Theorem xpima 5576

Description: The image by a constant function (or other Cartesian product). (Contributed by Thierry Arnoux, 4-Feb-2017.)

Assertion

Ref	Expression
xpima	|- ( ( A X. B ) " C ) = if ( ( A i^i C ) = (/) , (/) , B )

Proof of Theorem xpima

Step	Hyp	Ref	Expression
1		exmid 431	. . 3 |- ( ( A i^i C ) = (/) \/ -. ( A i^i C ) = (/) )
2		df-ima 5127	. . . . . . . 8 |- ( ( A X. B ) " C ) = ran ( ( A X. B ) |` C )
3		df-res 5126	. . . . . . . . 9 |- ( ( A X. B ) |` C ) = ( ( A X. B ) i^i ( C X. _V ) )
4	3	rneqi 5352	. . . . . . . 8 |- ran ( ( A X. B ) |` C ) = ran ( ( A X. B ) i^i ( C X. _V ) )
5	2, 4	eqtri 2644	. . . . . . 7 |- ( ( A X. B ) " C ) = ran ( ( A X. B ) i^i ( C X. _V ) )
6		inxp 5254	. . . . . . . 8 |- ( ( A X. B ) i^i ( C X. _V ) ) = ( ( A i^i C ) X. ( B i^i _V ) )
7	6	rneqi 5352	. . . . . . 7 |- ran ( ( A X. B ) i^i ( C X. _V ) ) = ran ( ( A i^i C ) X. ( B i^i _V ) )
8		inv1 3970	. . . . . . . . 9 |- ( B i^i _V ) = B
9	8	xpeq2i 5136	. . . . . . . 8 |- ( ( A i^i C ) X. ( B i^i _V ) ) = ( ( A i^i C ) X. B )
10	9	rneqi 5352	. . . . . . 7 |- ran ( ( A i^i C ) X. ( B i^i _V ) ) = ran ( ( A i^i C ) X. B )
11	5, 7, 10	3eqtri 2648	. . . . . 6 |- ( ( A X. B ) " C ) = ran ( ( A i^i C ) X. B )
12		xpeq1 5128	. . . . . . . . 9 |- ( ( A i^i C ) = (/) -> ( ( A i^i C ) X. B ) = ( (/) X. B ) )
13		0xp 5199	. . . . . . . . 9 |- ( (/) X. B ) = (/)
14	12, 13	syl6eq 2672	. . . . . . . 8 |- ( ( A i^i C ) = (/) -> ( ( A i^i C ) X. B ) = (/) )
15	14	rneqd 5353	. . . . . . 7 |- ( ( A i^i C ) = (/) -> ran ( ( A i^i C ) X. B ) = ran (/) )
16		rn0 5377	. . . . . . 7 |- ran (/) = (/)
17	15, 16	syl6eq 2672	. . . . . 6 |- ( ( A i^i C ) = (/) -> ran ( ( A i^i C ) X. B ) = (/) )
18	11, 17	syl5eq 2668	. . . . 5 |- ( ( A i^i C ) = (/) -> ( ( A X. B ) " C ) = (/) )
19	18	ancli 574	. . . 4 |- ( ( A i^i C ) = (/) -> ( ( A i^i C ) = (/) /\ ( ( A X. B ) " C ) = (/) ) )
20		df-ne 2795	. . . . . . 7 |- ( ( A i^i C ) =/= (/) <-> -. ( A i^i C ) = (/) )
21		rnxp 5564	. . . . . . 7 |- ( ( A i^i C ) =/= (/) -> ran ( ( A i^i C ) X. B ) = B )
22	20, 21	sylbir 225	. . . . . 6 |- ( -. ( A i^i C ) = (/) -> ran ( ( A i^i C ) X. B ) = B )
23	11, 22	syl5eq 2668	. . . . 5 |- ( -. ( A i^i C ) = (/) -> ( ( A X. B ) " C ) = B )
24	23	ancli 574	. . . 4 |- ( -. ( A i^i C ) = (/) -> ( -. ( A i^i C ) = (/) /\ ( ( A X. B ) " C ) = B ) )
25	19, 24	orim12i 538	. . 3 |- ( ( ( A i^i C ) = (/) \/ -. ( A i^i C ) = (/) ) -> ( ( ( A i^i C ) = (/) /\ ( ( A X. B ) " C ) = (/) ) \/ ( -. ( A i^i C ) = (/) /\ ( ( A X. B ) " C ) = B ) ) )
26	1, 25	ax-mp 5	. 2 |- ( ( ( A i^i C ) = (/) /\ ( ( A X. B ) " C ) = (/) ) \/ ( -. ( A i^i C ) = (/) /\ ( ( A X. B ) " C ) = B ) )
27		eqif 4126	. 2 |- ( ( ( A X. B ) " C ) = if ( ( A i^i C ) = (/) , (/) , B ) <-> ( ( ( A i^i C ) = (/) /\ ( ( A X. B ) " C ) = (/) ) \/ ( -. ( A i^i C ) = (/) /\ ( ( A X. B ) " C ) = B ) ) )
28	26, 27	mpbir 221	1 |- ( ( A X. B ) " C ) = if ( ( A i^i C ) = (/) , (/) , B )

Syntax hints:   -. wn 3   \/ wo 383   /\ wa 384   = wceq 1483   =/= wne 2794   _Vcvv 3200   i^i cin 3573   (/)c0 3915   ifcif 4086   X. cxp 5112   ran crn 5115   |` cres 5116   "cima 5117

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  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-ne 2795  df-ral 2917  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-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127

This theorem is referenced by:  xpima1  5577  xpima2  5578  imadifxp  29414  bj-xpimasn  32942