Theorem ssxpb 5056

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

Step	Hyp	Ref	Expression
1		xpnz 5046	. . . . . . . 8 ⊢ ((A ≠ ∅ ∧ B ≠ ∅) ↔ (A × B) ≠ ∅)
2		dmxp 4924	. . . . . . . . 9 ⊢ (B ≠ ∅ → dom (A × B) = A)
3	2	adantl 452	. . . . . . . 8 ⊢ ((A ≠ ∅ ∧ B ≠ ∅) → dom (A × B) = A)
4	1, 3	sylbir 204	. . . . . . 7 ⊢ ((A × B) ≠ ∅ → dom (A × B) = A)
5	4	adantr 451	. . . . . 6 ⊢ (((A × B) ≠ ∅ ∧ (A × B) ⊆ (C × D)) → dom (A × B) = A)
6		dmss 4907	. . . . . . 7 ⊢ ((A × B) ⊆ (C × D) → dom (A × B) ⊆ dom (C × D))
7	6	adantl 452	. . . . . 6 ⊢ (((A × B) ≠ ∅ ∧ (A × B) ⊆ (C × D)) → dom (A × B) ⊆ dom (C × D))
8	5, 7	eqsstr3d 3307	. . . . 5 ⊢ (((A × B) ≠ ∅ ∧ (A × B) ⊆ (C × D)) → A ⊆ dom (C × D))
9		dmxpss 5053	. . . . 5 ⊢ dom (C × D) ⊆ C
10	8, 9	syl6ss 3285	. . . 4 ⊢ (((A × B) ≠ ∅ ∧ (A × B) ⊆ (C × D)) → A ⊆ C)
11		rnxp 5052	. . . . . . . . 9 ⊢ (A ≠ ∅ → ran (A × B) = B)
12	11	adantr 451	. . . . . . . 8 ⊢ ((A ≠ ∅ ∧ B ≠ ∅) → ran (A × B) = B)
13	1, 12	sylbir 204	. . . . . . 7 ⊢ ((A × B) ≠ ∅ → ran (A × B) = B)
14	13	adantr 451	. . . . . 6 ⊢ (((A × B) ≠ ∅ ∧ (A × B) ⊆ (C × D)) → ran (A × B) = B)
15		rnss 4960	. . . . . . 7 ⊢ ((A × B) ⊆ (C × D) → ran (A × B) ⊆ ran (C × D))
16	15	adantl 452	. . . . . 6 ⊢ (((A × B) ≠ ∅ ∧ (A × B) ⊆ (C × D)) → ran (A × B) ⊆ ran (C × D))
17	14, 16	eqsstr3d 3307	. . . . 5 ⊢ (((A × B) ≠ ∅ ∧ (A × B) ⊆ (C × D)) → B ⊆ ran (C × D))
18		rnxpss 5054	. . . . 5 ⊢ ran (C × D) ⊆ D
19	17, 18	syl6ss 3285	. . . 4 ⊢ (((A × B) ≠ ∅ ∧ (A × B) ⊆ (C × D)) → B ⊆ D)
20	10, 19	jca 518	. . 3 ⊢ (((A × B) ≠ ∅ ∧ (A × B) ⊆ (C × D)) → (A ⊆ C ∧ B ⊆ D))
21	20	ex 423	. 2 ⊢ ((A × B) ≠ ∅ → ((A × B) ⊆ (C × D) → (A ⊆ C ∧ B ⊆ D)))
22		xpss12 4856	. 2 ⊢ ((A ⊆ C ∧ B ⊆ D) → (A × B) ⊆ (C × D))
23	21, 22	impbid1 194	1 ⊢ ((A × B) ≠ ∅ → ((A × B) ⊆ (C × D) ↔ (A ⊆ C ∧ B ⊆ D)))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ≠ wne 2517   ⊆ wss 3258  ∅c0 3551   × cxp 4771  dom cdm 4773  ran crn 4774

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:  xp11  5057