Theorem oprpiece1res2 22751

Description: Restriction to the second part of a piecewise defined function. (Contributed by Jeff Madsen, 11-Jun-2010.) (Proof shortened by Mario Carneiro, 3-Sep-2015.)

Hypotheses

Ref	Expression
oprpiece1.1	|- A e. RR
oprpiece1.2	|- B e. RR
oprpiece1.3	|- A <_ B
oprpiece1.4	|- R e. _V
oprpiece1.5	|- S e. _V
oprpiece1.6	|- K e. ( A [,] B )
oprpiece1.7	|- F = ( x e. ( A [,] B ) , y e. C |-> if ( x <_ K , R , S ) )
oprpiece1.9	|- ( x = K -> R = P )
oprpiece1.10	|- ( x = K -> S = Q )
oprpiece1.11	|- ( y e. C -> P = Q )
oprpiece1.12	|- G = ( x e. ( K [,] B ) , y e. C |-> S )

Assertion

Ref	Expression
oprpiece1res2	|- ( F |` ( ( K [,] B ) X. C ) ) = G

Distinct variable groups:   x, A, y   x, B, y   x, C, y   x, K, y   x, P   x, Q

Allowed substitution hints:   P( y)   Q( y)   R( x, y)   S( x, y)   F( x, y)   G( x, y)

Proof of Theorem oprpiece1res2

Step	Hyp	Ref	Expression
1		oprpiece1.6	. . . 4 |- K e. ( A [,] B )
2		oprpiece1.1	. . . . . 6 |- A e. RR
3	2	rexri 10097	. . . . 5 |- A e. RR*
4		oprpiece1.2	. . . . . 6 |- B e. RR
5	4	rexri 10097	. . . . 5 |- B e. RR*
6		oprpiece1.3	. . . . 5 |- A <_ B
7		ubicc2 12289	. . . . 5 |- ( ( A e. RR* /\ B e. RR* /\ A <_ B ) -> B e. ( A [,] B ) )
8	3, 5, 6, 7	mp3an 1424	. . . 4 |- B e. ( A [,] B )
9		iccss2 12244	. . . 4 |- ( ( K e. ( A [,] B ) /\ B e. ( A [,] B ) ) -> ( K [,] B ) C_ ( A [,] B ) )
10	1, 8, 9	mp2an 708	. . 3 |- ( K [,] B ) C_ ( A [,] B )
11		ssid 3624	. . 3 |- C C_ C
12		resmpt2 6758	. . 3 |- ( ( ( K [,] B ) C_ ( A [,] B ) /\ C C_ C ) -> ( ( x e. ( A [,] B ) , y e. C |-> if ( x <_ K , R , S ) ) |` ( ( K [,] B ) X. C ) ) = ( x e. ( K [,] B ) , y e. C |-> if ( x <_ K , R , S ) ) )
13	10, 11, 12	mp2an 708	. 2 |- ( ( x e. ( A [,] B ) , y e. C |-> if ( x <_ K , R , S ) ) |` ( ( K [,] B ) X. C ) ) = ( x e. ( K [,] B ) , y e. C |-> if ( x <_ K , R , S ) )
14		oprpiece1.7	. . 3 |- F = ( x e. ( A [,] B ) , y e. C |-> if ( x <_ K , R , S ) )
15	14	reseq1i 5392	. 2 |- ( F |` ( ( K [,] B ) X. C ) ) = ( ( x e. ( A [,] B ) , y e. C |-> if ( x <_ K , R , S ) ) |` ( ( K [,] B ) X. C ) )
16		oprpiece1.12	. . 3 |- G = ( x e. ( K [,] B ) , y e. C |-> S )
17		oprpiece1.11	. . . . . . 7 |- ( y e. C -> P = Q )
18	17	ad2antlr 763	. . . . . 6 |- ( ( ( x e. ( K [,] B ) /\ y e. C ) /\ x <_ K ) -> P = Q )
19		simpr 477	. . . . . . . 8 |- ( ( ( x e. ( K [,] B ) /\ y e. C ) /\ x <_ K ) -> x <_ K )
20	2, 4	elicc2i 12239	. . . . . . . . . . . . 13 |- ( K e. ( A [,] B ) <-> ( K e. RR /\ A <_ K /\ K <_ B ) )
21	20	simp1bi 1076	. . . . . . . . . . . 12 |- ( K e. ( A [,] B ) -> K e. RR )
22	1, 21	ax-mp 5	. . . . . . . . . . 11 |- K e. RR
23	22, 4	elicc2i 12239	. . . . . . . . . 10 |- ( x e. ( K [,] B ) <-> ( x e. RR /\ K <_ x /\ x <_ B ) )
24	23	simp2bi 1077	. . . . . . . . 9 |- ( x e. ( K [,] B ) -> K <_ x )
25	24	ad2antrr 762	. . . . . . . 8 |- ( ( ( x e. ( K [,] B ) /\ y e. C ) /\ x <_ K ) -> K <_ x )
26	23	simp1bi 1076	. . . . . . . . . 10 |- ( x e. ( K [,] B ) -> x e. RR )
27	26	ad2antrr 762	. . . . . . . . 9 |- ( ( ( x e. ( K [,] B ) /\ y e. C ) /\ x <_ K ) -> x e. RR )
28		letri3 10123	. . . . . . . . 9 |- ( ( x e. RR /\ K e. RR ) -> ( x = K <-> ( x <_ K /\ K <_ x ) ) )
29	27, 22, 28	sylancl 694	. . . . . . . 8 |- ( ( ( x e. ( K [,] B ) /\ y e. C ) /\ x <_ K ) -> ( x = K <-> ( x <_ K /\ K <_ x ) ) )
30	19, 25, 29	mpbir2and 957	. . . . . . 7 |- ( ( ( x e. ( K [,] B ) /\ y e. C ) /\ x <_ K ) -> x = K )
31		oprpiece1.9	. . . . . . 7 |- ( x = K -> R = P )
32	30, 31	syl 17	. . . . . 6 |- ( ( ( x e. ( K [,] B ) /\ y e. C ) /\ x <_ K ) -> R = P )
33		oprpiece1.10	. . . . . . 7 |- ( x = K -> S = Q )
34	30, 33	syl 17	. . . . . 6 |- ( ( ( x e. ( K [,] B ) /\ y e. C ) /\ x <_ K ) -> S = Q )
35	18, 32, 34	3eqtr4d 2666	. . . . 5 |- ( ( ( x e. ( K [,] B ) /\ y e. C ) /\ x <_ K ) -> R = S )
36		eqidd 2623	. . . . 5 |- ( ( ( x e. ( K [,] B ) /\ y e. C ) /\ -. x <_ K ) -> S = S )
37	35, 36	ifeqda 4121	. . . 4 |- ( ( x e. ( K [,] B ) /\ y e. C ) -> if ( x <_ K , R , S ) = S )
38	37	mpt2eq3ia 6720	. . 3 |- ( x e. ( K [,] B ) , y e. C |-> if ( x <_ K , R , S ) ) = ( x e. ( K [,] B ) , y e. C |-> S )
39	16, 38	eqtr4i 2647	. 2 |- G = ( x e. ( K [,] B ) , y e. C |-> if ( x <_ K , R , S ) )
40	13, 15, 39	3eqtr4i 2654	1 |- ( F |` ( ( K [,] B ) X. C ) ) = G

Syntax hints:   -. wn 3   -> wi 4   <-> wb 196   /\ wa 384   = wceq 1483   e. wcel 1990   _Vcvv 3200   C_ wss 3574   ifcif 4086   class class class wbr 4653   X. cxp 5112   |` cres 5116  (class class class)co 6650   |-> cmpt2 6652   RRcr 9935   RR*cxr 10073   <_ cle 10075   [,]cicc 12178

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-pow 4843  ax-pr 4906  ax-un 6949  ax-cnex 9992  ax-resscn 9993  ax-pre-lttri 10010  ax-pre-lttrn 10011

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  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-nel 2898  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-po 5035  df-so 5036  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-er 7742  df-en 7956  df-dom 7957  df-sdom 7958  df-pnf 10076  df-mnf 10077  df-xr 10078  df-ltxr 10079  df-le 10080  df-icc 12182

This theorem is referenced by: (None)