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Theorem ssxpb 5055
 Description: A cross-product subclass relationship is equivalent to the relationship for it components. (Contributed by set.mm contributors, 17-Dec-2008.)
Assertion
Ref Expression
ssxpb ((A × B) ≠ → ((A × B) (C × D) ↔ (A C B D)))

Proof of Theorem ssxpb
StepHypRef Expression
1 xpnz 5045 . . . . . . . 8 ((A B) ↔ (A × B) ≠ )
2 dmxp 4923 . . . . . . . . 9 (B → dom (A × B) = A)
32adantl 452 . . . . . . . 8 ((A B) → dom (A × B) = A)
41, 3sylbir 204 . . . . . . 7 ((A × B) ≠ → dom (A × B) = A)
54adantr 451 . . . . . 6 (((A × B) ≠ (A × B) (C × D)) → dom (A × B) = A)
6 dmss 4906 . . . . . . 7 ((A × B) (C × D) → dom (A × B) dom (C × D))
76adantl 452 . . . . . 6 (((A × B) ≠ (A × B) (C × D)) → dom (A × B) dom (C × D))
85, 7eqsstr3d 3306 . . . . 5 (((A × B) ≠ (A × B) (C × D)) → A dom (C × D))
9 dmxpss 5052 . . . . 5 dom (C × D) C
108, 9syl6ss 3284 . . . 4 (((A × B) ≠ (A × B) (C × D)) → A C)
11 rnxp 5051 . . . . . . . . 9 (A → ran (A × B) = B)
1211adantr 451 . . . . . . . 8 ((A B) → ran (A × B) = B)
131, 12sylbir 204 . . . . . . 7 ((A × B) ≠ → ran (A × B) = B)
1413adantr 451 . . . . . 6 (((A × B) ≠ (A × B) (C × D)) → ran (A × B) = B)
15 rnss 4959 . . . . . . 7 ((A × B) (C × D) → ran (A × B) ran (C × D))
1615adantl 452 . . . . . 6 (((A × B) ≠ (A × B) (C × D)) → ran (A × B) ran (C × D))
1714, 16eqsstr3d 3306 . . . . 5 (((A × B) ≠ (A × B) (C × D)) → B ran (C × D))
18 rnxpss 5053 . . . . 5 ran (C × D) D
1917, 18syl6ss 3284 . . . 4 (((A × B) ≠ (A × B) (C × D)) → B D)
2010, 19jca 518 . . 3 (((A × B) ≠ (A × B) (C × D)) → (A C B D))
2120ex 423 . 2 ((A × B) ≠ → ((A × B) (C × D) → (A C B D)))
22 xpss12 4855 . 2 ((A C B D) → (A × B) (C × D))
2321, 22impbid1 194 1 ((A × B) ≠ → ((A × B) (C × D) ↔ (A C B D)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ≠ wne 2516   ⊆ wss 3257  ∅c0 3550   × cxp 4770  dom cdm 4772  ran crn 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 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:  xp11  5056
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