Theorem bnj1097 31049

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
bnj1097.1	|- ( ph <-> ( f ` (/) ) = pred ( X , A , R ) )
bnj1097.3	|- ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) )
bnj1097.5	|- ( ta <-> ( B e. _V /\ TrFo ( B , A , R ) /\ pred ( X , A , R ) C_ B ) )

Assertion

Ref	Expression
bnj1097	|- ( ( i = (/) /\ ( ( th /\ ta /\ ch ) /\ ph0 ) ) -> ( f ` i ) C_ B )

Proof of Theorem bnj1097

Step	Hyp	Ref	Expression
1		bnj1097.3	. . . . . . . 8 |- ( ch <-> ( n e. D /\ f Fn n /\ ph /\ ps ) )
2		bnj1097.1	. . . . . . . . 9 |- ( ph <-> ( f ` (/) ) = pred ( X , A , R ) )
3	2	biimpi 206	. . . . . . . 8 |- ( ph -> ( f ` (/) ) = pred ( X , A , R ) )
4	1, 3	bnj771 30834	. . . . . . 7 |- ( ch -> ( f ` (/) ) = pred ( X , A , R ) )
5	4	3ad2ant3 1084	. . . . . 6 |- ( ( th /\ ta /\ ch ) -> ( f ` (/) ) = pred ( X , A , R ) )
6	5	adantr 481	. . . . 5 |- ( ( ( th /\ ta /\ ch ) /\ ph0 ) -> ( f ` (/) ) = pred ( X , A , R ) )
7		bnj1097.5	. . . . . . . 8 |- ( ta <-> ( B e. _V /\ TrFo ( B , A , R ) /\ pred ( X , A , R ) C_ B ) )
8	7	simp3bi 1078	. . . . . . 7 |- ( ta -> pred ( X , A , R ) C_ B )
9	8	3ad2ant2 1083	. . . . . 6 |- ( ( th /\ ta /\ ch ) -> pred ( X , A , R ) C_ B )
10	9	adantr 481	. . . . 5 |- ( ( ( th /\ ta /\ ch ) /\ ph0 ) -> pred ( X , A , R ) C_ B )
11	6, 10	jca 554	. . . 4 |- ( ( ( th /\ ta /\ ch ) /\ ph0 ) -> ( ( f ` (/) ) = pred ( X , A , R ) /\ pred ( X , A , R ) C_ B ) )
12	11	anim2i 593	. . 3 |- ( ( i = (/) /\ ( ( th /\ ta /\ ch ) /\ ph0 ) ) -> ( i = (/) /\ ( ( f ` (/) ) = pred ( X , A , R ) /\ pred ( X , A , R ) C_ B ) ) )
13		3anass 1042	. . 3 |- ( ( i = (/) /\ ( f ` (/) ) = pred ( X , A , R ) /\ pred ( X , A , R ) C_ B ) <-> ( i = (/) /\ ( ( f ` (/) ) = pred ( X , A , R ) /\ pred ( X , A , R ) C_ B ) ) )
14	12, 13	sylibr 224	. 2 |- ( ( i = (/) /\ ( ( th /\ ta /\ ch ) /\ ph0 ) ) -> ( i = (/) /\ ( f ` (/) ) = pred ( X , A , R ) /\ pred ( X , A , R ) C_ B ) )
15		fveq2 6191	. . . . . . 7 |- ( i = (/) -> ( f ` i ) = ( f ` (/) ) )
16	15	eqeq1d 2624	. . . . . 6 |- ( i = (/) -> ( ( f ` i ) = pred ( X , A , R ) <-> ( f ` (/) ) = pred ( X , A , R ) ) )
17	16	biimpar 502	. . . . 5 |- ( ( i = (/) /\ ( f ` (/) ) = pred ( X , A , R ) ) -> ( f ` i ) = pred ( X , A , R ) )
18	17	adantr 481	. . . 4 |- ( ( ( i = (/) /\ ( f ` (/) ) = pred ( X , A , R ) ) /\ pred ( X , A , R ) C_ B ) -> ( f ` i ) = pred ( X , A , R ) )
19		simpr 477	. . . 4 |- ( ( ( i = (/) /\ ( f ` (/) ) = pred ( X , A , R ) ) /\ pred ( X , A , R ) C_ B ) -> pred ( X , A , R ) C_ B )
20	18, 19	eqsstrd 3639	. . 3 |- ( ( ( i = (/) /\ ( f ` (/) ) = pred ( X , A , R ) ) /\ pred ( X , A , R ) C_ B ) -> ( f ` i ) C_ B )
21	20	3impa 1259	. 2 |- ( ( i = (/) /\ ( f ` (/) ) = pred ( X , A , R ) /\ pred ( X , A , R ) C_ B ) -> ( f ` i ) C_ B )
22	14, 21	syl 17	1 |- ( ( i = (/) /\ ( ( th /\ ta /\ ch ) /\ ph0 ) ) -> ( f ` i ) C_ B )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   _Vcvv 3200   C_ wss 3574   (/)c0 3915   Fn wfn 5883   ` cfv 5888   /\ w-bnj17 30752   predc-bnj14 30754   TrFow-bnj19 30762

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

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-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  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-uni 4437  df-br 4654  df-iota 5851  df-fv 5896  df-bnj17 30753

This theorem is referenced by:  bnj1030  31055