Theorem fmptdf2 33293

Description: Domain and codomain of the mapping operation; deduction form. This version of fmptd 7114 uses bound-variable hypothesis instead of distinct variable conditions. (Contributed by Thierry Arnoux, 28-Mar-2017.)

Hypotheses

Ref	Expression
fmptdf2.p	⊢ Ⅎ𝑥𝜑
fmptdf2.a	⊢ Ⅎ𝑥𝐴
fmptdf2.c	⊢ Ⅎ𝑥𝐶
fmptdf2.1	⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶)
fmptdf2.2	⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)

Assertion

Ref	Expression
fmptdf2	⊢ (𝜑 → 𝐹:𝐴⟶𝐶)

Proof of Theorem fmptdf2

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fmptdf2.1	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶)
2	1	sbimi 2110	. . . . 5 ⊢ ([𝑦 / 𝑥](𝜑 ∧ 𝑥 ∈ 𝐴) → [𝑦 / 𝑥]𝐵 ∈ 𝐶)
3		sban 2116	. . . . . 6 ⊢ ([𝑦 / 𝑥](𝜑 ∧ 𝑥 ∈ 𝐴) ↔ ([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝑥 ∈ 𝐴))
4		fmptdf2.p	. . . . . . . 8 ⊢ Ⅎ𝑥𝜑
5	4	sbf 2304	. . . . . . 7 ⊢ ([𝑦 / 𝑥]𝜑 ↔ 𝜑)
6		fmptdf2.a	. . . . . . . 8 ⊢ Ⅎ𝑥𝐴
7	6	clelsb1fw 2926	. . . . . . 7 ⊢ ([𝑦 / 𝑥]𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴)
8	5, 7	anbi12i 640	. . . . . 6 ⊢ (([𝑦 / 𝑥]𝜑 ∧ [𝑦 / 𝑥]𝑥 ∈ 𝐴) ↔ (𝜑 ∧ 𝑦 ∈ 𝐴))
9	3, 8	bitri 278	. . . . 5 ⊢ ([𝑦 / 𝑥](𝜑 ∧ 𝑥 ∈ 𝐴) ↔ (𝜑 ∧ 𝑦 ∈ 𝐴))
10		sbsbc 3742	. . . . . 6 ⊢ ([𝑦 / 𝑥]𝐵 ∈ 𝐶 ↔ [𝑦 / 𝑥]𝐵 ∈ 𝐶)
11		sbcel12 4368	. . . . . . 7 ⊢ ([𝑦 / 𝑥]𝐵 ∈ 𝐶 ↔ ⦋𝑦 / 𝑥⦌𝐵 ∈ ⦋𝑦 / 𝑥⦌𝐶)
12		vex 3454	. . . . . . . . 9 ⊢ 𝑦 ∈ V
13		fmptdf2.c	. . . . . . . . 9 ⊢ Ⅎ𝑥𝐶
14	12, 13	csbgfi 3866	. . . . . . . 8 ⊢ ⦋𝑦 / 𝑥⦌𝐶 = 𝐶
15	14	eleq2i 2852	. . . . . . 7 ⊢ (⦋𝑦 / 𝑥⦌𝐵 ∈ ⦋𝑦 / 𝑥⦌𝐶 ↔ ⦋𝑦 / 𝑥⦌𝐵 ∈ 𝐶)
16	11, 15	bitri 278	. . . . . 6 ⊢ ([𝑦 / 𝑥]𝐵 ∈ 𝐶 ↔ ⦋𝑦 / 𝑥⦌𝐵 ∈ 𝐶)
17	10, 16	bitri 278	. . . . 5 ⊢ ([𝑦 / 𝑥]𝐵 ∈ 𝐶 ↔ ⦋𝑦 / 𝑥⦌𝐵 ∈ 𝐶)
18	2, 9, 17	3imtr3i 294	. . . 4 ⊢ ((𝜑 ∧ 𝑦 ∈ 𝐴) → ⦋𝑦 / 𝑥⦌𝐵 ∈ 𝐶)
19	18	ralrimiva 3154	. . 3 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 ⦋𝑦 / 𝑥⦌𝐵 ∈ 𝐶)
20		nfcv 2922	. . . . 5 ⊢ Ⅎ𝑦𝐴
21		nfcv 2922	. . . . 5 ⊢ Ⅎ𝑦𝐵
22		nfcsb1v 3870	. . . . 5 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐵
23		csbeq1a 3860	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝐵 = ⦋𝑦 / 𝑥⦌𝐵)
24	6, 20, 21, 22, 23	cbvmptf 5204	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐵)
25	24	fmpt 7110	. . 3 ⊢ (∀𝑦 ∈ 𝐴 ⦋𝑦 / 𝑥⦌𝐵 ∈ 𝐶 ↔ (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶𝐶)
26	19, 25	sylib 221	. 2 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶𝐶)
27		fmptdf2.2	. . 3 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐵)
28	27	feq1i 6700	. 2 ⊢ (𝐹:𝐴⟶𝐶 ↔ (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶𝐶)
29	26, 28	sylibr 237	1 ⊢ (𝜑 → 𝐹:𝐴⟶𝐶)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569  Ⅎwnf 1815  [wsb 2098   ∈ wcel 2144  Ⅎwnfc 2907  ∀wral 3076  [wsbc 3738  ⦋csb 3846   ↦ cmpt 5185  ⟶wf 6534

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5545  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-fun 6540  df-fn 6541  df-f 6542

This theorem is used by:  fmptcof2  33294  esumcl  34705  esumid  34719  esumgsum  34720  esumval  34721  esumel  34722  esumsplit  34728  esumaddf  34736  esumss  34747  esumpfinvalf  34751