Theorem xpdom1 6729

Description: Dominance law for Cartesian product. Theorem 6L(c) of [Enderton] p. 149. (Contributed by NM, 28-Sep-2004.) (Revised by NM, 29-Mar-2006.) (Revised by Mario Carneiro, 7-May-2015.)

Hypothesis

Ref	Expression
xpdom1.2	|- C e. _V

Assertion

Ref	Expression
xpdom1	|- ( A ~<_ B -> ( A X. C ) ~<_ ( B X. C ) )

Proof of Theorem xpdom1

Step	Hyp	Ref	Expression
1		xpdom1.2	. 2 |- C e. _V
2		xpdom1g 6727	. 2 |- ( ( C e. _V /\ A ~<_ B ) -> ( A X. C ) ~<_ ( B X. C ) )
3	1, 2	mpan 420	1 |- ( A ~<_ B -> ( A X. C ) ~<_ ( B X. C ) )

Syntax hints:   -> wi 4   e. wcel 1480   _Vcvv 2686   class class class wbr 3929   X. cxp 4537   ~<_ cdom 6633

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-1st 6038  df-2nd 6039  df-en 6635  df-dom 6636

This theorem is referenced by: (None)