Theorem dmxpss 5053

Description: The domain of a cross product is a subclass of the first factor. (Contributed by set.mm contributors, 19-Mar-2007.)

Assertion

Ref	Expression
dmxpss	|- dom (A X. B) C_ A

Proof of Theorem dmxpss

Step	Hyp	Ref	Expression
1		0ss 3580	. . 3 |- (/) C_ A
2		xpeq2 4800	. . . . . . 7 |- (B = (/) -> (A X. B) = (A X. (/)))
3		xp0 5045	. . . . . . 7 |- (A X. (/)) = (/)
4	2, 3	syl6eq 2401	. . . . . 6 |- (B = (/) -> (A X. B) = (/))
5	4	dmeqd 4910	. . . . 5 |- (B = (/) -> dom (A X. B) = dom (/))
6		dm0 4919	. . . . 5 |- dom (/) = (/)
7	5, 6	syl6eq 2401	. . . 4 |- (B = (/) -> dom (A X. B) = (/))
8	7	sseq1d 3299	. . 3 |- (B = (/) -> (dom (A X. B) C_ A <-> (/) C_ A))
9	1, 8	mpbiri 224	. 2 |- (B = (/) -> dom (A X. B) C_ A)
10		dmxp 4924	. . 3 |- (B =/= (/) -> dom (A X. B) = A)
11		eqimss 3324	. . 3 |- (dom (A X. B) = A -> dom (A X. B) C_ A)
12	10, 11	syl 15	. 2 |- (B =/= (/) -> dom (A X. B) C_ A)
13	9, 12	pm2.61ine 2593	1 |- dom (A X. B) C_ A

Syntax hints:   = wceq 1642   =/= wne 2517   C_ wss 3258  (/)c0 3551   X. cxp 4771  dom cdm 4773

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-13 1712  ax-14 1714  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3or 935  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-eu 2208  df-mo 2209  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-reu 2622  df-rmo 2623  df-rab 2624  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-pss 3262  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-0c 4378  df-addc 4379  df-nnc 4380  df-fin 4381  df-lefin 4441  df-ltfin 4442  df-ncfin 4443  df-tfin 4444  df-evenfin 4445  df-oddfin 4446  df-sfin 4447  df-spfin 4448  df-phi 4566  df-op 4567  df-proj1 4568  df-proj2 4569  df-opab 4624  df-br 4641  df-ima 4728  df-xp 4785  df-cnv 4786  df-rn 4787  df-dm 4788

This theorem is referenced by:  rnxpss  5054  ssxpb  5056  funssxp  5234  dff3  5421  dmfrec  6317