Theorem ssxpb 6132

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 6117	. . . . . . . 8 ⊢ ((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ↔ (𝐴 × 𝐵) ≠ ∅)
2		dmxp 5878	. . . . . . . . 9 ⊢ (𝐵 ≠ ∅ → dom (𝐴 × 𝐵) = 𝐴)
3	2	adantl 481	. . . . . . . 8 ⊢ ((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) → dom (𝐴 × 𝐵) = 𝐴)
4	1, 3	sylbir 235	. . . . . . 7 ⊢ ((𝐴 × 𝐵) ≠ ∅ → dom (𝐴 × 𝐵) = 𝐴)
5	4	adantr 480	. . . . . 6 ⊢ (((𝐴 × 𝐵) ≠ ∅ ∧ (𝐴 × 𝐵) ⊆ (𝐶 × 𝐷)) → dom (𝐴 × 𝐵) = 𝐴)
6		dmss 5851	. . . . . . 7 ⊢ ((𝐴 × 𝐵) ⊆ (𝐶 × 𝐷) → dom (𝐴 × 𝐵) ⊆ dom (𝐶 × 𝐷))
7	6	adantl 481	. . . . . 6 ⊢ (((𝐴 × 𝐵) ≠ ∅ ∧ (𝐴 × 𝐵) ⊆ (𝐶 × 𝐷)) → dom (𝐴 × 𝐵) ⊆ dom (𝐶 × 𝐷))
8	5, 7	eqsstrrd 3969	. . . . 5 ⊢ (((𝐴 × 𝐵) ≠ ∅ ∧ (𝐴 × 𝐵) ⊆ (𝐶 × 𝐷)) → 𝐴 ⊆ dom (𝐶 × 𝐷))
9		dmxpss 6129	. . . . 5 ⊢ dom (𝐶 × 𝐷) ⊆ 𝐶
10	8, 9	sstrdi 3946	. . . 4 ⊢ (((𝐴 × 𝐵) ≠ ∅ ∧ (𝐴 × 𝐵) ⊆ (𝐶 × 𝐷)) → 𝐴 ⊆ 𝐶)
11		rnxp 6128	. . . . . . . . 9 ⊢ (𝐴 ≠ ∅ → ran (𝐴 × 𝐵) = 𝐵)
12	11	adantr 480	. . . . . . . 8 ⊢ ((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) → ran (𝐴 × 𝐵) = 𝐵)
13	1, 12	sylbir 235	. . . . . . 7 ⊢ ((𝐴 × 𝐵) ≠ ∅ → ran (𝐴 × 𝐵) = 𝐵)
14	13	adantr 480	. . . . . 6 ⊢ (((𝐴 × 𝐵) ≠ ∅ ∧ (𝐴 × 𝐵) ⊆ (𝐶 × 𝐷)) → ran (𝐴 × 𝐵) = 𝐵)
15		rnss 5888	. . . . . . 7 ⊢ ((𝐴 × 𝐵) ⊆ (𝐶 × 𝐷) → ran (𝐴 × 𝐵) ⊆ ran (𝐶 × 𝐷))
16	15	adantl 481	. . . . . 6 ⊢ (((𝐴 × 𝐵) ≠ ∅ ∧ (𝐴 × 𝐵) ⊆ (𝐶 × 𝐷)) → ran (𝐴 × 𝐵) ⊆ ran (𝐶 × 𝐷))
17	14, 16	eqsstrrd 3969	. . . . 5 ⊢ (((𝐴 × 𝐵) ≠ ∅ ∧ (𝐴 × 𝐵) ⊆ (𝐶 × 𝐷)) → 𝐵 ⊆ ran (𝐶 × 𝐷))
18		rnxpss 6130	. . . . 5 ⊢ ran (𝐶 × 𝐷) ⊆ 𝐷
19	17, 18	sstrdi 3946	. . . 4 ⊢ (((𝐴 × 𝐵) ≠ ∅ ∧ (𝐴 × 𝐵) ⊆ (𝐶 × 𝐷)) → 𝐵 ⊆ 𝐷)
20	10, 19	jca 511	. . 3 ⊢ (((𝐴 × 𝐵) ≠ ∅ ∧ (𝐴 × 𝐵) ⊆ (𝐶 × 𝐷)) → (𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐷))
21	20	ex 412	. 2 ⊢ ((𝐴 × 𝐵) ≠ ∅ → ((𝐴 × 𝐵) ⊆ (𝐶 × 𝐷) → (𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐷)))
22		xpss12 5639	. 2 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐷) → (𝐴 × 𝐵) ⊆ (𝐶 × 𝐷))
23	21, 22	impbid1 225	1 ⊢ ((𝐴 × 𝐵) ≠ ∅ → ((𝐴 × 𝐵) ⊆ (𝐶 × 𝐷) ↔ (𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐷)))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1541   ≠ wne 2932   ⊆ wss 3901  ∅c0 4285   × cxp 5622  dom cdm 5624  ran crn 5625

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-11 2162  ax-ext 2708  ax-sep 5241  ax-nul 5251  ax-pr 5377

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2715  df-cleq 2728  df-clel 2811  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3400  df-v 3442  df-dif 3904  df-un 3906  df-ss 3918  df-nul 4286  df-if 4480  df-sn 4581  df-pr 4583  df-op 4587  df-br 5099  df-opab 5161  df-xp 5630  df-rel 5631  df-cnv 5632  df-dm 5634  df-rn 5635

This theorem is referenced by:  xp11  6133  dibord  41419  aks6d1c2lem4  42381  aks6d1c2  42384