Theorem fmptdF 32638

Description: Domain and codomain of the mapping operation; deduction form. This version of fmptd 7047 uses bound-variable hypothesis instead of distinct variable conditions. (Contributed by Thierry Arnoux, 28-Mar-2017.)

Hypotheses

Ref	Expression
fmptdF.p	⊢ Ⅎ𝑥𝜑
fmptdF.a	⊢ Ⅎ𝑥𝐴
fmptdF.c	⊢ Ⅎ𝑥𝐶
fmptdF.1	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶)
fmptdF.2	⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)

Assertion

Ref	Expression
fmptdF	⊢ (𝜑 → 𝐹:𝐴⟶𝐶)

Proof of Theorem fmptdF

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fmptdF.1	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶)
2	1	sbimi 2077	. . . . 5 ⊢ ([𝑦 / 𝑥](𝜑 ∧ 𝑥 ∈ 𝐴) → [𝑦 / 𝑥]𝐵 ∈ 𝐶)
3		sban 2083	. . . . . 6 ⊢ ([𝑦 / 𝑥](𝜑 ∧ 𝑥 ∈ 𝐴) ↔ ([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝑥 ∈ 𝐴))
4		fmptdF.p	. . . . . . . 8 ⊢ Ⅎ𝑥𝜑
5	4	sbf 2273	. . . . . . 7 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜑)
6		fmptdF.a	. . . . . . . 8 ⊢ Ⅎ𝑥𝐴
7	6	clelsb1fw 2898	. . . . . . 7 ⊢ ([𝑦 / 𝑥]𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)
8	5, 7	anbi12i 628	. . . . . 6 ⊢ (([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝑥 ∈ 𝐴) ↔ (𝜑 ∧ 𝑦 ∈ 𝐴))
9	3, 8	bitri 275	. . . . 5 ⊢ ([𝑦 / 𝑥](𝜑 ∧ 𝑥 ∈ 𝐴) ↔ (𝜑 ∧ 𝑦 ∈ 𝐴))
10		sbsbc 3740	. . . . . 6 ⊢ ([𝑦 / 𝑥]𝐵 ∈ 𝐶 ↔ [𝑦 / 𝑥]𝐵 ∈ 𝐶)
11		sbcel12 4358	. . . . . . 7 ⊢ ([𝑦 / 𝑥]𝐵 ∈ 𝐶 ↔ ⦋𝑦 / 𝑥⦌𝐵 ∈ ⦋𝑦 / 𝑥⦌𝐶)
12		vex 3440	. . . . . . . . 9 ⊢ 𝑦 ∈ V
13		fmptdF.c	. . . . . . . . 9 ⊢ Ⅎ𝑥𝐶
14	12, 13	csbgfi 3865	. . . . . . . 8 ⊢ ⦋𝑦 / 𝑥⦌𝐶 = 𝐶
15	14	eleq2i 2823	. . . . . . 7 ⊢ (⦋𝑦 / 𝑥⦌𝐵 ∈ ⦋𝑦 / 𝑥⦌𝐶 ↔ ⦋𝑦 / 𝑥⦌𝐵 ∈ 𝐶)
16	11, 15	bitri 275	. . . . . 6 ⊢ ([𝑦 / 𝑥]𝐵 ∈ 𝐶 ↔ ⦋𝑦 / 𝑥⦌𝐵 ∈ 𝐶)
17	10, 16	bitri 275	. . . . 5 ⊢ ([𝑦 / 𝑥]𝐵 ∈ 𝐶 ↔ ⦋𝑦 / 𝑥⦌𝐵 ∈ 𝐶)
18	2, 9, 17	3imtr3i 291	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ⦋𝑦 / 𝑥⦌𝐵 ∈ 𝐶)
19	18	ralrimiva 3124	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 ⦋𝑦 / 𝑥⦌𝐵 ∈ 𝐶)
20		nfcv 2894	. . . . 5 ⊢ Ⅎ𝑦𝐴
21		nfcv 2894	. . . . 5 ⊢ Ⅎ𝑦𝐵
22		nfcsb1v 3869	. . . . 5 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵
23		csbeq1a 3859	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵)
24	6, 20, 21, 22, 23	cbvmptf 5189	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐵)
25	24	fmpt 7043	. . 3 ⊢ (∀𝑦 ∈ 𝐴 ⦋𝑦 / 𝑥⦌𝐵 ∈ 𝐶 ↔ (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶𝐶)
26	19, 25	sylib 218	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶𝐶)
27		fmptdF.2	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
28	27	feq1i 6642	. 2 ⊢ (𝐹:𝐴⟶𝐶 ↔ (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶𝐶)
29	26, 28	sylibr 234	1 ⊢ (𝜑 → 𝐹:𝐴⟶𝐶)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541  Ⅎwnf 1784  [wsb 2067   ∈ wcel 2111  Ⅎwnfc 2879  ∀wral 3047  [wsbc 3736  ⦋csb 3845   ↦ cmpt 5170  ⟶wf 6477

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-fun 6483  df-fn 6484  df-f 6485

This theorem is referenced by:  fmptcof2  32639  esumcl  34043  esumid  34057  esumgsum  34058  esumval  34059  esumel  34060  esumsplit  34066  esumaddf  34074  esumss  34085  esumpfinvalf  34089