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Theorem ffnfvf 6875
Description: A function maps to a class to which all values belong. This version of ffnfv 6874 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 6874 . 2 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑧𝐴 (𝐹𝑧) ∈ 𝐵))
2 nfcv 2974 . . . 4 𝑧𝐴
3 ffnfvf.1 . . . 4 𝑥𝐴
4 ffnfvf.3 . . . . . 6 𝑥𝐹
5 nfcv 2974 . . . . . 6 𝑥𝑧
64, 5nffv 6673 . . . . 5 𝑥(𝐹𝑧)
7 ffnfvf.2 . . . . 5 𝑥𝐵
86, 7nfel 2989 . . . 4 𝑥(𝐹𝑧) ∈ 𝐵
9 nfv 1906 . . . 4 𝑧(𝐹𝑥) ∈ 𝐵
10 fveq2 6663 . . . . 5 (𝑧 = 𝑥 → (𝐹𝑧) = (𝐹𝑥))
1110eleq1d 2894 . . . 4 (𝑧 = 𝑥 → ((𝐹𝑧) ∈ 𝐵 ↔ (𝐹𝑥) ∈ 𝐵))
122, 3, 8, 9, 11cbvralfw 3435 . . 3 (∀𝑧𝐴 (𝐹𝑧) ∈ 𝐵 ↔ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵)
1312anbi2i 622 . 2 ((𝐹 Fn 𝐴 ∧ ∀𝑧𝐴 (𝐹𝑧) ∈ 𝐵) ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
141, 13bitri 276 1 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 207  wa 396  wcel 2105  wnfc 2958  wral 3135   Fn wfn 6343  wf 6344  cfv 6348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fv 6356
This theorem is referenced by:  ixpf  8472  fconst7  41415
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