Theorem ssxpbm 4776

Description: A cross-product subclass relationship is equivalent to the relationship for its components. (Contributed by Jim Kingdon, 12-Dec-2018.)

Assertion

Ref	Expression
ssxpbm	⊢ (∃𝑥 𝑥 ∈ (𝐴 × 𝐵) → ((𝐴 × 𝐵) ⊆ (𝐶 × 𝐷) ↔ (𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐷)))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hints:   𝐶(𝑥)   𝐷(𝑥)

Proof of Theorem ssxpbm

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		xpm 4765	. . . . . . . 8 ⊢ ((∃𝑎 𝑎 ∈ 𝐴 ∧ ∃𝑏 𝑏 ∈ 𝐵) ↔ ∃𝑥 𝑥 ∈ (𝐴 × 𝐵))
2		dmxpm 4573	. . . . . . . . 9 ⊢ (∃𝑏 𝑏 ∈ 𝐵 → dom (𝐴 × 𝐵) = 𝐴)
3	2	adantl 271	. . . . . . . 8 ⊢ ((∃𝑎 𝑎 ∈ 𝐴 ∧ ∃𝑏 𝑏 ∈ 𝐵) → dom (𝐴 × 𝐵) = 𝐴)
4	1, 3	sylbir 133	. . . . . . 7 ⊢ (∃𝑥 𝑥 ∈ (𝐴 × 𝐵) → dom (𝐴 × 𝐵) = 𝐴)
5	4	adantr 270	. . . . . 6 ⊢ ((∃𝑥 𝑥 ∈ (𝐴 × 𝐵) ∧ (𝐴 × 𝐵) ⊆ (𝐶 × 𝐷)) → dom (𝐴 × 𝐵) = 𝐴)
6		dmss 4552	. . . . . . 7 ⊢ ((𝐴 × 𝐵) ⊆ (𝐶 × 𝐷) → dom (𝐴 × 𝐵) ⊆ dom (𝐶 × 𝐷))
7	6	adantl 271	. . . . . 6 ⊢ ((∃𝑥 𝑥 ∈ (𝐴 × 𝐵) ∧ (𝐴 × 𝐵) ⊆ (𝐶 × 𝐷)) → dom (𝐴 × 𝐵) ⊆ dom (𝐶 × 𝐷))
8	5, 7	eqsstr3d 3034	. . . . 5 ⊢ ((∃𝑥 𝑥 ∈ (𝐴 × 𝐵) ∧ (𝐴 × 𝐵) ⊆ (𝐶 × 𝐷)) → 𝐴 ⊆ dom (𝐶 × 𝐷))
9		dmxpss 4773	. . . . 5 ⊢ dom (𝐶 × 𝐷) ⊆ 𝐶
10	8, 9	syl6ss 3011	. . . 4 ⊢ ((∃𝑥 𝑥 ∈ (𝐴 × 𝐵) ∧ (𝐴 × 𝐵) ⊆ (𝐶 × 𝐷)) → 𝐴 ⊆ 𝐶)
11		rnxpm 4772	. . . . . . . . 9 ⊢ (∃𝑎 𝑎 ∈ 𝐴 → ran (𝐴 × 𝐵) = 𝐵)
12	11	adantr 270	. . . . . . . 8 ⊢ ((∃𝑎 𝑎 ∈ 𝐴 ∧ ∃𝑏 𝑏 ∈ 𝐵) → ran (𝐴 × 𝐵) = 𝐵)
13	1, 12	sylbir 133	. . . . . . 7 ⊢ (∃𝑥 𝑥 ∈ (𝐴 × 𝐵) → ran (𝐴 × 𝐵) = 𝐵)
14	13	adantr 270	. . . . . 6 ⊢ ((∃𝑥 𝑥 ∈ (𝐴 × 𝐵) ∧ (𝐴 × 𝐵) ⊆ (𝐶 × 𝐷)) → ran (𝐴 × 𝐵) = 𝐵)
15		rnss 4582	. . . . . . 7 ⊢ ((𝐴 × 𝐵) ⊆ (𝐶 × 𝐷) → ran (𝐴 × 𝐵) ⊆ ran (𝐶 × 𝐷))
16	15	adantl 271	. . . . . 6 ⊢ ((∃𝑥 𝑥 ∈ (𝐴 × 𝐵) ∧ (𝐴 × 𝐵) ⊆ (𝐶 × 𝐷)) → ran (𝐴 × 𝐵) ⊆ ran (𝐶 × 𝐷))
17	14, 16	eqsstr3d 3034	. . . . 5 ⊢ ((∃𝑥 𝑥 ∈ (𝐴 × 𝐵) ∧ (𝐴 × 𝐵) ⊆ (𝐶 × 𝐷)) → 𝐵 ⊆ ran (𝐶 × 𝐷))
18		rnxpss 4774	. . . . 5 ⊢ ran (𝐶 × 𝐷) ⊆ 𝐷
19	17, 18	syl6ss 3011	. . . 4 ⊢ ((∃𝑥 𝑥 ∈ (𝐴 × 𝐵) ∧ (𝐴 × 𝐵) ⊆ (𝐶 × 𝐷)) → 𝐵 ⊆ 𝐷)
20	10, 19	jca 300	. . 3 ⊢ ((∃𝑥 𝑥 ∈ (𝐴 × 𝐵) ∧ (𝐴 × 𝐵) ⊆ (𝐶 × 𝐷)) → (𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐷))
21	20	ex 113	. 2 ⊢ (∃𝑥 𝑥 ∈ (𝐴 × 𝐵) → ((𝐴 × 𝐵) ⊆ (𝐶 × 𝐷) → (𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐷)))
22		xpss12 4463	. 2 ⊢ ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐷) → (𝐴 × 𝐵) ⊆ (𝐶 × 𝐷))
23	21, 22	impbid1 140	1 ⊢ (∃𝑥 𝑥 ∈ (𝐴 × 𝐵) → ((𝐴 × 𝐵) ⊆ (𝐶 × 𝐷) ↔ (𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐷)))

Syntax hints:   → wi 4   ∧ wa 102   ↔ wb 103   = wceq 1284  ∃wex 1421   ∈ wcel 1433   ⊆ wss 2973   × cxp 4361  dom cdm 4363  ran crn 4364

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-un 2977  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-xp 4369  df-rel 4370  df-cnv 4371  df-dm 4373  df-rn 4374

This theorem is referenced by:  xp11m  4779