Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  ffnfvf Unicode version

Theorem ffnfvf 6058
 Description: A function maps to a class to which all values belong. This version of ffnfv 6057 uses bound-variable hypotheses instead of distinct variable conditions. (Contributed by NM, 28-Sep-2006.)
Hypotheses
Ref Expression
ffnfvf.1
ffnfvf.2
ffnfvf.3
Assertion
Ref Expression
ffnfvf

Proof of Theorem ffnfvf
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 ffnfv 6057 . 2
2 nfcv 2619 . . . 4
3 ffnfvf.1 . . . 4
4 ffnfvf.3 . . . . . 6
5 nfcv 2619 . . . . . 6
64, 5nffv 5878 . . . . 5
7 ffnfvf.2 . . . . 5
86, 7nfel 2632 . . . 4
9 nfv 1707 . . . 4
10 fveq2 5871 . . . . 5
1110eleq1d 2526 . . . 4
122, 3, 8, 9, 11cbvralf 3078 . . 3
1312anbi2i 694 . 2
141, 13bitri 249 1
 Colors of variables: wff setvar class Syntax hints:  <->wb 184  /\wa 369  e.wcel 1818  F/_wnfc 2605  A.wral 2807  Fnwfn 5588  -->wf 5589  `cfv 5593 This theorem is referenced by:  ixpf  7511 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1618  ax-4 1631  ax-5 1704  ax-6 1747  ax-7 1790  ax-9 1822  ax-10 1837  ax-11 1842  ax-12 1854  ax-13 1999  ax-ext 2435  ax-sep 4573  ax-nul 4581  ax-pr 4691 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1613  df-nf 1617  df-sb 1740  df-eu 2286  df-mo 2287  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-sbc 3328  df-dif 3478  df-un 3480  df-in 3482  df-ss 3489  df-nul 3785  df-if 3942  df-sn 4030  df-pr 4032  df-op 4036  df-uni 4250  df-br 4453  df-opab 4511  df-mpt 4512  df-id 4800  df-xp 5010  df-rel 5011  df-cnv 5012  df-co 5013  df-dm 5014  df-rn 5015  df-iota 5556  df-fun 5595  df-fn 5596  df-f 5597  df-fv 5601
 Copyright terms: Public domain W3C validator