Theorem xpima2 5578

Description: The image by a Cartesian product. (Contributed by Thierry Arnoux, 16-Dec-2017.)

Assertion

Ref	Expression
xpima2	|- ( ( A i^i C ) =/= (/) -> ( ( A X. B ) " C ) = B )

Proof of Theorem xpima2

Step	Hyp	Ref	Expression
1		xpima 5576	. 2 |- ( ( A X. B ) " C ) = if ( ( A i^i C ) = (/) , (/) , B )
2		ifnefalse 4098	. 2 |- ( ( A i^i C ) =/= (/) -> if ( ( A i^i C ) = (/) , (/) , B ) = B )
3	1, 2	syl5eq 2668	1 |- ( ( A i^i C ) =/= (/) -> ( ( A X. B ) " C ) = B )

Syntax hints:   -> wi 4   = wceq 1483   =/= wne 2794   i^i cin 3573   (/)c0 3915   ifcif 4086   X. cxp 5112   "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:  xpimasn  5579  ustuqtop1  22045  ustuqtop5  22049  brtrclfv2  38019  aacllem  42547