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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) ↔ (A𝐶 B𝐷)))
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.
StepHypRef 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)
32adantl 262 . . . . . . . 8 ((𝑎 𝑎 A 𝑏 𝑏 B) → dom (A × B) = A)
41, 3sylbir 125 . . . . . . 7 (x x (A × B) → dom (A × B) = A)
54adantr 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))
76adantl 262 . . . . . 6 ((x x (A × B) (A × B) ⊆ (𝐶 × 𝐷)) → dom (A × B) ⊆ dom (𝐶 × 𝐷))
85, 7eqsstr3d 2974 . . . . 5 ((x x (A × B) (A × B) ⊆ (𝐶 × 𝐷)) → A ⊆ dom (𝐶 × 𝐷))
9 dmxpss 4696 . . . . 5 dom (𝐶 × 𝐷) ⊆ 𝐶
108, 9syl6ss 2951 . . . 4 ((x x (A × B) (A × B) ⊆ (𝐶 × 𝐷)) → A𝐶)
11 rnxpm 4695 . . . . . . . . 9 (𝑎 𝑎 A → ran (A × B) = B)
1211adantr 261 . . . . . . . 8 ((𝑎 𝑎 A 𝑏 𝑏 B) → ran (A × B) = B)
131, 12sylbir 125 . . . . . . 7 (x x (A × B) → ran (A × B) = B)
1413adantr 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))
1615adantl 262 . . . . . 6 ((x x (A × B) (A × B) ⊆ (𝐶 × 𝐷)) → ran (A × B) ⊆ ran (𝐶 × 𝐷))
1714, 16eqsstr3d 2974 . . . . 5 ((x x (A × B) (A × B) ⊆ (𝐶 × 𝐷)) → B ⊆ ran (𝐶 × 𝐷))
18 rnxpss 4697 . . . . 5 ran (𝐶 × 𝐷) ⊆ 𝐷
1917, 18syl6ss 2951 . . . 4 ((x x (A × B) (A × B) ⊆ (𝐶 × 𝐷)) → B𝐷)
2010, 19jca 290 . . 3 ((x x (A × B) (A × B) ⊆ (𝐶 × 𝐷)) → (A𝐶 B𝐷))
2120ex 108 . 2 (x x (A × B) → ((A × B) ⊆ (𝐶 × 𝐷) → (A𝐶 B𝐷)))
22 xpss12 4388 . 2 ((A𝐶 B𝐷) → (A × B) ⊆ (𝐶 × 𝐷))
2321, 22impbid1 130 1 (x x (A × B) → ((A × B) ⊆ (𝐶 × 𝐷) ↔ (A𝐶 B𝐷)))
Colors of variables: wff set class
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-bnd 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
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