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Theorem ffnfvf 6296
Description: A function maps to a class to which all values belong. This version of ffnfv 6295 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 6295 . 2 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑧𝐴 (𝐹𝑧) ∈ 𝐵))
2 nfcv 2751 . . . 4 𝑧𝐴
3 ffnfvf.1 . . . 4 𝑥𝐴
4 ffnfvf.3 . . . . . 6 𝑥𝐹
5 nfcv 2751 . . . . . 6 𝑥𝑧
64, 5nffv 6110 . . . . 5 𝑥(𝐹𝑧)
7 ffnfvf.2 . . . . 5 𝑥𝐵
86, 7nfel 2763 . . . 4 𝑥(𝐹𝑧) ∈ 𝐵
9 nfv 1830 . . . 4 𝑧(𝐹𝑥) ∈ 𝐵
10 fveq2 6103 . . . . 5 (𝑧 = 𝑥 → (𝐹𝑧) = (𝐹𝑥))
1110eleq1d 2672 . . . 4 (𝑧 = 𝑥 → ((𝐹𝑧) ∈ 𝐵 ↔ (𝐹𝑥) ∈ 𝐵))
122, 3, 8, 9, 11cbvralf 3141 . . 3 (∀𝑧𝐴 (𝐹𝑧) ∈ 𝐵 ↔ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵)
1312anbi2i 726 . 2 ((𝐹 Fn 𝐴 ∧ ∀𝑧𝐴 (𝐹𝑧) ∈ 𝐵) ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
141, 13bitri 263 1 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 195  wa 383  wcel 1977  wnfc 2738  wral 2896   Fn wfn 5799  wf 5800  cfv 5804
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fv 5812
This theorem is referenced by:  ixpf  7816
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