Theorem xpexgALT 6098

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 4717 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 3920	. . . 4 |- U_ y e. B { y } = B
2	1	xpeq2i 4624	. . 3 |- ( A X. U_ y e. B { y } ) = ( A X. B )
3		xpiundi 4661	. . 3 |- ( A X. U_ y e. B { y } ) = U_ y e. B ( A X. { y } )
4	2, 3	eqtr3i 2188	. 2 |- ( A X. B ) = U_ y e. B ( A X. { y } )
5		id 19	. . 3 |- ( B e. W -> B e. W )
6		fconstmpt 4650	. . . . 5 |- ( A X. { y } ) = ( x e. A |-> y )
7		mptexg 5709	. . . . 5 |- ( A e. V -> ( x e. A |-> y ) e. _V )
8	6, 7	eqeltrid 2252	. . . 4 |- ( A e. V -> ( A X. { y } ) e. _V )
9	8	ralrimivw 2539	. . 3 |- ( A e. V -> A. y e. B ( A X. { y } ) e. _V )
10		iunexg 6084	. . 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 2252	1 |- ( ( A e. V /\ B e. W ) -> ( A X. B ) e. _V )

Syntax hints:   -> wi 4   /\ wa 103   e. wcel 2136   A.wral 2443   _Vcvv 2725   {csn 3575   U_ciun 3865   |-> cmpt 4042   X. cxp 4601

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-coll 4096  ax-sep 4099  ax-pow 4152  ax-pr 4186  ax-un 4410

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2296  df-ral 2448  df-rex 2449  df-reu 2450  df-rab 2452  df-v 2727  df-sbc 2951  df-csb 3045  df-un 3119  df-in 3121  df-ss 3128  df-pw 3560  df-sn 3581  df-pr 3582  df-op 3584  df-uni 3789  df-iun 3867  df-br 3982  df-opab 4043  df-mpt 4044  df-id 4270  df-xp 4609  df-rel 4610  df-cnv 4611  df-co 4612  df-dm 4613  df-rn 4614  df-res 4615  df-ima 4616  df-iota 5152  df-fun 5189  df-fn 5190  df-f 5191  df-f1 5192  df-fo 5193  df-f1o 5194  df-fv 5195

This theorem is referenced by: (None)