Theorem unxpdomlem1 8164

Description: Lemma for unxpdom 8167. (Trivial substitution proof.) (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 >. )

Assertion

Ref	Expression
unxpdomlem1	|- ( z e. ( a u. b ) -> ( F ` z ) = if ( z e. a , <. z , if ( z = m , t , s ) >. , <. if ( z = t , n , m ) , z >. ) )

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

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

Proof of Theorem unxpdomlem1

Step	Hyp	Ref	Expression
1		unxpdomlem1.2	. . 3 |- G = if ( x e. a , <. x , if ( x = m , t , s ) >. , <. if ( x = t , n , m ) , x >. )
2		elequ1 1997	. . . 4 |- ( x = z -> ( x e. a <-> z e. a ) )
3		opeq1 4402	. . . . 5 |- ( x = z -> <. x , if ( x = m , t , s ) >. = <. z , if ( x = m , t , s ) >. )
4		equequ1 1952	. . . . . . 7 |- ( x = z -> ( x = m <-> z = m ) )
5	4	ifbid 4108	. . . . . 6 |- ( x = z -> if ( x = m , t , s ) = if ( z = m , t , s ) )
6	5	opeq2d 4409	. . . . 5 |- ( x = z -> <. z , if ( x = m , t , s ) >. = <. z , if ( z = m , t , s ) >. )
7	3, 6	eqtrd 2656	. . . 4 |- ( x = z -> <. x , if ( x = m , t , s ) >. = <. z , if ( z = m , t , s ) >. )
8		equequ1 1952	. . . . . . 7 |- ( x = z -> ( x = t <-> z = t ) )
9	8	ifbid 4108	. . . . . 6 |- ( x = z -> if ( x = t , n , m ) = if ( z = t , n , m ) )
10	9	opeq1d 4408	. . . . 5 |- ( x = z -> <. if ( x = t , n , m ) , x >. = <. if ( z = t , n , m ) , x >. )
11		opeq2 4403	. . . . 5 |- ( x = z -> <. if ( z = t , n , m ) , x >. = <. if ( z = t , n , m ) , z >. )
12	10, 11	eqtrd 2656	. . . 4 |- ( x = z -> <. if ( x = t , n , m ) , x >. = <. if ( z = t , n , m ) , z >. )
13	2, 7, 12	ifbieq12d 4113	. . 3 |- ( x = z -> if ( x e. a , <. x , if ( x = m , t , s ) >. , <. if ( x = t , n , m ) , x >. ) = if ( z e. a , <. z , if ( z = m , t , s ) >. , <. if ( z = t , n , m ) , z >. ) )
14	1, 13	syl5eq 2668	. 2 |- ( x = z -> G = if ( z e. a , <. z , if ( z = m , t , s ) >. , <. if ( z = t , n , m ) , z >. ) )
15		unxpdomlem1.1	. 2 |- F = ( x e. ( a u. b ) |-> G )
16		opex 4932	. . 3 |- <. z , if ( z = m , t , s ) >. e. _V
17		opex 4932	. . 3 |- <. if ( z = t , n , m ) , z >. e. _V
18	16, 17	ifex 4156	. 2 |- if ( z e. a , <. z , if ( z = m , t , s ) >. , <. if ( z = t , n , m ) , z >. ) e. _V
19	14, 15, 18	fvmpt 6282	1 |- ( z e. ( a u. b ) -> ( F ` z ) = if ( z e. a , <. z , if ( z = m , t , s ) >. , <. if ( z = t , n , m ) , z >. ) )

Syntax hints:   -> wi 4   = 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:  unxpdomlem2  8165  unxpdomlem3  8166