Theorem ffnfvf 7115

Description: A function maps to a class to which all values belong. This version of ffnfv 7114 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 28-Sep-2006.)

Hypotheses

Ref	Expression
ffnfvf.1	⊢ Ⅎ𝑥𝐴
ffnfvf.2	⊢ Ⅎ𝑥𝐵
ffnfvf.3	⊢ Ⅎ𝑥𝐹

Assertion

Ref	Expression
ffnfvf	⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵))

Proof of Theorem ffnfvf

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ffnfv 7114	. 2 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑧 ∈ 𝐴 (𝐹‘𝑧) ∈ 𝐵))
2		nfcv 2899	. . . 4 ⊢ Ⅎ𝑧𝐴
3		ffnfvf.1	. . . 4 ⊢ Ⅎ𝑥𝐴
4		ffnfvf.3	. . . . . 6 ⊢ Ⅎ𝑥𝐹
5		nfcv 2899	. . . . . 6 ⊢ Ⅎ𝑥𝑧
6	4, 5	nffv 6891	. . . . 5 ⊢ Ⅎ𝑥(𝐹‘𝑧)
7		ffnfvf.2	. . . . 5 ⊢ Ⅎ𝑥𝐵
8	6, 7	nfel 2914	. . . 4 ⊢ Ⅎ𝑥(𝐹‘𝑧) ∈ 𝐵
9		nfv 1914	. . . 4 ⊢ Ⅎ𝑧(𝐹‘𝑥) ∈ 𝐵
10		fveq2 6881	. . . . 5 ⊢ (𝑧 = 𝑥 → (𝐹‘𝑧) = (𝐹‘𝑥))
11	10	eleq1d 2820	. . . 4 ⊢ (𝑧 = 𝑥 → ((𝐹‘𝑧) ∈ 𝐵 ↔ (𝐹‘𝑥) ∈ 𝐵))
12	2, 3, 8, 9, 11	cbvralfw 3288	. . 3 ⊢ (∀𝑧 ∈ 𝐴 (𝐹‘𝑧) ∈ 𝐵 ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵)
13	12	anbi2i 623	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑧 ∈ 𝐴 (𝐹‘𝑧) ∈ 𝐵) ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵))
14	1, 13	bitri 275	1 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵))

Syntax hints:   ↔ wb 206   ∧ wa 395   ∈ wcel 2109  Ⅎwnfc 2884  ∀wral 3052   Fn wfn 6531  ⟶wf 6532  ‘cfv 6536

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2708  ax-sep 5271  ax-nul 5281  ax-pr 5407

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2728  df-clel 2810  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-rab 3421  df-v 3466  df-dif 3934  df-un 3936  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4889  df-br 5125  df-opab 5187  df-mpt 5207  df-id 5553  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-iota 6489  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544

This theorem is referenced by:  ixpf  8939  fconst7  45255