Theorem ssxpbm 4699

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 4688	. . . . . . . 8 ⊢ ((∃𝑎 𝑎 ∈ A ∧ ∃𝑏 𝑏 ∈ B) ↔ ∃x x ∈ (A × B))
2		dmxpm 4498	. . . . . . . . 9 ⊢ (∃𝑏 𝑏 ∈ B → dom (A × B) = A)
3	2	adantl 262	. . . . . . . 8 ⊢ ((∃𝑎 𝑎 ∈ A ∧ ∃𝑏 𝑏 ∈ B) → dom (A × B) = A)
4	1, 3	sylbir 125	. . . . . . 7 ⊢ (∃x x ∈ (A × B) → dom (A × B) = A)
5	4	adantr 261	. . . . . 6 ⊢ ((∃x x ∈ (A × B) ∧ (A × B) ⊆ (𝐶 × 𝐷)) → dom (A × B) = A)
6		dmss 4477	. . . . . . 7 ⊢ ((A × B) ⊆ (𝐶 × 𝐷) → dom (A × B) ⊆ dom (𝐶 × 𝐷))
7	6	adantl 262	. . . . . 6 ⊢ ((∃x x ∈ (A × B) ∧ (A × B) ⊆ (𝐶 × 𝐷)) → dom (A × B) ⊆ dom (𝐶 × 𝐷))
8	5, 7	eqsstr3d 2974	. . . . 5 ⊢ ((∃x x ∈ (A × B) ∧ (A × B) ⊆ (𝐶 × 𝐷)) → A ⊆ dom (𝐶 × 𝐷))
9		dmxpss 4696	. . . . 5 ⊢ dom (𝐶 × 𝐷) ⊆ 𝐶
10	8, 9	syl6ss 2951	. . . 4 ⊢ ((∃x x ∈ (A × B) ∧ (A × B) ⊆ (𝐶 × 𝐷)) → A ⊆ 𝐶)
11		rnxpm 4695	. . . . . . . . 9 ⊢ (∃𝑎 𝑎 ∈ A → ran (A × B) = B)
12	11	adantr 261	. . . . . . . 8 ⊢ ((∃𝑎 𝑎 ∈ A ∧ ∃𝑏 𝑏 ∈ B) → ran (A × B) = B)
13	1, 12	sylbir 125	. . . . . . 7 ⊢ (∃x x ∈ (A × B) → ran (A × B) = B)
14	13	adantr 261	. . . . . 6 ⊢ ((∃x x ∈ (A × B) ∧ (A × B) ⊆ (𝐶 × 𝐷)) → ran (A × B) = B)
15		rnss 4507	. . . . . . 7 ⊢ ((A × B) ⊆ (𝐶 × 𝐷) → ran (A × B) ⊆ ran (𝐶 × 𝐷))
16	15	adantl 262	. . . . . 6 ⊢ ((∃x x ∈ (A × B) ∧ (A × B) ⊆ (𝐶 × 𝐷)) → ran (A × B) ⊆ ran (𝐶 × 𝐷))
17	14, 16	eqsstr3d 2974	. . . . 5 ⊢ ((∃x x ∈ (A × B) ∧ (A × B) ⊆ (𝐶 × 𝐷)) → B ⊆ ran (𝐶 × 𝐷))
18		rnxpss 4697	. . . . 5 ⊢ ran (𝐶 × 𝐷) ⊆ 𝐷
19	17, 18	syl6ss 2951	. . . 4 ⊢ ((∃x x ∈ (A × B) ∧ (A × B) ⊆ (𝐶 × 𝐷)) → B ⊆ 𝐷)
20	10, 19	jca 290	. . 3 ⊢ ((∃x x ∈ (A × B) ∧ (A × B) ⊆ (𝐶 × 𝐷)) → (A ⊆ 𝐶 ∧ B ⊆ 𝐷))
21	20	ex 108	. 2 ⊢ (∃x x ∈ (A × B) → ((A × B) ⊆ (𝐶 × 𝐷) → (A ⊆ 𝐶 ∧ B ⊆ 𝐷)))
22		xpss12 4388	. 2 ⊢ ((A ⊆ 𝐶 ∧ B ⊆ 𝐷) → (A × B) ⊆ (𝐶 × 𝐷))
23	21, 22	impbid1 130	1 ⊢ (∃x x ∈ (A × B) → ((A × B) ⊆ (𝐶 × 𝐷) ↔ (A ⊆ 𝐶 ∧ B ⊆ 𝐷)))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242  ∃wex 1378   ∈ wcel 1390   ⊆ wss 2911   × cxp 4286  dom cdm 4288  ran crn 4289

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296  df-dm 4298  df-rn 4299

This theorem is referenced by:  xp11m  4702