Theorem ffnfvf 7108

Description: A function maps to a class to which all values belong. This version of ffnfv 7107 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 7107	. 2 ⊢ (𝐹:𝐴⟶𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑧 ∈ 𝐴 (𝐹‘𝑧) ∈ 𝐵))
2		nfcv 2922	. . . 4 ⊢ Ⅎ𝑧𝐴
3		ffnfvf.1	. . . 4 ⊢ Ⅎ𝑥𝐴
4		ffnfvf.3	. . . . . 6 ⊢ Ⅎ𝑥𝐹
5		nfcv 2922	. . . . . 6 ⊢ Ⅎ𝑥𝑧
6	4, 5	nffv 6883	. . . . 5 ⊢ Ⅎ𝑥(𝐹‘𝑧)
7		ffnfvf.2	. . . . 5 ⊢ Ⅎ𝑥𝐵
8	6, 7	nfel 2936	. . . 4 ⊢ Ⅎ𝑥(𝐹‘𝑧) ∈ 𝐵
9		nfv 1947	. . . 4 ⊢ Ⅎ𝑧(𝐹‘𝑥) ∈ 𝐵
10		fveq2 6873	. . . . 5 ⊢ (𝑧 = 𝑥 → (𝐹‘𝑧) = (𝐹‘𝑥))
11	10	eleq1d 2845	. . . 4 ⊢ (𝑧 = 𝑥 → ((𝐹‘𝑧) ∈ 𝐵 ↔ (𝐹‘𝑥) ∈ 𝐵))
12	2, 3, 8, 9, 11	cbvralfw 3302	. . 3 ⊢ (∀𝑧 ∈ 𝐴 (𝐹‘𝑧) ∈ 𝐵 ↔ ∀𝑥 ∈ 𝐴 (𝐹‘𝑥) ∈ 𝐵)
13	12	anbi2i 635	. 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))

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   ∈ wcel 2145  Ⅎwnfc 2907  ∀wral 3076   Fn wfn 6522  ⟶wf 6523  ‘cfv 6527

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-mpt 5186  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-rn 5658  df-iota 6483  df-fun 6529  df-fn 6530  df-f 6531  df-fv 6535

This theorem is used by:  ixpf  8926  fconst7v  33148  fconst7  46197