Theorem fmptff 45698

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 1916	. . 3 ⊢ Ⅎ𝑦 𝐶 ∈ 𝐵
4		nfcsb1v 3861	. . . 4 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐶
5		fmptff.2	. . . 4 ⊢ Ⅎ𝑥𝐵
6	4, 5	nfel 2913	. . 3 ⊢ Ⅎ𝑥⦋𝑦 / 𝑥⦌𝐶 ∈ 𝐵
7		csbeq1a 3851	. . . 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 3277	. 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 5185	. . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐶) = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐶)
13	10, 12	eqtri 2759	. . 3 ⊢ 𝐹 = (𝑦 ∈ 𝐴 ↦ ⦋𝑦 / 𝑥⦌𝐶)
14	13	fmpt 7062	. 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 1542   ∈ wcel 2114  Ⅎwnfc 2883  ∀wral 3051  ⦋csb 3837   ↦ cmpt 5166  ⟶wf 6494

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2708  ax-sep 5231  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2539  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2811  df-nfc 2885  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-sbc 3729  df-csb 3838  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-br 5086  df-opab 5148  df-mpt 5167  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-fun 6500  df-fn 6501  df-f 6502

This theorem is referenced by:  fvmptelcdmf  45699  fmptdff  45700  rnmptssff  45703