Theorem unxpdomlem2 8165

Description: Lemma for unxpdom 8167. (Contributed by Mario Carneiro, 13-Jan-2013.)

Hypotheses

Ref	Expression
unxpdomlem1.1	|- F = ( x e. ( a u. b ) |-> G )
unxpdomlem1.2	|- G = if ( x e. a , <. x , if ( x = m , t , s ) >. , <. if ( x = t , n , m ) , x >. )
unxpdomlem2.1	|- ( ph -> w e. ( a u. b ) )
unxpdomlem2.2	|- ( ph -> -. m = n )
unxpdomlem2.3	|- ( ph -> -. s = t )

Assertion

Ref	Expression
unxpdomlem2	|- ( ( ph /\ ( z e. a /\ -. w e. a ) ) -> -. ( F ` z ) = ( F ` w ) )

Distinct variable groups:   w, F, z   a, b, m, n, s, t, w, x, z

Allowed substitution hints:   ph( x, z, w, t, m, n, s, a, b)   F( x, t, m, n, s, a, b)   G( x, z, w, t, m, n, s, a, b)

Proof of Theorem unxpdomlem2

Step	Hyp	Ref	Expression
1		unxpdomlem2.3	. . 3 |- ( ph -> -. s = t )
2	1	adantr 481	. 2 |- ( ( ph /\ ( z e. a /\ -. w e. a ) ) -> -. s = t )
3		elun1 3780	. . . . . . . . . 10 |- ( z e. a -> z e. ( a u. b ) )
4	3	ad2antrl 764	. . . . . . . . 9 |- ( ( ph /\ ( z e. a /\ -. w e. a ) ) -> z e. ( a u. b ) )
5		unxpdomlem1.1	. . . . . . . . . 10 |- F = ( x e. ( a u. b ) |-> G )
6		unxpdomlem1.2	. . . . . . . . . 10 |- G = if ( x e. a , <. x , if ( x = m , t , s ) >. , <. if ( x = t , n , m ) , x >. )
7	5, 6	unxpdomlem1 8164	. . . . . . . . 9 |- ( z e. ( a u. b ) -> ( F ` z ) = if ( z e. a , <. z , if ( z = m , t , s ) >. , <. if ( z = t , n , m ) , z >. ) )
8	4, 7	syl 17	. . . . . . . 8 |- ( ( ph /\ ( z e. a /\ -. w e. a ) ) -> ( F ` z ) = if ( z e. a , <. z , if ( z = m , t , s ) >. , <. if ( z = t , n , m ) , z >. ) )
9		iftrue 4092	. . . . . . . . 9 |- ( z e. a -> if ( z e. a , <. z , if ( z = m , t , s ) >. , <. if ( z = t , n , m ) , z >. ) = <. z , if ( z = m , t , s ) >. )
10	9	ad2antrl 764	. . . . . . . 8 |- ( ( ph /\ ( z e. a /\ -. w e. a ) ) -> if ( z e. a , <. z , if ( z = m , t , s ) >. , <. if ( z = t , n , m ) , z >. ) = <. z , if ( z = m , t , s ) >. )
11	8, 10	eqtrd 2656	. . . . . . 7 |- ( ( ph /\ ( z e. a /\ -. w e. a ) ) -> ( F ` z ) = <. z , if ( z = m , t , s ) >. )
12		unxpdomlem2.1	. . . . . . . . . 10 |- ( ph -> w e. ( a u. b ) )
13	12	adantr 481	. . . . . . . . 9 |- ( ( ph /\ ( z e. a /\ -. w e. a ) ) -> w e. ( a u. b ) )
14	5, 6	unxpdomlem1 8164	. . . . . . . . 9 |- ( w e. ( a u. b ) -> ( F ` w ) = if ( w e. a , <. w , if ( w = m , t , s ) >. , <. if ( w = t , n , m ) , w >. ) )
15	13, 14	syl 17	. . . . . . . 8 |- ( ( ph /\ ( z e. a /\ -. w e. a ) ) -> ( F ` w ) = if ( w e. a , <. w , if ( w = m , t , s ) >. , <. if ( w = t , n , m ) , w >. ) )
16		iffalse 4095	. . . . . . . . 9 |- ( -. w e. a -> if ( w e. a , <. w , if ( w = m , t , s ) >. , <. if ( w = t , n , m ) , w >. ) = <. if ( w = t , n , m ) , w >. )
17	16	ad2antll 765	. . . . . . . 8 |- ( ( ph /\ ( z e. a /\ -. w e. a ) ) -> if ( w e. a , <. w , if ( w = m , t , s ) >. , <. if ( w = t , n , m ) , w >. ) = <. if ( w = t , n , m ) , w >. )
18	15, 17	eqtrd 2656	. . . . . . 7 |- ( ( ph /\ ( z e. a /\ -. w e. a ) ) -> ( F ` w ) = <. if ( w = t , n , m ) , w >. )
19	11, 18	eqeq12d 2637	. . . . . 6 |- ( ( ph /\ ( z e. a /\ -. w e. a ) ) -> ( ( F ` z ) = ( F ` w ) <-> <. z , if ( z = m , t , s ) >. = <. if ( w = t , n , m ) , w >. ) )
20	19	biimpa 501	. . . . 5 |- ( ( ( ph /\ ( z e. a /\ -. w e. a ) ) /\ ( F ` z ) = ( F ` w ) ) -> <. z , if ( z = m , t , s ) >. = <. if ( w = t , n , m ) , w >. )
21		vex 3203	. . . . . 6 |- z e. _V
22		vex 3203	. . . . . . 7 |- t e. _V
23		vex 3203	. . . . . . 7 |- s e. _V
24	22, 23	ifex 4156	. . . . . 6 |- if ( z = m , t , s ) e. _V
25	21, 24	opth 4945	. . . . 5 |- ( <. z , if ( z = m , t , s ) >. = <. if ( w = t , n , m ) , w >. <-> ( z = if ( w = t , n , m ) /\ if ( z = m , t , s ) = w ) )
26	20, 25	sylib 208	. . . 4 |- ( ( ( ph /\ ( z e. a /\ -. w e. a ) ) /\ ( F ` z ) = ( F ` w ) ) -> ( z = if ( w = t , n , m ) /\ if ( z = m , t , s ) = w ) )
27	26	simprd 479	. . 3 |- ( ( ( ph /\ ( z e. a /\ -. w e. a ) ) /\ ( F ` z ) = ( F ` w ) ) -> if ( z = m , t , s ) = w )
28		iftrue 4092	. . . . . . 7 |- ( z = m -> if ( z = m , t , s ) = t )
29	27	eqeq1d 2624	. . . . . . 7 |- ( ( ( ph /\ ( z e. a /\ -. w e. a ) ) /\ ( F ` z ) = ( F ` w ) ) -> ( if ( z = m , t , s ) = t <-> w = t ) )
30	28, 29	syl5ib 234	. . . . . 6 |- ( ( ( ph /\ ( z e. a /\ -. w e. a ) ) /\ ( F ` z ) = ( F ` w ) ) -> ( z = m -> w = t ) )
31		iftrue 4092	. . . . . . 7 |- ( w = t -> if ( w = t , n , m ) = n )
32	26	simpld 475	. . . . . . . 8 |- ( ( ( ph /\ ( z e. a /\ -. w e. a ) ) /\ ( F ` z ) = ( F ` w ) ) -> z = if ( w = t , n , m ) )
33	32	eqeq1d 2624	. . . . . . 7 |- ( ( ( ph /\ ( z e. a /\ -. w e. a ) ) /\ ( F ` z ) = ( F ` w ) ) -> ( z = n <-> if ( w = t , n , m ) = n ) )
34	31, 33	syl5ibr 236	. . . . . 6 |- ( ( ( ph /\ ( z e. a /\ -. w e. a ) ) /\ ( F ` z ) = ( F ` w ) ) -> ( w = t -> z = n ) )
35	30, 34	syld 47	. . . . 5 |- ( ( ( ph /\ ( z e. a /\ -. w e. a ) ) /\ ( F ` z ) = ( F ` w ) ) -> ( z = m -> z = n ) )
36		unxpdomlem2.2	. . . . . . 7 |- ( ph -> -. m = n )
37	36	ad2antrr 762	. . . . . 6 |- ( ( ( ph /\ ( z e. a /\ -. w e. a ) ) /\ ( F ` z ) = ( F ` w ) ) -> -. m = n )
38		equequ1 1952	. . . . . . 7 |- ( z = m -> ( z = n <-> m = n ) )
39	38	notbid 308	. . . . . 6 |- ( z = m -> ( -. z = n <-> -. m = n ) )
40	37, 39	syl5ibrcom 237	. . . . 5 |- ( ( ( ph /\ ( z e. a /\ -. w e. a ) ) /\ ( F ` z ) = ( F ` w ) ) -> ( z = m -> -. z = n ) )
41	35, 40	pm2.65d 187	. . . 4 |- ( ( ( ph /\ ( z e. a /\ -. w e. a ) ) /\ ( F ` z ) = ( F ` w ) ) -> -. z = m )
42	41	iffalsed 4097	. . 3 |- ( ( ( ph /\ ( z e. a /\ -. w e. a ) ) /\ ( F ` z ) = ( F ` w ) ) -> if ( z = m , t , s ) = s )
43		iffalse 4095	. . . . 5 |- ( -. w = t -> if ( w = t , n , m ) = m )
44	32	eqeq1d 2624	. . . . 5 |- ( ( ( ph /\ ( z e. a /\ -. w e. a ) ) /\ ( F ` z ) = ( F ` w ) ) -> ( z = m <-> if ( w = t , n , m ) = m ) )
45	43, 44	syl5ibr 236	. . . 4 |- ( ( ( ph /\ ( z e. a /\ -. w e. a ) ) /\ ( F ` z ) = ( F ` w ) ) -> ( -. w = t -> z = m ) )
46	41, 45	mt3d 140	. . 3 |- ( ( ( ph /\ ( z e. a /\ -. w e. a ) ) /\ ( F ` z ) = ( F ` w ) ) -> w = t )
47	27, 42, 46	3eqtr3d 2664	. 2 |- ( ( ( ph /\ ( z e. a /\ -. w e. a ) ) /\ ( F ` z ) = ( F ` w ) ) -> s = t )
48	2, 47	mtand 691	1 |- ( ( ph /\ ( z e. a /\ -. w e. a ) ) -> -. ( F ` z ) = ( F ` w ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 384   = wceq 1483   e. wcel 1990   u. cun 3572   ifcif 4086   <.cop 4183   |-> cmpt 4729   ` cfv 5888

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

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-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-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-iota 5851  df-fun 5890  df-fv 5896

This theorem is referenced by:  unxpdomlem3  8166