Theorem bnj941 30843

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

Hypothesis

Ref	Expression
bnj941.1	|- G = ( f u. { <. n , C >. } )

Assertion

Ref	Expression
bnj941	|- ( C e. _V -> ( ( p = suc n /\ f Fn n ) -> G Fn p ) )

Proof of Theorem bnj941

Step	Hyp	Ref	Expression
1		bnj941.1	. . . . 5 |- G = ( f u. { <. n , C >. } )
2		opeq2 4403	. . . . . . 7 |- ( C = if ( C e. _V , C , (/) ) -> <. n , C >. = <. n , if ( C e. _V , C , (/) ) >. )
3	2	sneqd 4189	. . . . . 6 |- ( C = if ( C e. _V , C , (/) ) -> { <. n , C >. } = { <. n , if ( C e. _V , C , (/) ) >. } )
4	3	uneq2d 3767	. . . . 5 |- ( C = if ( C e. _V , C , (/) ) -> ( f u. { <. n , C >. } ) = ( f u. { <. n , if ( C e. _V , C , (/) ) >. } ) )
5	1, 4	syl5eq 2668	. . . 4 |- ( C = if ( C e. _V , C , (/) ) -> G = ( f u. { <. n , if ( C e. _V , C , (/) ) >. } ) )
6	5	fneq1d 5981	. . 3 |- ( C = if ( C e. _V , C , (/) ) -> ( G Fn p <-> ( f u. { <. n , if ( C e. _V , C , (/) ) >. } ) Fn p ) )
7	6	imbi2d 330	. 2 |- ( C = if ( C e. _V , C , (/) ) -> ( ( ( p = suc n /\ f Fn n ) -> G Fn p ) <-> ( ( p = suc n /\ f Fn n ) -> ( f u. { <. n , if ( C e. _V , C , (/) ) >. } ) Fn p ) ) )
8		eqid 2622	. . 3 |- ( f u. { <. n , if ( C e. _V , C , (/) ) >. } ) = ( f u. { <. n , if ( C e. _V , C , (/) ) >. } )
9		0ex 4790	. . . 4 |- (/) e. _V
10	9	elimel 4150	. . 3 |- if ( C e. _V , C , (/) ) e. _V
11	8, 10	bnj927 30839	. 2 |- ( ( p = suc n /\ f Fn n ) -> ( f u. { <. n , if ( C e. _V , C , (/) ) >. } ) Fn p )
12	7, 11	dedth 4139	1 |- ( C e. _V -> ( ( 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   (/)c0 3915   ifcif 4086   {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:  bnj945  30844  bnj910  31018