Theorem xpexgALT 6290

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 4838 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 4024	. . . 4 |- U_ y e. B { y } = B
2	1	xpeq2i 4744	. . 3 |- ( A X. U_ y e. B { y } ) = ( A X. B )
3		xpiundi 4782	. . 3 |- ( A X. U_ y e. B { y } ) = U_ y e. B ( A X. { y } )
4	2, 3	eqtr3i 2252	. 2 |- ( A X. B ) = U_ y e. B ( A X. { y } )
5		id 19	. . 3 |- ( B e. W -> B e. W )
6		fconstmpt 4771	. . . . 5 |- ( A X. { y } ) = ( x e. A |-> y )
7		mptexg 5874	. . . . 5 |- ( A e. V -> ( x e. A |-> y ) e. _V )
8	6, 7	eqeltrid 2316	. . . 4 |- ( A e. V -> ( A X. { y } ) e. _V )
9	8	ralrimivw 2604	. . 3 |- ( A e. V -> A. y e. B ( A X. { y } ) e. _V )
10		iunexg 6276	. . 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 290	. 2 |- ( ( A e. V /\ B e. W ) -> U_ y e. B ( A X. { y } ) e. _V )
12	4, 11	eqeltrid 2316	1 |- ( ( A e. V /\ B e. W ) -> ( A X. B ) e. _V )

Syntax hints:   -> wi 4   /\ wa 104   e. wcel 2200   A.wral 2508   _Vcvv 2800   {csn 3667   U_ciun 3968   |-> cmpt 4148   X. cxp 4721

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4202  ax-sep 4205  ax-pow 4262  ax-pr 4297  ax-un 4528

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2802  df-sbc 3030  df-csb 3126  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-uni 3892  df-iun 3970  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-res 4735  df-ima 4736  df-iota 5284  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331  df-fv 5332

This theorem is referenced by: (None)