Theorem fmptff 45869

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 2931	. . 3 ⊢ Ⅎ𝑦𝐴
3		nfv 1941	. . 3 ⊢ Ⅎ𝑦 𝐶 ∈ 𝐵
4		nfcsb1v 3885	. . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐶
5		fmptff.2	. . . 4 ⊢ Ⅎ𝑥𝐵
6	4, 5	nfel 2945	. . 3 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐶 ∈ 𝐵
7		csbeq1a 3875	. . . 4 ⊢ (𝑥 = 𝑦 → 𝐶 = ⦋𝑦 / 𝑥⦌𝐶)
8	7	eleq1d 2854	. . 3 ⊢ (𝑥 = 𝑦 → (𝐶 ∈ 𝐵 ↔ ⦋𝑦 / 𝑥⦌𝐶 ∈ 𝐵))
9	1, 2, 3, 6, 8	cbvralfw 3311	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ↔ ∀𝑦 ∈ 𝐴 ⦋𝑦 / 𝑥⦌𝐶 ∈ 𝐵)
10		fmptff.3	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶)
11		nfcv 2931	. . . . 5 ⊢ Ⅎ𝑦𝐶
12	1, 2, 11, 4, 7	cbvmptf 5212	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐶)
13	10, 12	eqtri 2792	. . 3 ⊢ 𝐹 = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐶)
14	13	fmpt 7103	. 2 ⊢ (∀𝑦 ∈ 𝐴 ⦋𝑦 / 𝑥⦌𝐶 ∈ 𝐵 ↔ 𝐹:𝐴⟶𝐵)
15	9, 14	bitri 278	1 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ↔ 𝐹:𝐴⟶𝐵)

Syntax hints:   ↔ wb 209   = wceq 1567   ∈ wcel 2149  Ⅎwnfc 2916  ∀wral 3085  ⦋csb 3861   ↦ cmpt 5193  ⟶wf 6529

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5258  ax-pr 5402

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-sbc 3754  df-csb 3862  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111  df-opab 5175  df-mpt 5194  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-fun 6535  df-fn 6536  df-f 6537

This theorem is referenced by:  fvmptelcdmf  45870  fmptdff  45871  rnmptssff  45874