Theorem bnj229 30954

Description: Technical lemma for bnj517 30955. 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.)

Hypothesis

Ref	Expression
bnj229.1	|- ( ps <-> A. i e. om ( suc i e. N -> ( F ` suc i ) = U_ y e. ( F ` i ) pred ( y , A , R ) ) )

Assertion

Ref	Expression
bnj229	|- ( ( n e. N /\ ( suc m = n /\ m e. om /\ ps ) ) -> ( F ` n ) C_ A )

Distinct variable groups:   A, i, m, y   i, F, m, y   i, N, m   R, i, m

Allowed substitution hints:   ps( y, i, m, n)   A( n)   R( y, n)   F( n)   N( y, n)

Proof of Theorem bnj229

Step	Hyp	Ref	Expression
1		bnj213 30952	. . 3 |- pred ( y , A , R ) C_ A
2	1	bnj226 30802	. 2 |- U_ y e. ( F ` m ) pred ( y , A , R ) C_ A
3		bnj229.1	. . . . . . . 8 |- ( ps <-> A. i e. om ( suc i e. N -> ( F ` suc i ) = U_ y e. ( F ` i ) pred ( y , A , R ) ) )
4	3	bnj222 30953	. . . . . . 7 |- ( ps <-> A. m e. om ( suc m e. N -> ( F ` suc m ) = U_ y e. ( F ` m ) pred ( y , A , R ) ) )
5	4	bnj228 30803	. . . . . 6 |- ( ( m e. om /\ ps ) -> ( suc m e. N -> ( F ` suc m ) = U_ y e. ( F ` m ) pred ( y , A , R ) ) )
6	5	adantl 482	. . . . 5 |- ( ( suc m = n /\ ( m e. om /\ ps ) ) -> ( suc m e. N -> ( F ` suc m ) = U_ y e. ( F ` m ) pred ( y , A , R ) ) )
7		eleq1 2689	. . . . . . 7 |- ( suc m = n -> ( suc m e. N <-> n e. N ) )
8		fveq2 6191	. . . . . . . 8 |- ( suc m = n -> ( F ` suc m ) = ( F ` n ) )
9	8	eqeq1d 2624	. . . . . . 7 |- ( suc m = n -> ( ( F ` suc m ) = U_ y e. ( F ` m ) pred ( y , A , R ) <-> ( F ` n ) = U_ y e. ( F ` m ) pred ( y , A , R ) ) )
10	7, 9	imbi12d 334	. . . . . 6 |- ( suc m = n -> ( ( suc m e. N -> ( F ` suc m ) = U_ y e. ( F ` m ) pred ( y , A , R ) ) <-> ( n e. N -> ( F ` n ) = U_ y e. ( F ` m ) pred ( y , A , R ) ) ) )
11	10	adantr 481	. . . . 5 |- ( ( suc m = n /\ ( m e. om /\ ps ) ) -> ( ( suc m e. N -> ( F ` suc m ) = U_ y e. ( F ` m ) pred ( y , A , R ) ) <-> ( n e. N -> ( F ` n ) = U_ y e. ( F ` m ) pred ( y , A , R ) ) ) )
12	6, 11	mpbid 222	. . . 4 |- ( ( suc m = n /\ ( m e. om /\ ps ) ) -> ( n e. N -> ( F ` n ) = U_ y e. ( F ` m ) pred ( y , A , R ) ) )
13	12	3impb 1260	. . 3 |- ( ( suc m = n /\ m e. om /\ ps ) -> ( n e. N -> ( F ` n ) = U_ y e. ( F ` m ) pred ( y , A , R ) ) )
14	13	impcom 446	. 2 |- ( ( n e. N /\ ( suc m = n /\ m e. om /\ ps ) ) -> ( F ` n ) = U_ y e. ( F ` m ) pred ( y , A , R ) )
15	2, 14	bnj1262 30881	1 |- ( ( n e. N /\ ( suc m = n /\ m e. om /\ ps ) ) -> ( F ` n ) C_ A )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   A.wral 2912   C_ wss 3574   U_ciun 4520   suc csuc 5725   ` cfv 5888   omcom 7065   predc-bnj14 30754

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-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-uni 4437  df-iun 4522  df-br 4654  df-suc 5729  df-iota 5851  df-fv 5896  df-bnj14 30755

This theorem is referenced by: (None)