Theorem xpexgALT 5760

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 4452 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 3712	. . . 4 |- U_ y e. B { y } = B
2	1	xpeq2i 4366	. . 3 |- ( A X. U_ y e. B { y } ) = ( A X. B )
3		xpiundi 4398	. . 3 |- ( A X. U_ y e. B { y } ) = U_ y e. B ( A X. { y } )
4	2, 3	eqtr3i 2062	. 2 |- ( A X. B ) = U_ y e. B ( A X. { y } )
5		id 19	. . 3 |- ( B e. W -> B e. W )
6		fconstmpt 4387	. . . . 5 |- ( A X. { y } ) = ( x e. A |-> y )
7		mptexg 5386	. . . . 5 |- ( A e. V -> ( x e. A |-> y ) e. _V )
8	6, 7	syl5eqel 2124	. . . 4 |- ( A e. V -> ( A X. { y } ) e. _V )
9	8	ralrimivw 2393	. . 3 |- ( A e. V -> A. y e. B ( A X. { y } ) e. _V )
10		iunexg 5746	. . 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 274	. 2 |- ( ( A e. V /\ B e. W ) -> U_ y e. B ( A X. { y } ) e. _V )
12	4, 11	syl5eqel 2124	1 |- ( ( A e. V /\ B e. W ) -> ( A X. B ) e. _V )

Syntax hints:   -> wi 4   /\ wa 97   e. wcel 1393   A.wral 2306   _Vcvv 2557   {csn 3375   U_ciun 3657   |-> cmpt 3818   X. cxp 4343

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-coll 3872  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-reu 2313  df-rab 2315  df-v 2559  df-sbc 2765  df-csb 2853  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-iun 3659  df-br 3765  df-opab 3819  df-mpt 3820  df-id 4030  df-xp 4351  df-rel 4352  df-cnv 4353  df-co 4354  df-dm 4355  df-rn 4356  df-res 4357  df-ima 4358  df-iota 4867  df-fun 4904  df-fn 4905  df-f 4906  df-f1 4907  df-fo 4908  df-f1o 4909  df-fv 4910

This theorem is referenced by: (None)