Theorem bnj554 30969

Description: Technical lemma for bnj852 30991. 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
bnj554.19	|- ( et <-> ( m e. D /\ n = suc m /\ p e. om /\ m = suc p ) )
bnj554.20	|- ( ze <-> ( i e. om /\ suc i e. n /\ m = suc i ) )
bnj554.21	|- K = U_ y e. ( G ` i ) pred ( y , A , R )
bnj554.22	|- L = U_ y e. ( G ` p ) pred ( y , A , R )
bnj554.23	|- K = U_ y e. ( G ` i ) pred ( y , A , R )
bnj554.24	|- L = U_ y e. ( G ` p ) pred ( y , A , R )

Assertion

Ref	Expression
bnj554	|- ( ( et /\ ze ) -> ( ( G ` m ) = L <-> ( G ` suc i ) = K ) )

Distinct variable groups:   y, G   y, i   y, p

Allowed substitution hints:   et( y, i, m, n, p)   ze( y, i, m, n, p)   A( y, i, m, n, p)   D( y, i, m, n, p)   R( y, i, m, n, p)   G( i, m, n, p)   K( y, i, m, n, p)   L( y, i, m, n, p)

Proof of Theorem bnj554

Step	Hyp	Ref	Expression
1		bnj554.19	. . 3 |- ( et <-> ( m e. D /\ n = suc m /\ p e. om /\ m = suc p ) )
2	1	bnj1254 30880	. 2 |- ( et -> m = suc p )
3		bnj554.20	. . 3 |- ( ze <-> ( i e. om /\ suc i e. n /\ m = suc i ) )
4	3	simp3bi 1078	. 2 |- ( ze -> m = suc i )
5		simpr 477	. . 3 |- ( ( m = suc p /\ m = suc i ) -> m = suc i )
6		bnj551 30812	. . 3 |- ( ( m = suc p /\ m = suc i ) -> p = i )
7		fveq2 6191	. . . 4 |- ( m = suc i -> ( G ` m ) = ( G ` suc i ) )
8		fveq2 6191	. . . . 5 |- ( p = i -> ( G ` p ) = ( G ` i ) )
9		iuneq1 4534	. . . . . 6 |- ( ( G ` p ) = ( G ` i ) -> U_ y e. ( G ` p ) pred ( y , A , R ) = U_ y e. ( G ` i ) pred ( y , A , R ) )
10		bnj554.24	. . . . . 6 |- L = U_ y e. ( G ` p ) pred ( y , A , R )
11		bnj554.23	. . . . . 6 |- K = U_ y e. ( G ` i ) pred ( y , A , R )
12	9, 10, 11	3eqtr4g 2681	. . . . 5 |- ( ( G ` p ) = ( G ` i ) -> L = K )
13	8, 12	syl 17	. . . 4 |- ( p = i -> L = K )
14	7, 13	eqeqan12d 2638	. . 3 |- ( ( m = suc i /\ p = i ) -> ( ( G ` m ) = L <-> ( G ` suc i ) = K ) )
15	5, 6, 14	syl2anc 693	. 2 |- ( ( m = suc p /\ m = suc i ) -> ( ( G ` m ) = L <-> ( G ` suc i ) = K ) )
16	2, 4, 15	syl2an 494	1 |- ( ( et /\ ze ) -> ( ( G ` m ) = L <-> ( G ` suc i ) = K ) )

Syntax hints:   -> wi 4   <-> wb 196   /\ wa 384   /\ w3a 1037   = wceq 1483   e. wcel 1990   U_ciun 4520   suc csuc 5725   ` cfv 5888   omcom 7065   /\ w-bnj17 30752   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-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  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-sbc 3436  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-opab 4713  df-eprel 5029  df-fr 5073  df-suc 5729  df-iota 5851  df-fv 5896  df-bnj17 30753

This theorem is referenced by:  bnj558  30972