New Foundations Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  NFE Home  >  Th. List  >  dmxpss GIF version

Theorem dmxpss 5052
 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 × B) A

Proof of Theorem dmxpss
StepHypRef Expression
1 0ss 3579 . . 3 A
2 xpeq2 4799 . . . . . . 7 (B = → (A × B) = (A × ))
3 xp0 5044 . . . . . . 7 (A × ) =
42, 3syl6eq 2401 . . . . . 6 (B = → (A × B) = )
54dmeqd 4909 . . . . 5 (B = → dom (A × B) = dom )
6 dm0 4918 . . . . 5 dom =
75, 6syl6eq 2401 . . . 4 (B = → dom (A × B) = )
87sseq1d 3298 . . 3 (B = → (dom (A × B) A A))
91, 8mpbiri 224 . 2 (B = → dom (A × B) A)
10 dmxp 4923 . . 3 (B → dom (A × B) = A)
11 eqimss 3323 . . 3 (dom (A × B) = A → dom (A × B) A)
1210, 11syl 15 . 2 (B → dom (A × B) A)
139, 12pm2.61ine 2592 1 dom (A × B) A
 Colors of variables: wff setvar class Syntax hints:   = wceq 1642   ≠ wne 2516   ⊆ wss 3257  ∅c0 3550   × cxp 4770  dom cdm 4772 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 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 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 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-reu 2621  df-rmo 2622  df-rab 2623  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-pss 3261  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-0c 4377  df-addc 4378  df-nnc 4379  df-fin 4380  df-lefin 4440  df-ltfin 4441  df-ncfin 4442  df-tfin 4443  df-evenfin 4444  df-oddfin 4445  df-sfin 4446  df-spfin 4447  df-phi 4565  df-op 4566  df-proj1 4567  df-proj2 4568  df-opab 4623  df-br 4640  df-ima 4727  df-xp 4784  df-cnv 4785  df-rn 4786  df-dm 4787 This theorem is referenced by:  rnxpss  5053  ssxpb  5055  funssxp  5233  dff3  5420  dmfrec  6316
 Copyright terms: Public domain W3C validator