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Theorem csbrng 4855
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 2978 . . 3 (𝐴𝑉𝐴 / 𝑥{𝑦 ∣ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐵} = {𝑦[𝐴 / 𝑥]𝑤𝑤, 𝑦⟩ ∈ 𝐵})
2 sbcexg 2882 . . . . 5 (𝐴𝑉 → ([𝐴 / 𝑥]𝑤𝑤, 𝑦⟩ ∈ 𝐵 ↔ ∃𝑤[𝐴 / 𝑥]𝑤, 𝑦⟩ ∈ 𝐵))
3 sbcel2g 2941 . . . . . 6 (𝐴𝑉 → ([𝐴 / 𝑥]𝑤, 𝑦⟩ ∈ 𝐵 ↔ ⟨𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵))
43exbidv 1750 . . . . 5 (𝐴𝑉 → (∃𝑤[𝐴 / 𝑥]𝑤, 𝑦⟩ ∈ 𝐵 ↔ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵))
52, 4bitrd 186 . . . 4 (𝐴𝑉 → ([𝐴 / 𝑥]𝑤𝑤, 𝑦⟩ ∈ 𝐵 ↔ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵))
65abbidv 2202 . . 3 (𝐴𝑉 → {𝑦[𝐴 / 𝑥]𝑤𝑤, 𝑦⟩ ∈ 𝐵} = {𝑦 ∣ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵})
71, 6eqtrd 2117 . 2 (𝐴𝑉𝐴 / 𝑥{𝑦 ∣ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐵} = {𝑦 ∣ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵})
8 dfrn3 4592 . . 3 ran 𝐵 = {𝑦 ∣ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐵}
98csbeq2i 2946 . 2 𝐴 / 𝑥ran 𝐵 = 𝐴 / 𝑥{𝑦 ∣ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐵}
10 dfrn3 4592 . 2 ran 𝐴 / 𝑥𝐵 = {𝑦 ∣ ∃𝑤𝑤, 𝑦⟩ ∈ 𝐴 / 𝑥𝐵}
117, 9, 103eqtr4g 2142 1 (𝐴𝑉𝐴 / 𝑥ran 𝐵 = ran 𝐴 / 𝑥𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1287  wex 1424  wcel 1436  {cab 2071  [wsbc 2829  csb 2922  cop 3434  ran crn 4411
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1379  ax-7 1380  ax-gen 1381  ax-ie1 1425  ax-ie2 1426  ax-8 1438  ax-10 1439  ax-11 1440  ax-i12 1441  ax-bndl 1442  ax-4 1443  ax-14 1448  ax-17 1462  ax-i9 1466  ax-ial 1470  ax-i5r 1471  ax-ext 2067  ax-sep 3931  ax-pow 3983  ax-pr 4009
This theorem depends on definitions:  df-bi 115  df-3an 924  df-tru 1290  df-nf 1393  df-sb 1690  df-eu 1948  df-mo 1949  df-clab 2072  df-cleq 2078  df-clel 2081  df-nfc 2214  df-v 2617  df-sbc 2830  df-csb 2923  df-un 2992  df-in 2994  df-ss 3001  df-pw 3417  df-sn 3437  df-pr 3438  df-op 3440  df-br 3821  df-opab 3875  df-cnv 4418  df-dm 4420  df-rn 4421
This theorem is referenced by:  sbcfg  5122
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