Theorem findfvcl 32451

Description: Please add description here. (Contributed by Jeff Hoffman, 12-Feb-2008.)

Hypotheses

Ref	Expression
findfvcl.1	|- ( ph -> ( F ` (/) ) e. P )
findfvcl.2	|- ( y e. om -> ( ph -> ( ( F ` y ) e. P -> ( F ` suc y ) e. P ) ) )

Assertion

Ref	Expression
findfvcl	|- ( A e. om -> ( ph -> ( F ` A ) e. P ) )

Distinct variable groups:   y, F   y, P   ph, y

Allowed substitution hint:   A( y)

Proof of Theorem findfvcl

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveleq 32450	. 2 |- ( x = (/) -> ( ( ph -> ( F ` x ) e. P ) <-> ( ph -> ( F ` (/) ) e. P ) ) )
2		fveleq 32450	. 2 |- ( x = y -> ( ( ph -> ( F ` x ) e. P ) <-> ( ph -> ( F ` y ) e. P ) ) )
3		fveleq 32450	. 2 |- ( x = suc y -> ( ( ph -> ( F ` x ) e. P ) <-> ( ph -> ( F ` suc y ) e. P ) ) )
4		fveleq 32450	. 2 |- ( x = A -> ( ( ph -> ( F ` x ) e. P ) <-> ( ph -> ( F ` A ) e. P ) ) )
5		findfvcl.1	. 2 |- ( ph -> ( F ` (/) ) e. P )
6		findfvcl.2	. . 3 |- ( y e. om -> ( ph -> ( ( F ` y ) e. P -> ( F ` suc y ) e. P ) ) )
7	6	a2d 29	. 2 |- ( y e. om -> ( ( ph -> ( F ` y ) e. P ) -> ( ph -> ( F ` suc y ) e. P ) ) )
8	1, 2, 3, 4, 5, 7	finds 7092	1 |- ( A e. om -> ( ph -> ( F ` A ) e. P ) )

Syntax hints:   -> wi 4   e. wcel 1990   (/)c0 3915   suc csuc 5725   ` cfv 5888   omcom 7065

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-tr 4753  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fv 5896  df-om 7066

This theorem is referenced by:  findreccl  32452