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Theorem ffnfvf 6706
Description: A function maps to a class to which all values belong. This version of ffnfv 6705 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 6705 . 2 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑧𝐴 (𝐹𝑧) ∈ 𝐵))
2 nfcv 2932 . . . 4 𝑧𝐴
3 ffnfvf.1 . . . 4 𝑥𝐴
4 ffnfvf.3 . . . . . 6 𝑥𝐹
5 nfcv 2932 . . . . . 6 𝑥𝑧
64, 5nffv 6509 . . . . 5 𝑥(𝐹𝑧)
7 ffnfvf.2 . . . . 5 𝑥𝐵
86, 7nfel 2944 . . . 4 𝑥(𝐹𝑧) ∈ 𝐵
9 nfv 1873 . . . 4 𝑧(𝐹𝑥) ∈ 𝐵
10 fveq2 6499 . . . . 5 (𝑧 = 𝑥 → (𝐹𝑧) = (𝐹𝑥))
1110eleq1d 2850 . . . 4 (𝑧 = 𝑥 → ((𝐹𝑧) ∈ 𝐵 ↔ (𝐹𝑥) ∈ 𝐵))
122, 3, 8, 9, 11cbvralf 3377 . . 3 (∀𝑧𝐴 (𝐹𝑧) ∈ 𝐵 ↔ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵)
1312anbi2i 613 . 2 ((𝐹 Fn 𝐴 ∧ ∀𝑧𝐴 (𝐹𝑧) ∈ 𝐵) ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
141, 13bitri 267 1 (𝐹:𝐴𝐵 ↔ (𝐹 Fn 𝐴 ∧ ∀𝑥𝐴 (𝐹𝑥) ∈ 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wb 198  wa 387  wcel 2050  wnfc 2916  wral 3088   Fn wfn 6183  wf 6184  cfv 6188
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-sep 5060  ax-nul 5067  ax-pr 5186
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ral 3093  df-rex 3094  df-rab 3097  df-v 3417  df-sbc 3682  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-nul 4179  df-if 4351  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-br 4930  df-opab 4992  df-mpt 5009  df-id 5312  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-fv 6196
This theorem is referenced by:  ixpf  8281  fconst7  40974
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