Theorem fmptff 45513

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 2898	. . 3 ⊢ Ⅎ𝑦𝐴
3		nfv 1915	. . 3 ⊢ Ⅎ𝑦 𝐶 ∈ 𝐵
4		nfcsb1v 3873	. . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐶
5		fmptff.2	. . . 4 ⊢ Ⅎ𝑥𝐵
6	4, 5	nfel 2913	. . 3 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐶 ∈ 𝐵
7		csbeq1a 3863	. . . 4 ⊢ (𝑥 = 𝑦 → 𝐶 = ⦋𝑦 / 𝑥⦌𝐶)
8	7	eleq1d 2821	. . 3 ⊢ (𝑥 = 𝑦 → (𝐶 ∈ 𝐵 ↔ ⦋𝑦 / 𝑥⦌𝐶 ∈ 𝐵))
9	1, 2, 3, 6, 8	cbvralfw 3276	. 2 ⊢ (∀𝑥 ∈ 𝐴 𝐶 ∈ 𝐵 ↔ ∀𝑦 ∈ 𝐴 ⦋𝑦 / 𝑥⦌𝐶 ∈ 𝐵)
10		fmptff.3	. . . 4 ⊢ 𝐹 = (𝑥 ∈ 𝐴 ↦ 𝐶)
11		nfcv 2898	. . . . 5 ⊢ Ⅎ𝑦𝐶
12	1, 2, 11, 4, 7	cbvmptf 5198	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐶)
13	10, 12	eqtri 2759	. . 3 ⊢ 𝐹 = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐶)
14	13	fmpt 7055	. 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 206   = wceq 1541   ∈ wcel 2113  Ⅎwnfc 2883  ∀wral 3051  ⦋csb 3849   ↦ cmpt 5179  ⟶wf 6488

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2184  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-sbc 3741  df-csb 3850  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-br 5099  df-opab 5161  df-mpt 5180  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-fun 6494  df-fn 6495  df-f 6496

This theorem is referenced by:  fvmptelcdmf  45514  fmptdff  45515  rnmptssff  45518