Theorem ssxpb 6167

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 6151	. . . . . . . 8 ⊢ ((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) ↔ (𝐴 × 𝐵) ≠ ∅)
2		dmxp 5913	. . . . . . . . 9 ⊢ (𝐵 ≠ ∅ → dom (𝐴 × 𝐵) = 𝐴)
3	2	adantl 487	. . . . . . . 8 ⊢ ((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) → dom (𝐴 × 𝐵) = 𝐴)
4	1, 3	sylbir 238	. . . . . . 7 ⊢ ((𝐴 × 𝐵) ≠ ∅ → dom (𝐴 × 𝐵) = 𝐴)
5	4	adantr 486	. . . . . 6 ⊢ (((𝐴 × 𝐵) ≠ ∅ ∧ (𝐴 × 𝐵) ⊆ (𝐶 × 𝐷)) → dom (𝐴 × 𝐵) = 𝐴)
6		dmss 5886	. . . . . . 7 ⊢ ((𝐴 × 𝐵) ⊆ (𝐶 × 𝐷) → dom (𝐴 × 𝐵) ⊆ dom (𝐶 × 𝐷))
7	6	adantl 487	. . . . . 6 ⊢ (((𝐴 × 𝐵) ≠ ∅ ∧ (𝐴 × 𝐵) ⊆ (𝐶 × 𝐷)) → dom (𝐴 × 𝐵) ⊆ dom (𝐶 × 𝐷))
8	5, 7	eqsstrrd 3966	. . . . 5 ⊢ (((𝐴 × 𝐵) ≠ ∅ ∧ (𝐴 × 𝐵) ⊆ (𝐶 × 𝐷)) → 𝐴 ⊆ dom (𝐶 × 𝐷))
9		dmxpss 6164	. . . . 5 ⊢ dom (𝐶 × 𝐷) ⊆ 𝐶
10	8, 9	sstrdi 3943	. . . 4 ⊢ (((𝐴 × 𝐵) ≠ ∅ ∧ (𝐴 × 𝐵) ⊆ (𝐶 × 𝐷)) → 𝐴 ⊆ 𝐶)
11		rnxp 6163	. . . . . . . . 9 ⊢ (𝐴 ≠ ∅ → ran (𝐴 × 𝐵) = 𝐵)
12	11	adantr 486	. . . . . . . 8 ⊢ ((𝐴 ≠ ∅ ∧ 𝐵 ≠ ∅) → ran (𝐴 × 𝐵) = 𝐵)
13	1, 12	sylbir 238	. . . . . . 7 ⊢ ((𝐴 × 𝐵) ≠ ∅ → ran (𝐴 × 𝐵) = 𝐵)
14	13	adantr 486	. . . . . 6 ⊢ (((𝐴 × 𝐵) ≠ ∅ ∧ (𝐴 × 𝐵) ⊆ (𝐶 × 𝐷)) → ran (𝐴 × 𝐵) = 𝐵)
15		rnss 5923	. . . . . . 7 ⊢ ((𝐴 × 𝐵) ⊆ (𝐶 × 𝐷) → ran (𝐴 × 𝐵) ⊆ ran (𝐶 × 𝐷))
16	15	adantl 487	. . . . . 6 ⊢ (((𝐴 × 𝐵) ≠ ∅ ∧ (𝐴 × 𝐵) ⊆ (𝐶 × 𝐷)) → ran (𝐴 × 𝐵) ⊆ ran (𝐶 × 𝐷))
17	14, 16	eqsstrrd 3966	. . . . 5 ⊢ (((𝐴 × 𝐵) ≠ ∅ ∧ (𝐴 × 𝐵) ⊆ (𝐶 × 𝐷)) → 𝐵 ⊆ ran (𝐶 × 𝐷))
18		rnxpss 6165	. . . . 5 ⊢ ran (𝐶 × 𝐷) ⊆ 𝐷
19	17, 18	sstrdi 3943	. . . 4 ⊢ (((𝐴 × 𝐵) ≠ ∅ ∧ (𝐴 × 𝐵) ⊆ (𝐶 × 𝐷)) → 𝐵 ⊆ 𝐷)
20	10, 19	jca 521	. . 3 ⊢ (((𝐴 × 𝐵) ≠ ∅ ∧ (𝐴 × 𝐵) ⊆ (𝐶 × 𝐷)) → (𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐷))
21	20	ex 418	. 2 ⊢ ((𝐴 × 𝐵) ≠ ∅ → ((𝐴 × 𝐵) ⊆ (𝐶 × 𝐷) → (𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐷)))
22		xpss12 5670	. 2 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐷) → (𝐴 × 𝐵) ⊆ (𝐶 × 𝐷))
23	21, 22	impbid1 228	1 ⊢ ((𝐴 × 𝐵) ≠ ∅ → ((𝐴 × 𝐵) ⊆ (𝐶 × 𝐷) ↔ (𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐷)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ≠ wne 2955   ⊆ wss 3899  ∅c0 4279   × cxp 5653  dom cdm 5655  ran crn 5656

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5251  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5168  df-xp 5661  df-rel 5662  df-cnv 5663  df-dm 5665  df-rn 5666

This theorem is used by:  xp11  6168  dibord  42035  aks6d1c2lem4  42996  aks6d1c2  42999