Theorem fvmptex 7006

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 6896.) 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 3857	. . . 4 ⊢ (𝑦 = 𝐶 → ⦋𝑦 / 𝑥⦌𝐵 = ⦋𝐶 / 𝑥⦌𝐵)
2		fvmptex.1	. . . . 5 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
3		nfcv 2925	. . . . . 6 ⊢ Ⅎ𝑦𝐵
4		nfcsb1v 3878	. . . . . 6 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵
5		csbeq1a 3868	. . . . . 6 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵)
6	3, 4, 5	cbvmpt 5214	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐵)
7	2, 6	eqtri 2786	. . . 4 ⊢ 𝐹 = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐵)
8	1, 7	fvmpti 6990	. . 3 ⊢ (𝐶 ∈ 𝐴 → (𝐹‘𝐶) = ( I ‘⦋𝐶 / 𝑥⦌𝐵))
9	1	fveq2d 6887	. . . 4 ⊢ (𝑦 = 𝐶 → ( I ‘⦋𝑦 / 𝑥⦌𝐵) = ( I ‘⦋𝐶 / 𝑥⦌𝐵))
10		fvmptex.2	. . . . 5 ⊢ 𝐺 = (𝑥 ∈ 𝐴 ↦ ( I ‘𝐵))
11		nfcv 2925	. . . . . 6 ⊢ Ⅎ𝑦( I ‘𝐵)
12		nfcv 2925	. . . . . . 7 ⊢ Ⅎ𝑥 I
13	12, 4	nffv 6893	. . . . . 6 ⊢ Ⅎ𝑥( I ‘⦋𝑦 / 𝑥⦌𝐵)
14	5	fveq2d 6887	. . . . . 6 ⊢ (𝑥 = 𝑦 → ( I ‘𝐵) = ( I ‘⦋𝑦 / 𝑥⦌𝐵))
15	11, 13, 14	cbvmpt 5214	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ ( I ‘𝐵)) = (𝑦 ∈ 𝐴 ↦ ( I ‘⦋𝑦 / 𝑥⦌𝐵))
16	10, 15	eqtri 2786	. . . 4 ⊢ 𝐺 = (𝑦 ∈ 𝐴 ↦ ( I ‘⦋𝑦 / 𝑥⦌𝐵))
17		fvex 6896	. . . 4 ⊢ ( I ‘⦋𝐶 / 𝑥⦌𝐵) ∈ V
18	9, 16, 17	fvmpt 6991	. . 3 ⊢ (𝐶 ∈ 𝐴 → (𝐺‘𝐶) = ( I ‘⦋𝐶 / 𝑥⦌𝐵))
19	8, 18	eqtr4d 2801	. 2 ⊢ (𝐶 ∈ 𝐴 → (𝐹‘𝐶) = (𝐺‘𝐶))
20	2	dmmptss 6244	. . . . 5 ⊢ dom 𝐹 ⊆ 𝐴
21	20	sseli 3934	. . . 4 ⊢ (𝐶 ∈ dom 𝐹 → 𝐶 ∈ 𝐴)
22		ndmfv 6915	. . . 4 ⊢ (¬ 𝐶 ∈ dom 𝐹 → (𝐹‘𝐶) = ∅)
23	21, 22	nsyl5 160	. . 3 ⊢ (¬ 𝐶 ∈ 𝐴 → (𝐹‘𝐶) = ∅)
24		fvex 6896	. . . . . 6 ⊢ ( I ‘𝐵) ∈ V
25	24, 10	dmmpti 6681	. . . . 5 ⊢ dom 𝐺 = 𝐴
26	25	eleq2i 2855	. . . 4 ⊢ (𝐶 ∈ dom 𝐺 ↔ 𝐶 ∈ 𝐴)
27		ndmfv 6915	. . . 4 ⊢ (¬ 𝐶 ∈ dom 𝐺 → (𝐺‘𝐶) = ∅)
28	26, 27	sylnbir 334	. . 3 ⊢ (¬ 𝐶 ∈ 𝐴 → (𝐺‘𝐶) = ∅)
29	23, 28	eqtr4d 2801	. 2 ⊢ (¬ 𝐶 ∈ 𝐴 → (𝐹‘𝐶) = (𝐺‘𝐶))
30	19, 29	pm2.61i 184	1 ⊢ (𝐹‘𝐶) = (𝐺‘𝐶)

Syntax hints:  ¬ wn 3   = wceq 1570   ∈ wcel 2143  ⦋csb 3854  ∅c0 4287   ↦ cmpt 5193   I cid 5557  dom cdm 5663  ‘cfv 6538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-sbc 3746  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5558  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fun 6540  df-fn 6541  df-fv 6546

This theorem is referenced by:  fvmptnf  7014  sumeq2ii  15746  prodeq2ii  15967