Theorem bnj927 30839

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj927.1	|- G = ( f u. { <. n , C >. } )
bnj927.2	|- C e. _V

Assertion

Ref	Expression
bnj927	|- ( ( p = suc n /\ f Fn n ) -> G Fn p )

Proof of Theorem bnj927

Step	Hyp	Ref	Expression
1		simpr 477	. . . 4 |- ( ( p = suc n /\ f Fn n ) -> f Fn n )
2		vex 3203	. . . . . 6 |- n e. _V
3		bnj927.2	. . . . . 6 |- C e. _V
4	2, 3	fnsn 5946	. . . . 5 |- { <. n , C >. } Fn { n }
5	4	a1i 11	. . . 4 |- ( ( p = suc n /\ f Fn n ) -> { <. n , C >. } Fn { n } )
6		bnj521 30805	. . . . 5 |- ( n i^i { n } ) = (/)
7	6	a1i 11	. . . 4 |- ( ( p = suc n /\ f Fn n ) -> ( n i^i { n } ) = (/) )
8		fnun 5997	. . . 4 |- ( ( ( f Fn n /\ { <. n , C >. } Fn { n } ) /\ ( n i^i { n } ) = (/) ) -> ( f u. { <. n , C >. } ) Fn ( n u. { n } ) )
9	1, 5, 7, 8	syl21anc 1325	. . 3 |- ( ( p = suc n /\ f Fn n ) -> ( f u. { <. n , C >. } ) Fn ( n u. { n } ) )
10		bnj927.1	. . . 4 |- G = ( f u. { <. n , C >. } )
11	10	fneq1i 5985	. . 3 |- ( G Fn ( n u. { n } ) <-> ( f u. { <. n , C >. } ) Fn ( n u. { n } ) )
12	9, 11	sylibr 224	. 2 |- ( ( p = suc n /\ f Fn n ) -> G Fn ( n u. { n } ) )
13		df-suc 5729	. . . . . 6 |- suc n = ( n u. { n } )
14	13	eqeq2i 2634	. . . . 5 |- ( p = suc n <-> p = ( n u. { n } ) )
15	14	biimpi 206	. . . 4 |- ( p = suc n -> p = ( n u. { n } ) )
16	15	adantr 481	. . 3 |- ( ( p = suc n /\ f Fn n ) -> p = ( n u. { n } ) )
17	16	fneq2d 5982	. 2 |- ( ( p = suc n /\ f Fn n ) -> ( G Fn p <-> G Fn ( n u. { n } ) ) )
18	12, 17	mpbird 247	1 |- ( ( p = suc n /\ f Fn n ) -> G Fn p )

Syntax hints:   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   u. cun 3572   i^i cin 3573   (/)c0 3915   {csn 4177   <.cop 4183   suc csuc 5725   Fn wfn 5883

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-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-reg 8497

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  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-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-suc 5729  df-fun 5890  df-fn 5891

This theorem is referenced by:  bnj941  30843  bnj929  31006