Theorem xpexr2m 4988

Description: If a nonempty cross product is a set, so are both of its components. (Contributed by Jim Kingdon, 14-Dec-2018.)

Assertion

Ref	Expression
xpexr2m	⊢ (((𝐴 × 𝐵) ∈ 𝐶 ∧ ∃𝑥 𝑥 ∈ (𝐴 × 𝐵)) → (𝐴 ∈ V ∧ 𝐵 ∈ V))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem xpexr2m

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

Step	Hyp	Ref	Expression
1		xpm 4968	. 2 ⊢ ((∃𝑎 𝑎 ∈ 𝐴 ∧ ∃𝑏 𝑏 ∈ 𝐵) ↔ ∃𝑥 𝑥 ∈ (𝐴 × 𝐵))
2		dmxpm 4767	. . . . . 6 ⊢ (∃𝑏 𝑏 ∈ 𝐵 → dom (𝐴 × 𝐵) = 𝐴)
3	2	adantl 275	. . . . 5 ⊢ (((𝐴 × 𝐵) ∈ 𝐶 ∧ ∃𝑏 𝑏 ∈ 𝐵) → dom (𝐴 × 𝐵) = 𝐴)
4		dmexg 4811	. . . . . 6 ⊢ ((𝐴 × 𝐵) ∈ 𝐶 → dom (𝐴 × 𝐵) ∈ V)
5	4	adantr 274	. . . . 5 ⊢ (((𝐴 × 𝐵) ∈ 𝐶 ∧ ∃𝑏 𝑏 ∈ 𝐵) → dom (𝐴 × 𝐵) ∈ V)
6	3, 5	eqeltrrd 2218	. . . 4 ⊢ (((𝐴 × 𝐵) ∈ 𝐶 ∧ ∃𝑏 𝑏 ∈ 𝐵) → 𝐴 ∈ V)
7		rnxpm 4976	. . . . . 6 ⊢ (∃𝑎 𝑎 ∈ 𝐴 → ran (𝐴 × 𝐵) = 𝐵)
8	7	adantl 275	. . . . 5 ⊢ (((𝐴 × 𝐵) ∈ 𝐶 ∧ ∃𝑎 𝑎 ∈ 𝐴) → ran (𝐴 × 𝐵) = 𝐵)
9		rnexg 4812	. . . . . 6 ⊢ ((𝐴 × 𝐵) ∈ 𝐶 → ran (𝐴 × 𝐵) ∈ V)
10	9	adantr 274	. . . . 5 ⊢ (((𝐴 × 𝐵) ∈ 𝐶 ∧ ∃𝑎 𝑎 ∈ 𝐴) → ran (𝐴 × 𝐵) ∈ V)
11	8, 10	eqeltrrd 2218	. . . 4 ⊢ (((𝐴 × 𝐵) ∈ 𝐶 ∧ ∃𝑎 𝑎 ∈ 𝐴) → 𝐵 ∈ V)
12	6, 11	anim12dan 590	. . 3 ⊢ (((𝐴 × 𝐵) ∈ 𝐶 ∧ (∃𝑏 𝑏 ∈ 𝐵 ∧ ∃𝑎 𝑎 ∈ 𝐴)) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
13	12	ancom2s 556	. 2 ⊢ (((𝐴 × 𝐵) ∈ 𝐶 ∧ (∃𝑎 𝑎 ∈ 𝐴 ∧ ∃𝑏 𝑏 ∈ 𝐵)) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
14	1, 13	sylan2br 286	1 ⊢ (((𝐴 × 𝐵) ∈ 𝐶 ∧ ∃𝑥 𝑥 ∈ (𝐴 × 𝐵)) → (𝐴 ∈ V ∧ 𝐵 ∈ V))

Syntax hints:   → wi 4   ∧ wa 103   = wceq 1332  ∃wex 1469   ∈ wcel 1481  Vcvv 2689   × cxp 4545  dom cdm 4547  ran crn 4548

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363

This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-xp 4553  df-rel 4554  df-cnv 4555  df-dm 4557  df-rn 4558

This theorem is referenced by: (None)