Theorem fmptff 46017

Description: Functionality of the mapping operation. (Contributed by Glauco Siliprandi, 5-Jan-2025.)

Hypotheses

Ref	Expression
fmptff.1	⊢ Ⅎ𝑥𝐴
fmptff.2	⊢ Ⅎ𝑥𝐵
fmptff.3	⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶)

Assertion

Ref	Expression
fmptff	⊢ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ↔ 𝐹:𝐴⟶𝐵)

Proof of Theorem fmptff

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fmptff.1	. . 3 ⊢ Ⅎ𝑥𝐴
2		nfcv 2927	. . 3 ⊢ Ⅎ𝑦𝐴
3		nfv 1947	. . 3 ⊢ Ⅎ𝑦 𝐶 ∈ 𝐵
4		nfcsb1v 3878	. . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐶
5		fmptff.2	. . . 4 ⊢ Ⅎ𝑥𝐵
6	4, 5	nfel 2941	. . 3 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐶 ∈ 𝐵
7		csbeq1a 3868	. . . 4 ⊢ (𝑥 = 𝑦 → 𝐶 = ⦋𝑦 / 𝑥⦌𝐶)
8	7	eleq1d 2850	. . 3 ⊢ (𝑥 = 𝑦 → (𝐶 ∈ 𝐵 ↔ ⦋𝑦 / 𝑥⦌𝐶 ∈ 𝐵))
9	1, 2, 3, 6, 8	cbvralfw 3307	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ↔ ∀𝑦 ∈ 𝐴 ⦋𝑦 / 𝑥⦌𝐶 ∈ 𝐵)
10		fmptff.3	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶)
11		nfcv 2927	. . . . 5 ⊢ Ⅎ𝑦𝐶
12	1, 2, 11, 4, 7	cbvmptf 5213	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐶)
13	10, 12	eqtri 2788	. . 3 ⊢ 𝐹 = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐶)
14	13	fmpt 7109	. 2 ⊢ (∀𝑦 ∈ 𝐴 ⦋𝑦 / 𝑥⦌𝐶 ∈ 𝐵 ↔ 𝐹:𝐴⟶𝐵)
15	9, 14	bitri 278	1 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ↔ 𝐹:𝐴⟶𝐵)

This proof depends on syntax axioms:   ↔ wb 209   = wceq 1570   ∈ wcel 2146  Ⅎwnfc 2912  ∀wral 3081  ⦋csb 3854   ↦ cmpt 5194  ⟶wf 6536

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2737  ax-sep 5259  ax-pr 5406

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ral 3082  df-rex 3092  df-rab 3419  df-v 3459  df-sbc 3747  df-csb 3855  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4287  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5112  df-opab 5176  df-mpt 5195  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-fun 6542  df-fn 6543  df-f 6544

This theorem is used by:  fvmptelcdmf  46018  fmptdff  46019  rnmptssff  46022