3 sbcel2g 3070 . . . . . 6 (𝐴𝑉 → ([𝐴 / 𝑥]𝑤, 𝑦⟩ ∈ 𝐵 ↔ ⟨𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵))
43exbidv 1818 . . . . 5 (𝐴𝑉 → (∃𝑤[𝐴 / 𝑥]𝑤, 𝑦⟩ ∈ 𝐵 ↔ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵))
52, 4bitrd 187 . . . 4 (𝐴𝑉 → ([𝐴 / 𝑥]𝑤𝑤, 𝑦⟩ ∈ 𝐵 ↔ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵))
65abbidv 2288 . . 3 (𝐴𝑉 → {𝑦[𝐴 / 𝑥]𝑤𝑤, 𝑦⟩ ∈ 𝐵} = {𝑦 ∣ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵})
71, 6eqtrd 2203 . 2 (𝐴𝑉𝐴 / 𝑥{𝑦 ∣ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐵} = {𝑦 ∣ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵})
8 dfrn3 4800 . . 3 ran 𝐵 = {𝑦 ∣ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐵}
98csbeq2i 3076 . 2 𝐴 / 𝑥ran 𝐵 = 𝐴 / 𝑥{𝑦 ∣ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐵}
10 dfrn3 4800 . 2 ran 𝐴 / 𝑥𝐵 = {𝑦 ∣ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵}
117, 9, 103eqtr4g 2228 1 (𝐴𝑉𝐴 / 𝑥ran 𝐵 = ran 𝐴 / 𝑥𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1348  wex 1485  wcel 2141  {cab 2156  [wsbc 2955  csb 3049  cop 3586  ran crn 4612
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-br 3990  df-opab 4051  df-cnv 4619  df-dm 4621  df-rn 4622
This theorem is referenced by:  sbcfg  5346
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