Theorem xpexr2m 5132

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 5112	. 2 ⊢ ((∃𝑎 𝑎 ∈ 𝐴 ∧ ∃𝑏 𝑏 ∈ 𝐵) ↔ ∃𝑥 𝑥 ∈ (𝐴 × 𝐵))
2		dmxpm 4906	. . . . . 6 ⊢ (∃𝑏 𝑏 ∈ 𝐵 → dom (𝐴 × 𝐵) = 𝐴)
3	2	adantl 277	. . . . 5 ⊢ (((𝐴 × 𝐵) ∈ 𝐶 ∧ ∃𝑏 𝑏 ∈ 𝐵) → dom (𝐴 × 𝐵) = 𝐴)
4		dmexg 4950	. . . . . 6 ⊢ ((𝐴 × 𝐵) ∈ 𝐶 → dom (𝐴 × 𝐵) ∈ V)
5	4	adantr 276	. . . . 5 ⊢ (((𝐴 × 𝐵) ∈ 𝐶 ∧ ∃𝑏 𝑏 ∈ 𝐵) → dom (𝐴 × 𝐵) ∈ V)
6	3, 5	eqeltrrd 2284	. . . 4 ⊢ (((𝐴 × 𝐵) ∈ 𝐶 ∧ ∃𝑏 𝑏 ∈ 𝐵) → 𝐴 ∈ V)
7		rnxpm 5120	. . . . . 6 ⊢ (∃𝑎 𝑎 ∈ 𝐴 → ran (𝐴 × 𝐵) = 𝐵)
8	7	adantl 277	. . . . 5 ⊢ (((𝐴 × 𝐵) ∈ 𝐶 ∧ ∃𝑎 𝑎 ∈ 𝐴) → ran (𝐴 × 𝐵) = 𝐵)
9		rnexg 4951	. . . . . 6 ⊢ ((𝐴 × 𝐵) ∈ 𝐶 → ran (𝐴 × 𝐵) ∈ V)
10	9	adantr 276	. . . . 5 ⊢ (((𝐴 × 𝐵) ∈ 𝐶 ∧ ∃𝑎 𝑎 ∈ 𝐴) → ran (𝐴 × 𝐵) ∈ V)
11	8, 10	eqeltrrd 2284	. . . 4 ⊢ (((𝐴 × 𝐵) ∈ 𝐶 ∧ ∃𝑎 𝑎 ∈ 𝐴) → 𝐵 ∈ V)
12	6, 11	anim12dan 600	. . 3 ⊢ (((𝐴 × 𝐵) ∈ 𝐶 ∧ (∃𝑏 𝑏 ∈ 𝐵 ∧ ∃𝑎 𝑎 ∈ 𝐴)) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
13	12	ancom2s 566	. 2 ⊢ (((𝐴 × 𝐵) ∈ 𝐶 ∧ (∃𝑎 𝑎 ∈ 𝐴 ∧ ∃𝑏 𝑏 ∈ 𝐵)) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
14	1, 13	sylan2br 288	1 ⊢ (((𝐴 × 𝐵) ∈ 𝐶 ∧ ∃𝑥 𝑥 ∈ (𝐴 × 𝐵)) → (𝐴 ∈ V ∧ 𝐵 ∈ V))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1373  ∃wex 1516   ∈ wcel 2177  Vcvv 2773   × cxp 4680  dom cdm 4682  ran crn 4683

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4169  ax-pow 4225  ax-pr 4260  ax-un 4487

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-un 3174  df-in 3176  df-ss 3183  df-pw 3622  df-sn 3643  df-pr 3644  df-op 3646  df-uni 3856  df-br 4051  df-opab 4113  df-xp 4688  df-rel 4689  df-cnv 4690  df-dm 4692  df-rn 4693

This theorem is referenced by: (None)