Theorem ssxpb 6160

Description: A Cartesian product subclass relationship is equivalent to the conjunction of the analogous relationships for the factors. (Contributed by NM, 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 6144	. . . . . . . 8 ⊢ ((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ↔ (𝐴 × 𝐵) ≠ ∅)
2		dmxp 5905	. . . . . . . . 9 ⊢ (𝐵 ≠ ∅ → dom (𝐴 × 𝐵) = 𝐴)
3	2	adantl 485	. . . . . . . 8 ⊢ ((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) → dom (𝐴 × 𝐵) = 𝐴)
4	1, 3	sylbir 237	. . . . . . 7 ⊢ ((𝐴 × 𝐵) ≠ ∅ → dom (𝐴 × 𝐵) = 𝐴)
5	4	adantr 484	. . . . . 6 ⊢ (((𝐴 × 𝐵) ≠ ∅ ∧ (𝐴 × 𝐵) ⊆ (𝐶 × 𝐷)) → dom (𝐴 × 𝐵) = 𝐴)
6		dmss 5878	. . . . . . 7 ⊢ ((𝐴 × 𝐵) ⊆ (𝐶 × 𝐷) → dom (𝐴 × 𝐵) ⊆ dom (𝐶 × 𝐷))
7	6	adantl 485	. . . . . 6 ⊢ (((𝐴 × 𝐵) ≠ ∅ ∧ (𝐴 × 𝐵) ⊆ (𝐶 × 𝐷)) → dom (𝐴 × 𝐵) ⊆ dom (𝐶 × 𝐷))
8	5, 7	eqsstrrd 3971	. . . . 5 ⊢ (((𝐴 × 𝐵) ≠ ∅ ∧ (𝐴 × 𝐵) ⊆ (𝐶 × 𝐷)) → 𝐴 ⊆ dom (𝐶 × 𝐷))
9		dmxpss 6157	. . . . 5 ⊢ dom (𝐶 × 𝐷) ⊆ 𝐶
10	8, 9	sstrdi 3948	. . . 4 ⊢ (((𝐴 × 𝐵) ≠ ∅ ∧ (𝐴 × 𝐵) ⊆ (𝐶 × 𝐷)) → 𝐴 ⊆ 𝐶)
11		rnxp 6156	. . . . . . . . 9 ⊢ (𝐴 ≠ ∅ → ran (𝐴 × 𝐵) = 𝐵)
12	11	adantr 484	. . . . . . . 8 ⊢ ((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) → ran (𝐴 × 𝐵) = 𝐵)
13	1, 12	sylbir 237	. . . . . . 7 ⊢ ((𝐴 × 𝐵) ≠ ∅ → ran (𝐴 × 𝐵) = 𝐵)
14	13	adantr 484	. . . . . 6 ⊢ (((𝐴 × 𝐵) ≠ ∅ ∧ (𝐴 × 𝐵) ⊆ (𝐶 × 𝐷)) → ran (𝐴 × 𝐵) = 𝐵)
15		rnss 5915	. . . . . . 7 ⊢ ((𝐴 × 𝐵) ⊆ (𝐶 × 𝐷) → ran (𝐴 × 𝐵) ⊆ ran (𝐶 × 𝐷))
16	15	adantl 485	. . . . . 6 ⊢ (((𝐴 × 𝐵) ≠ ∅ ∧ (𝐴 × 𝐵) ⊆ (𝐶 × 𝐷)) → ran (𝐴 × 𝐵) ⊆ ran (𝐶 × 𝐷))
17	14, 16	eqsstrrd 3971	. . . . 5 ⊢ (((𝐴 × 𝐵) ≠ ∅ ∧ (𝐴 × 𝐵) ⊆ (𝐶 × 𝐷)) → 𝐵 ⊆ ran (𝐶 × 𝐷))
18		rnxpss 6158	. . . . 5 ⊢ ran (𝐶 × 𝐷) ⊆ 𝐷
19	17, 18	sstrdi 3948	. . . 4 ⊢ (((𝐴 × 𝐵) ≠ ∅ ∧ (𝐴 × 𝐵) ⊆ (𝐶 × 𝐷)) → 𝐵 ⊆ 𝐷)
20	10, 19	jca 519	. . 3 ⊢ (((𝐴 × 𝐵) ≠ ∅ ∧ (𝐴 × 𝐵) ⊆ (𝐶 × 𝐷)) → (𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐷))
21	20	ex 416	. 2 ⊢ ((𝐴 × 𝐵) ≠ ∅ → ((𝐴 × 𝐵) ⊆ (𝐶 × 𝐷) → (𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐷)))
22		xpss12 5662	. 2 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐷) → (𝐴 × 𝐵) ⊆ (𝐶 × 𝐷))
23	21, 22	impbid1 227	1 ⊢ ((𝐴 × 𝐵) ≠ ∅ → ((𝐴 × 𝐵) ⊆ (𝐶 × 𝐷) ↔ (𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐷)))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 399   = wceq 1560   ≠ wne 2957   ⊆ wss 3904  ∅c0 4285   × cxp 5645  dom cdm 5647  ran crn 5648

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-11 2191  ax-ext 2734  ax-sep 5246  ax-pr 5390

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1100  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-ral 3077  df-rex 3087  df-rab 3415  df-v 3456  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4481  df-sn 4583  df-pr 4585  df-op 4589  df-br 5101  df-opab 5163  df-xp 5653  df-rel 5654  df-cnv 5655  df-dm 5657  df-rn 5658

This theorem is referenced by:  xp11  6161  dibord  41783  aks6d1c2lem4  42744  aks6d1c2  42747