Theorem fmptdF 32673

Description: Domain and codomain of the mapping operation; deduction form. This version of fmptd 7134 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 2072	. . . . 5 ⊢ ([𝑦 / 𝑥](𝜑 ∧ 𝑥 ∈ 𝐴) → [𝑦 / 𝑥]𝐵 ∈ 𝐶)
3		sban 2078	. . . . . 6 ⊢ ([𝑦 / 𝑥](𝜑 ∧ 𝑥 ∈ 𝐴) ↔ ([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝑥 ∈ 𝐴))
4		fmptdF.p	. . . . . . . 8 ⊢ Ⅎ𝑥𝜑
5	4	sbf 2269	. . . . . . 7 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜑)
6		fmptdF.a	. . . . . . . 8 ⊢ Ⅎ𝑥𝐴
7	6	clelsb1fw 2907	. . . . . . 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 3795	. . . . . 6 ⊢ ([𝑦 / 𝑥]𝐵 ∈ 𝐶 ↔ [𝑦 / 𝑥]𝐵 ∈ 𝐶)
11		sbcel12 4417	. . . . . . 7 ⊢ ([𝑦 / 𝑥]𝐵 ∈ 𝐶 ↔ ⦋𝑦 / 𝑥⦌𝐵 ∈ ⦋𝑦 / 𝑥⦌𝐶)
12		vex 3482	. . . . . . . . 9 ⊢ 𝑦 ∈ V
13		fmptdF.c	. . . . . . . . 9 ⊢ Ⅎ𝑥𝐶
14	12, 13	csbgfi 3929	. . . . . . . 8 ⊢ ⦋𝑦 / 𝑥⦌𝐶 = 𝐶
15	14	eleq2i 2831	. . . . . . 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 3144	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 ⦋𝑦 / 𝑥⦌𝐵 ∈ 𝐶)
20		nfcv 2903	. . . . 5 ⊢ Ⅎ𝑦𝐴
21		nfcv 2903	. . . . 5 ⊢ Ⅎ𝑦𝐵
22		nfcsb1v 3933	. . . . 5 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵
23		csbeq1a 3922	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵)
24	6, 20, 21, 22, 23	cbvmptf 5257	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐵)
25	24	fmpt 7130	. . 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 6728	. 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 1537  Ⅎwnf 1780  [wsb 2062   ∈ wcel 2106  Ⅎwnfc 2888  ∀wral 3059  [wsbc 3791  ⦋csb 3908   ↦ cmpt 5231  ⟶wf 6559

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-fun 6565  df-fn 6566  df-f 6567

This theorem is referenced by:  fmptcof2  32674  esumcl  34011  esumid  34025  esumgsum  34026  esumval  34027  esumel  34028  esumsplit  34034  esumaddf  34042  esumss  34053  esumpfinvalf  34057