Theorem fmptdF 32752

Description: Domain and codomain of the mapping operation; deduction form. This version of fmptd 7059 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 2086	. . . . 5 ⊢ ([𝑦 / 𝑥](𝜑 ∧ 𝑥 ∈ 𝐴) → [𝑦 / 𝑥]𝐵 ∈ 𝐶)
3		sban 2092	. . . . . 6 ⊢ ([𝑦 / 𝑥](𝜑 ∧ 𝑥 ∈ 𝐴) ↔ ([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝑥 ∈ 𝐴))
4		fmptdF.p	. . . . . . . 8 ⊢ Ⅎ𝑥𝜑
5	4	sbf 2284	. . . . . . 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 635	. . . . . 6 ⊢ (([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝑥 ∈ 𝐴) ↔ (𝜑 ∧ 𝑦 ∈ 𝐴))
9	3, 8	bitri 277	. . . . 5 ⊢ ([𝑦 / 𝑥](𝜑 ∧ 𝑥 ∈ 𝐴) ↔ (𝜑 ∧ 𝑦 ∈ 𝐴))
10		sbsbc 3729	. . . . . 6 ⊢ ([𝑦 / 𝑥]𝐵 ∈ 𝐶 ↔ [𝑦 / 𝑥]𝐵 ∈ 𝐶)
11		sbcel12 4342	. . . . . . 7 ⊢ ([𝑦 / 𝑥]𝐵 ∈ 𝐶 ↔ ⦋𝑦 / 𝑥⦌𝐵 ∈ ⦋𝑦 / 𝑥⦌𝐶)
12		vex 3437	. . . . . . . . 9 ⊢ 𝑦 ∈ V
13		fmptdF.c	. . . . . . . . 9 ⊢ Ⅎ𝑥𝐶
14	12, 13	csbgfi 3853	. . . . . . . 8 ⊢ ⦋𝑦 / 𝑥⦌𝐶 = 𝐶
15	14	eleq2i 2833	. . . . . . 7 ⊢ (⦋𝑦 / 𝑥⦌𝐵 ∈ ⦋𝑦 / 𝑥⦌𝐶 ↔ ⦋𝑦 / 𝑥⦌𝐵 ∈ 𝐶)
16	11, 15	bitri 277	. . . . . 6 ⊢ ([𝑦 / 𝑥]𝐵 ∈ 𝐶 ↔ ⦋𝑦 / 𝑥⦌𝐵 ∈ 𝐶)
17	10, 16	bitri 277	. . . . 5 ⊢ ([𝑦 / 𝑥]𝐵 ∈ 𝐶 ↔ ⦋𝑦 / 𝑥⦌𝐵 ∈ 𝐶)
18	2, 9, 17	3imtr3i 293	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ⦋𝑦 / 𝑥⦌𝐵 ∈ 𝐶)
19	18	ralrimiva 3133	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 ⦋𝑦 / 𝑥⦌𝐵 ∈ 𝐶)
20		nfcv 2903	. . . . 5 ⊢ Ⅎ𝑦𝐴
21		nfcv 2903	. . . . 5 ⊢ Ⅎ𝑦𝐵
22		nfcsb1v 3857	. . . . 5 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵
23		csbeq1a 3847	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵)
24	6, 20, 21, 22, 23	cbvmptf 5175	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐵)
25	24	fmpt 7055	. . 3 ⊢ (∀𝑦 ∈ 𝐴 ⦋𝑦 / 𝑥⦌𝐵 ∈ 𝐶 ↔ (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶𝐶)
26	19, 25	sylib 220	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶𝐶)
27		fmptdF.2	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
28	27	feq1i 6650	. 2 ⊢ (𝐹:𝐴⟶𝐶 ↔ (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶𝐶)
29	26, 28	sylibr 236	1 ⊢ (𝜑 → 𝐹:𝐴⟶𝐶)

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1548  Ⅎwnf 1791  [wsb 2074   ∈ wcel 2121  Ⅎwnfc 2888  ∀wral 3055  [wsbc 3725  ⦋csb 3833   ↦ cmpt 5156  ⟶wf 6485

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-sep 5221  ax-pr 5365

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-sn 4559  df-pr 4561  df-op 4565  df-br 5076  df-opab 5138  df-mpt 5157  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-fun 6491  df-fn 6492  df-f 6493

This theorem is referenced by:  fmptcof2  32753  esumcl  34226  esumid  34240  esumgsum  34241  esumval  34242  esumel  34243  esumsplit  34249  esumaddf  34257  esumss  34268  esumpfinvalf  34272