Theorem fmptff 44527

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 2897	. . 3 ⊢ Ⅎ𝑦𝐴
3		nfv 1909	. . 3 ⊢ Ⅎ𝑦 𝐶 ∈ 𝐵
4		nfcsb1v 3913	. . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐶
5		fmptff.2	. . . 4 ⊢ Ⅎ𝑥𝐵
6	4, 5	nfel 2911	. . 3 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐶 ∈ 𝐵
7		csbeq1a 3902	. . . 4 ⊢ (𝑥 = 𝑦 → 𝐶 = ⦋𝑦 / 𝑥⦌𝐶)
8	7	eleq1d 2812	. . 3 ⊢ (𝑥 = 𝑦 → (𝐶 ∈ 𝐵 ↔ ⦋𝑦 / 𝑥⦌𝐶 ∈ 𝐵))
9	1, 2, 3, 6, 8	cbvralfw 3295	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ↔ ∀𝑦 ∈ 𝐴 ⦋𝑦 / 𝑥⦌𝐶 ∈ 𝐵)
10		fmptff.3	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶)
11		nfcv 2897	. . . . 5 ⊢ Ⅎ𝑦𝐶
12	1, 2, 11, 4, 7	cbvmptf 5250	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐶)
13	10, 12	eqtri 2754	. . 3 ⊢ 𝐹 = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐶)
14	13	fmpt 7104	. 2 ⊢ (∀𝑦 ∈ 𝐴 ⦋𝑦 / 𝑥⦌𝐶 ∈ 𝐵 ↔ 𝐹:𝐴⟶𝐵)
15	9, 14	bitri 275	1 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ↔ 𝐹:𝐴⟶𝐵)

Syntax hints:   ↔ wb 205   = wceq 1533   ∈ wcel 2098  Ⅎwnfc 2877  ∀wral 3055  ⦋csb 3888   ↦ cmpt 5224  ⟶wf 6532

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-fun 6538  df-fn 6539  df-f 6540

This theorem is referenced by:  fvmptelcdmf  44528  fmptdff  44529  rnmptssff  44532