Theorem fvmptex 6984

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 6874.) 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 3853	. . . 4 ⊢ (𝑦 = 𝐶 → ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝐶 / 𝑥⦌𝐵)
2		fvmptex.1	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
3		nfcv 2923	. . . . . 6 ⊢ Ⅎ𝑦𝐵
4		nfcsb1v 3874	. . . . . 6 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵
5		csbeq1a 3864	. . . . . 6 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵)
6	3, 4, 5	cbvmpt 5199	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐵)
7	2, 6	eqtri 2784	. . . 4 ⊢ 𝐹 = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐵)
8	1, 7	fvmpti 6968	. . 3 ⊢ (𝐶 ∈ 𝐴 → (𝐹‘𝐶) = ( I ‘⦋𝐶 / 𝑥⦌𝐵))
9	1	fveq2d 6865	. . . 4 ⊢ (𝑦 = 𝐶 → ( I ‘⦋𝑦 / 𝑥⦌𝐵) = ( I ‘⦋𝐶 / 𝑥⦌𝐵))
10		fvmptex.2	. . . . 5 ⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ ( I ‘𝐵))
11		nfcv 2923	. . . . . 6 ⊢ Ⅎ𝑦( I ‘𝐵)
12		nfcv 2923	. . . . . . 7 ⊢ Ⅎ𝑥 I
13	12, 4	nffv 6871	. . . . . 6 ⊢ Ⅎ𝑥( I ‘⦋𝑦 / 𝑥⦌𝐵)
14	5	fveq2d 6865	. . . . . 6 ⊢ (𝑥 = 𝑦 → ( I ‘𝐵) = ( I ‘⦋𝑦 / 𝑥⦌𝐵))
15	11, 13, 14	cbvmpt 5199	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ ( I ‘𝐵)) = (𝑦 ∈ 𝐴 ↦ ( I ‘⦋𝑦 / 𝑥⦌𝐵))
16	10, 15	eqtri 2784	. . . 4 ⊢ 𝐺 = (𝑦 ∈ 𝐴 ↦ ( I ‘⦋𝑦 / 𝑥⦌𝐵))
17		fvex 6874	. . . 4 ⊢ ( I ‘⦋𝐶 / 𝑥⦌𝐵) ∈ V
18	9, 16, 17	fvmpt 6969	. . 3 ⊢ (𝐶 ∈ 𝐴 → (𝐺‘𝐶) = ( I ‘⦋𝐶 / 𝑥⦌𝐵))
19	8, 18	eqtr4d 2799	. 2 ⊢ (𝐶 ∈ 𝐴 → (𝐹‘𝐶) = (𝐺‘𝐶))
20	2	dmmptss 6222	. . . . 5 ⊢ dom 𝐹 ⊆ 𝐴
21	20	sseli 3930	. . . 4 ⊢ (𝐶 ∈ dom 𝐹 → 𝐶 ∈ 𝐴)
22		ndmfv 6893	. . . 4 ⊢ (¬ 𝐶 ∈ dom 𝐹 → (𝐹‘𝐶) = ∅)
23	21, 22	nsyl5 159	. . 3 ⊢ (¬ 𝐶 ∈ 𝐴 → (𝐹‘𝐶) = ∅)
24		fvex 6874	. . . . . 6 ⊢ ( I ‘𝐵) ∈ V
25	24, 10	dmmpti 6659	. . . . 5 ⊢ dom 𝐺 = 𝐴
26	25	eleq2i 2853	. . . 4 ⊢ (𝐶 ∈ dom 𝐺 ↔ 𝐶 ∈ 𝐴)
27		ndmfv 6893	. . . 4 ⊢ (¬ 𝐶 ∈ dom 𝐺 → (𝐺‘𝐶) = ∅)
28	26, 27	sylnbir 333	. . 3 ⊢ (¬ 𝐶 ∈ 𝐴 → (𝐺‘𝐶) = ∅)
29	23, 28	eqtr4d 2799	. 2 ⊢ (¬ 𝐶 ∈ 𝐴 → (𝐹‘𝐶) = (𝐺‘𝐶))
30	19, 29	pm2.61i 183	1 ⊢ (𝐹‘𝐶) = (𝐺‘𝐶)

Syntax hints:  ¬ wn 3   = wceq 1559   ∈ wcel 2141  ⦋csb 3850  ∅c0 4283   ↦ cmpt 5178   I cid 5537  dom cdm 5643  ‘cfv 6515

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5243  ax-nul 5253  ax-pr 5387

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1099  df-tru 1562  df-fal 1572  df-ex 1799  df-nf 1803  df-sb 2090  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3076  df-rex 3086  df-rab 3414  df-v 3455  df-sbc 3743  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4284  df-if 4478  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-br 5098  df-opab 5160  df-mpt 5179  df-id 5538  df-xp 5649  df-rel 5650  df-cnv 5651  df-co 5652  df-dm 5653  df-rn 5654  df-res 5655  df-ima 5656  df-iota 6471  df-fun 6517  df-fn 6518  df-fv 6523

This theorem is referenced by:  fvmptnf  6992  sumeq2ii  15710  prodeq2ii  15931