Theorem xpexgALT 6112

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 4725 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 3928	. . . 4 |- U_ y e. B { y } = B
2	1	xpeq2i 4632	. . 3 |- ( A X. U_ y e. B { y } ) = ( A X. B )
3		xpiundi 4669	. . 3 |- ( A X. U_ y e. B { y } ) = U_ y e. B ( A X. { y } )
4	2, 3	eqtr3i 2193	. 2 |- ( A X. B ) = U_ y e. B ( A X. { y } )
5		id 19	. . 3 |- ( B e. W -> B e. W )
6		fconstmpt 4658	. . . . 5 |- ( A X. { y } ) = ( x e. A |-> y )
7		mptexg 5721	. . . . 5 |- ( A e. V -> ( x e. A |-> y ) e. _V )
8	6, 7	eqeltrid 2257	. . . 4 |- ( A e. V -> ( A X. { y } ) e. _V )
9	8	ralrimivw 2544	. . 3 |- ( A e. V -> A. y e. B ( A X. { y } ) e. _V )
10		iunexg 6098	. . 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 2257	1 |- ( ( A e. V /\ B e. W ) -> ( A X. B ) e. _V )

Syntax hints:   -> wi 4   /\ wa 103   e. wcel 2141   A.wral 2448   _Vcvv 2730   {csn 3583   U_ciun 3873   |-> cmpt 4050   X. cxp 4609

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-coll 4104  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  df-un 3125  df-in 3127  df-ss 3134  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-id 4278  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206

This theorem is referenced by: (None)