Theorem csbima12g 3411

Description: Move class substitution in and out of the image of a function. (Contributed by FL, 15-Dec-2006.)

Assertion

Ref	Expression
csbima12g	⊢ (A ∈ C → [A / x](F “ B) = ([A / x]F “ [A / x]B))

Proof of Theorem csbima12g

Step	Hyp	Ref	Expression
1		ax-17 969	. . . 4 ⊢ (A ∈ C → ∀y A ∈ C)
2		ax-17 969	. . . . 5 ⊢ (z ∈ A → ∀y z ∈ A)
3	2	hbcsb1g 2020	. . . 4 ⊢ (A ∈ C → (z ∈ [A / y][y / x]F → ∀y z ∈ [A / y][y / x]F))
4	2	hbcsb1g 2020	. . . 4 ⊢ (A ∈ C → (z ∈ [A / y][y / x]B → ∀y z ∈ [A / y][y / x]B))
5	1, 3, 4	hbimad 3410	. . 3 ⊢ (A ∈ C → (z ∈ ([A / y][y / x]F “ [A / y][y / x]B) → ∀y z ∈ ([A / y][y / x]F “ [A / y][y / x]B)))
6		a9e 1123	. . . . . 6 ⊢ ∃x x = y
7		visset 1809	. . . . . . . . 9 ⊢ y ∈ V
8		ax-17 969	. . . . . . . . 9 ⊢ (z ∈ y → ∀x z ∈ y)
9	7, 8	hbcsb1 2021	. . . . . . . 8 ⊢ (z ∈ [y / x](F “ B) → ∀x z ∈ [y / x](F “ B))
10	7, 8	hbcsb1 2021	. . . . . . . . 9 ⊢ (z ∈ [y / x]F → ∀x z ∈ [y / x]F)
11	7, 8	hbcsb1 2021	. . . . . . . . 9 ⊢ (z ∈ [y / x]B → ∀x z ∈ [y / x]B)
12	10, 11	hbima 3409	. . . . . . . 8 ⊢ (z ∈ ([y / x]F “ [y / x]B) → ∀x z ∈ ([y / x]F “ [y / x]B))
13	9, 12	hbeq 1562	. . . . . . 7 ⊢ ([y / x](F “ B) = ([y / x]F “ [y / x]B) → ∀x[y / x](F “ B) = ([y / x]F “ [y / x]B))
14		csbeq1a 2002	. . . . . . . . 9 ⊢ (x = y → F = [y / x]F)
15	14	imaeq1d 3401	. . . . . . . 8 ⊢ (x = y → (F “ B) = ([y / x]F “ B))
16		csbeq1a 2002	. . . . . . . 8 ⊢ (x = y → (F “ B) = [y / x](F “ B))
17		csbeq1a 2002	. . . . . . . . 9 ⊢ (x = y → B = [y / x]B)
18	17	imaeq2d 3402	. . . . . . . 8 ⊢ (x = y → ([y / x]F “ B) = ([y / x]F “ [y / x]B))
19	15, 16, 18	3eqtr3d 1512	. . . . . . 7 ⊢ (x = y → [y / x](F “ B) = ([y / x]F “ [y / x]B))
20	13, 19	19.23ai 1062	. . . . . 6 ⊢ (∃x x = y → [y / x](F “ B) = ([y / x]F “ [y / x]B))
21	6, 20	ax-mp 7	. . . . 5 ⊢ [y / x](F “ B) = ([y / x]F “ [y / x]B)
22	21	a1i 8	. . . 4 ⊢ (y = A → [y / x](F “ B) = ([y / x]F “ [y / x]B))
23		csbeq1a 2002	. . . . 5 ⊢ (y = A → [y / x]F = [A / y][y / x]F)
24	23	imaeq1d 3401	. . . 4 ⊢ (y = A → ([y / x]F “ [y / x]B) = ([A / y][y / x]F “ [y / x]B))
25		csbeq1a 2002	. . . . 5 ⊢ (y = A → [y / x]B = [A / y][y / x]B)
26	25	imaeq2d 3402	. . . 4 ⊢ (y = A → ([A / y][y / x]F “ [y / x]B) = ([A / y][y / x]F “ [A / y][y / x]B))
27	22, 24, 26	3eqtrd 1508	. . 3 ⊢ (y = A → [y / x](F “ B) = ([A / y][y / x]F “ [A / y][y / x]B))
28	5, 27	csbiegf 2027	. 2 ⊢ (A ∈ C → [A / y][y / x](F “ B) = ([A / y][y / x]F “ [A / y][y / x]B))
29		csbcog 2003	. 2 ⊢ (A ∈ C → [A / y][y / x](F “ B) = [A / x](F “ B))
30		csbcog 2003	. . . 4 ⊢ (A ∈ C → [A / y][y / x]F = [A / x]F)
31	30	imaeq1d 3401	. . 3 ⊢ (A ∈ C → ([A / y][y / x]F “ [A / y][y / x]B) = ([A / x]F “ [A / y][y / x]B))
32		csbcog 2003	. . . 4 ⊢ (A ∈ C → [A / y][y / x]B = [A / x]B)
33	32	imaeq2d 3402	. . 3 ⊢ (A ∈ C → ([A / x]F “ [A / y][y / x]B) = ([A / x]F “ [A / x]B))
34	31, 33	eqtrd 1504	. 2 ⊢ (A ∈ C → ([A / y][y / x]F “ [A / y][y / x]B) = ([A / x]F “ [A / x]B))
35	28, 29, 34	3eqtr3d 1512	1 ⊢ (A ∈ C → [A / x](F “ B) = ([A / x]F “ [A / x]B))

Syntax hints:   → wi 3   = wceq 954   ∈ wcel 956  ∃wex 978  [csb 1997   “ cima 3173

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-9 963  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2699  ax-pow 2738  ax-pr 2775

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 776  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-rex 1647  df-v 1808  df-sbc 1938  df-csb 1998  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-br 2616  df-opab 2663  df-xp 3184  df-cnv 3186  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191

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