Theorem xpexgALT 6031

Description: The cross product of two sets is a set. Proposition 6.2 of [TakeutiZaring] p. 23. This version is proven using Replacement; see xpexg 4653 for a version that uses the Power Set axiom instead. (Contributed by Mario Carneiro, 20-May-2013.) (Proof modification is discouraged.) (New usage is discouraged.)

Assertion

Ref	Expression
xpexgALT	|- ( ( A e. V /\ B e. W ) -> ( A X. B ) e. _V )

Proof of Theorem xpexgALT

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		iunid 3868	. . . 4 |- U_ y e. B { y } = B
2	1	xpeq2i 4560	. . 3 |- ( A X. U_ y e. B { y } ) = ( A X. B )
3		xpiundi 4597	. . 3 |- ( A X. U_ y e. B { y } ) = U_ y e. B ( A X. { y } )
4	2, 3	eqtr3i 2162	. 2 |- ( A X. B ) = U_ y e. B ( A X. { y } )
5		id 19	. . 3 |- ( B e. W -> B e. W )
6		fconstmpt 4586	. . . . 5 |- ( A X. { y } ) = ( x e. A |-> y )
7		mptexg 5645	. . . . 5 |- ( A e. V -> ( x e. A |-> y ) e. _V )
8	6, 7	eqeltrid 2226	. . . 4 |- ( A e. V -> ( A X. { y } ) e. _V )
9	8	ralrimivw 2506	. . 3 |- ( A e. V -> A. y e. B ( A X. { y } ) e. _V )
10		iunexg 6017	. . 3 |- ( ( B e. W /\ A. y e. B ( A X. { y } ) e. _V ) -> U_ y e. B ( A X. { y } ) e. _V )
11	5, 9, 10	syl2anr 288	. 2 |- ( ( A e. V /\ B e. W ) -> U_ y e. B ( A X. { y } ) e. _V )
12	4, 11	eqeltrid 2226	1 |- ( ( A e. V /\ B e. W ) -> ( A X. B ) e. _V )

Syntax hints:   -> wi 4   /\ wa 103   e. wcel 1480   A.wral 2416   _Vcvv 2686   {csn 3527   U_ciun 3813   |-> cmpt 3989   X. cxp 4537

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-coll 4043  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-reu 2423  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-iun 3815  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

This theorem is referenced by: (None)