Theorem csbfv12g 3733

Description: Move class substitution in and out of a function value.

Assertion

Ref	Expression
csbfv12g	⊢ (A ∈ C → [A / x](F ‘B) = ([A / x]F ‘[A / x]B))

Proof of Theorem csbfv12g

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	hbfvd 3721	. . 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	hbfv 3720	. . . . . . . 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	fveq1d 3717	. . . . . . . 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	fveq2d 3719	. . . . . . . 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	fveq1d 3717	. . . 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	fveq2d 3719	. . . 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	fveq1d 3717	. . 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	fveq2d 3719	. . 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   ‘cfv 3177

This theorem is referenced by:  csbfv2g 3734  fsumcnlem 7939

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 2698  ax-pow 2737  ax-pr 2774

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-uni 2499  df-br 2615  df-opab 2662  df-xp 3179  df-cnv 3181  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fv 3193

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