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Theorem csbima12g 4629
Description: Move class substitution in and out of the image of a function. (Contributed by FL, 15-Dec-2006.) (Proof shortened by Mario Carneiro, 4-Dec-2016.)
Assertion
Ref Expression
csbima12g (A 𝐶A / x(𝐹B) = (A / x𝐹A / xB))

Proof of Theorem csbima12g
Dummy variable y is distinct from all other variables.
StepHypRef Expression
1 csbeq1 2849 . . 3 (y = Ay / x(𝐹B) = A / x(𝐹B))
2 csbeq1 2849 . . . 4 (y = Ay / x𝐹 = A / x𝐹)
3 csbeq1 2849 . . . 4 (y = Ay / xB = A / xB)
42, 3imaeq12d 4612 . . 3 (y = A → (y / x𝐹y / xB) = (A / x𝐹A / xB))
51, 4eqeq12d 2051 . 2 (y = A → (y / x(𝐹B) = (y / x𝐹y / xB) ↔ A / x(𝐹B) = (A / x𝐹A / xB)))
6 vex 2554 . . 3 y V
7 nfcsb1v 2876 . . . 4 xy / x𝐹
8 nfcsb1v 2876 . . . 4 xy / xB
97, 8nfima 4619 . . 3 x(y / x𝐹y / xB)
10 csbeq1a 2854 . . . 4 (x = y𝐹 = y / x𝐹)
11 csbeq1a 2854 . . . 4 (x = yB = y / xB)
1210, 11imaeq12d 4612 . . 3 (x = y → (𝐹B) = (y / x𝐹y / xB))
136, 9, 12csbief 2885 . 2 y / x(𝐹B) = (y / x𝐹y / xB)
145, 13vtoclg 2607 1 (A 𝐶A / x(𝐹B) = (A / x𝐹A / xB))
Colors of variables: wff set class
Syntax hints:  wi 4   = wceq 1242   wcel 1390  csb 2846  cima 4291
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-cnv 4296  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301
This theorem is referenced by: (None)
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