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Theorem ffnfvf 6864
 Description: A function maps to a class to which all values belong. This version of ffnfv 6863 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 28-Sep-2006.)
Hypotheses
Ref Expression
ffnfvf.1 𝑥𝐴
ffnfvf.2 𝑥𝐵
ffnfvf.3 𝑥𝐹
Assertion
Ref Expression
ffnfvf (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))

Proof of Theorem ffnfvf
Dummy variable 𝑧 is distinct from all other variables.
StepHypRef Expression
1 ffnfv 6863 . 2 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑧𝐴 (𝐹𝑧) ∈ 𝐵))
2 nfcv 2958 . . . 4 𝑧𝐴
3 ffnfvf.1 . . . 4 𝑥𝐴
4 ffnfvf.3 . . . . . 6 𝑥𝐹
5 nfcv 2958 . . . . . 6 𝑥𝑧
64, 5nffv 6659 . . . . 5 𝑥(𝐹𝑧)
7 ffnfvf.2 . . . . 5 𝑥𝐵
86, 7nfel 2972 . . . 4 𝑥(𝐹𝑧) ∈ 𝐵
9 nfv 1915 . . . 4 𝑧(𝐹𝑥) ∈ 𝐵
10 fveq2 6649 . . . . 5 (𝑧 = 𝑥 → (𝐹𝑧) = (𝐹𝑥))
1110eleq1d 2877 . . . 4 (𝑧 = 𝑥 → ((𝐹𝑧) ∈ 𝐵 ↔ (𝐹𝑥) ∈ 𝐵))
122, 3, 8, 9, 11cbvralfw 3385 . . 3 (∀𝑧𝐴 (𝐹𝑧) ∈ 𝐵 ↔ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵)
1312anbi2i 625 . 2 ((𝐹 Fn 𝐴 ∧ ∀𝑧𝐴 (𝐹𝑧) ∈ 𝐵) ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
141, 13bitri 278 1 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209   ∧ wa 399   ∈ wcel 2112  Ⅎwnfc 2939  ∀wral 3109   Fn wfn 6323  ⟶wf 6324  ‘cfv 6328 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 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pr 5298 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-v 3446  df-sbc 3724  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-fv 6336 This theorem is referenced by:  ixpf  8471  fconst7  41901
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