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 2923	. . . 4 ⊢ Ⅎ𝑧𝐴
3		ffnfvf.1	. . . 4 ⊢ Ⅎ𝑥𝐴
4		ffnfvf.3	. . . . . 6 ⊢ Ⅎ𝑥𝐹
5		nfcv 2923	. . . . . 6 ⊢ Ⅎ𝑥𝑧
6	4, 5	nffv 6891	. . . . 5 ⊢ Ⅎ𝑥(𝐹‘𝑧)
7		ffnfvf.2	. . . . 5 ⊢ Ⅎ𝑥𝐵
8	6, 7	nfel 2937	. . . 4 ⊢ Ⅎ𝑥(𝐹‘𝑧) ∈ 𝐵
9		nfv 1942	. . . 4 ⊢ Ⅎ𝑧(𝐹‘𝑥) ∈ 𝐵
10		fveq2 6881	. . . . 5 ⊢ (𝑧 = 𝑥 → (𝐹‘𝑧) = (𝐹‘𝑥))
11	10	eleq1d 2846	. . . 4 ⊢ (𝑧 = 𝑥 → ((𝐹‘𝑧) ∈ 𝐵 ↔ (𝐹‘𝑥) ∈ 𝐵))
12	2, 3, 8, 9, 11	cbvralfw 3303	. . 3 ⊢ (∀𝑧 ∈ 𝐴 (𝐹‘𝑧) ∈ 𝐵 ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵)
13	12	anbi2i 634	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ ∀𝑧 ∈ 𝐴 (𝐹‘𝑧) ∈ 𝐵) ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵))
14	1, 13	bitri 278	1 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵))

Syntax hints:   ↔ wb 209   ∧ wa 400   ∈ wcel 2141  Ⅎwnfc 2908  ∀wral 3077   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 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5256  ax-nul 5268  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3415  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-mpt 5192  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-co 5670  df-dm 5671  df-rn 5672  df-iota 6492  df-fun 6538  df-fn 6539  df-f 6540  df-fv 6544

This theorem is referenced by:  ixpf  8917  fconst7v  32931  fconst7  45927