Theorem bnj1133 31057

Description: Technical lemma for bnj69 31078. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj1133.3	|- D = ( om \ { (/) } )
bnj1133.5	|- ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) )
bnj1133.7	|- ( ta <-> A. j e. n ( j _E i -> [. j / i ]. th ) )
bnj1133.8	|- ( ( i e. n /\ ta ) -> th )

Assertion

Ref	Expression
bnj1133	|- ( ch -> A. i e. n th )

Distinct variable groups:   i, j, n   th, j

Allowed substitution hints:   ph( f, i, j, n)   ps( f, i, j, n)   ch( f, i, j, n)   th( f, i, n)   ta( f, i, j, n)   D( f, i, j, n)

Proof of Theorem bnj1133

Step	Hyp	Ref	Expression
1		bnj1133.5	. . 3 |- ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) )
2		bnj1133.3	. . . 4 |- D = ( om \ { (/) } )
3	2	bnj1071 31045	. . 3 |- ( n e. D -> _E Fr n )
4	1, 3	bnj769 30832	. 2 |- ( ch -> _E Fr n )
5		impexp 462	. . . . . 6 |- ( ( ( i e. n /\ ta ) -> th ) <-> ( i e. n -> ( ta -> th ) ) )
6	5	bicomi 214	. . . . 5 |- ( ( i e. n -> ( ta -> th ) ) <-> ( ( i e. n /\ ta ) -> th ) )
7	6	albii 1747	. . . 4 |- ( A. i ( i e. n -> ( ta -> th ) ) <-> A. i ( ( i e. n /\ ta ) -> th ) )
8		bnj1133.8	. . . 4 |- ( ( i e. n /\ ta ) -> th )
9	7, 8	mpgbir 1726	. . 3 |- A. i ( i e. n -> ( ta -> th ) )
10		df-ral 2917	. . 3 |- ( A. i e. n ( ta -> th ) <-> A. i ( i e. n -> ( ta -> th ) ) )
11	9, 10	mpbir 221	. 2 |- A. i e. n ( ta -> th )
12		vex 3203	. . 3 |- n e. _V
13		bnj1133.7	. . 3 |- ( ta <-> A. j e. n ( j _E i -> [. j / i ]. th ) )
14	12, 13	bnj110 30928	. 2 |- ( ( _E Fr n /\ A. i e. n ( ta -> th ) ) -> A. i e. n th )
15	4, 11, 14	sylancl 694	1 |- ( ch -> A. i e. n th )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   A.wal 1481   = wceq 1483   e. wcel 1990   A.wral 2912   [.wsbc 3435   \ cdif 3571   (/)c0 3915   {csn 4177   class class class wbr 4653   _E cep 5028   Fr wfr 5070   Fn wfn 5883   omcom 7065   /\ w-bnj17 30752

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-fal 1489  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-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  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-om 7066  df-bnj17 30753

This theorem is referenced by:  bnj1128  31058