Theorem fvmptex 7000

Description: Express a function 𝐹 whose value 𝐵 may not always be a set in terms of another function 𝐺 for which sethood is guaranteed. (Note that ( I ‘𝐵) is just shorthand for if(𝐵 ∈ V, 𝐵, ∅), and it is always a set by fvex 6889.) Note also that these functions are not the same; wherever 𝐵(𝐶) is not a set, 𝐶 is not in the domain of 𝐹 (so it evaluates to the empty set), but 𝐶 is in the domain of 𝐺, and 𝐺(𝐶) is defined to be the empty set. (Contributed by Mario Carneiro, 14-Jul-2013.) (Revised by Mario Carneiro, 23-Apr-2014.)

Hypotheses

Ref	Expression
fvmptex.1	⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
fvmptex.2	⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ ( I ‘𝐵))

Assertion

Ref	Expression
fvmptex	⊢ (𝐹‘𝐶) = (𝐺‘𝐶)

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥)   𝐹(𝑥)   𝐺(𝑥)

Proof of Theorem fvmptex

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		csbeq1 3877	. . . 4 ⊢ (𝑦 = 𝐶 → ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝐶 / 𝑥⦌𝐵)
2		fvmptex.1	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
3		nfcv 2898	. . . . . 6 ⊢ Ⅎ𝑦𝐵
4		nfcsb1v 3898	. . . . . 6 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵
5		csbeq1a 3888	. . . . . 6 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵)
6	3, 4, 5	cbvmpt 5223	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐵)
7	2, 6	eqtri 2758	. . . 4 ⊢ 𝐹 = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐵)
8	1, 7	fvmpti 6985	. . 3 ⊢ (𝐶 ∈ 𝐴 → (𝐹‘𝐶) = ( I ‘⦋𝐶 / 𝑥⦌𝐵))
9	1	fveq2d 6880	. . . 4 ⊢ (𝑦 = 𝐶 → ( I ‘⦋𝑦 / 𝑥⦌𝐵) = ( I ‘⦋𝐶 / 𝑥⦌𝐵))
10		fvmptex.2	. . . . 5 ⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ ( I ‘𝐵))
11		nfcv 2898	. . . . . 6 ⊢ Ⅎ𝑦( I ‘𝐵)
12		nfcv 2898	. . . . . . 7 ⊢ Ⅎ𝑥 I
13	12, 4	nffv 6886	. . . . . 6 ⊢ Ⅎ𝑥( I ‘⦋𝑦 / 𝑥⦌𝐵)
14	5	fveq2d 6880	. . . . . 6 ⊢ (𝑥 = 𝑦 → ( I ‘𝐵) = ( I ‘⦋𝑦 / 𝑥⦌𝐵))
15	11, 13, 14	cbvmpt 5223	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ ( I ‘𝐵)) = (𝑦 ∈ 𝐴 ↦ ( I ‘⦋𝑦 / 𝑥⦌𝐵))
16	10, 15	eqtri 2758	. . . 4 ⊢ 𝐺 = (𝑦 ∈ 𝐴 ↦ ( I ‘⦋𝑦 / 𝑥⦌𝐵))
17		fvex 6889	. . . 4 ⊢ ( I ‘⦋𝐶 / 𝑥⦌𝐵) ∈ V
18	9, 16, 17	fvmpt 6986	. . 3 ⊢ (𝐶 ∈ 𝐴 → (𝐺‘𝐶) = ( I ‘⦋𝐶 / 𝑥⦌𝐵))
19	8, 18	eqtr4d 2773	. 2 ⊢ (𝐶 ∈ 𝐴 → (𝐹‘𝐶) = (𝐺‘𝐶))
20	2	dmmptss 6230	. . . . 5 ⊢ dom 𝐹 ⊆ 𝐴
21	20	sseli 3954	. . . 4 ⊢ (𝐶 ∈ dom 𝐹 → 𝐶 ∈ 𝐴)
22		ndmfv 6911	. . . 4 ⊢ (¬ 𝐶 ∈ dom 𝐹 → (𝐹‘𝐶) = ∅)
23	21, 22	nsyl5 159	. . 3 ⊢ (¬ 𝐶 ∈ 𝐴 → (𝐹‘𝐶) = ∅)
24		fvex 6889	. . . . . 6 ⊢ ( I ‘𝐵) ∈ V
25	24, 10	dmmpti 6682	. . . . 5 ⊢ dom 𝐺 = 𝐴
26	25	eleq2i 2826	. . . 4 ⊢ (𝐶 ∈ dom 𝐺 ↔ 𝐶 ∈ 𝐴)
27		ndmfv 6911	. . . 4 ⊢ (¬ 𝐶 ∈ dom 𝐺 → (𝐺‘𝐶) = ∅)
28	26, 27	sylnbir 331	. . 3 ⊢ (¬ 𝐶 ∈ 𝐴 → (𝐺‘𝐶) = ∅)
29	23, 28	eqtr4d 2773	. 2 ⊢ (¬ 𝐶 ∈ 𝐴 → (𝐹‘𝐶) = (𝐺‘𝐶))
30	19, 29	pm2.61i 182	1 ⊢ (𝐹‘𝐶) = (𝐺‘𝐶)

Syntax hints:  ¬ wn 3   = wceq 1540   ∈ wcel 2108  ⦋csb 3874  ∅c0 4308   ↦ cmpt 5201   I cid 5547  dom cdm 5654  ‘cfv 6531

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-iota 6484  df-fun 6533  df-fn 6534  df-fv 6539

This theorem is referenced by:  fvmptnf  7008  sumeq2ii  15709  prodeq2ii  15927