Theorem xpexr2m 5045

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 5025	. 2 ⊢ ((∃𝑎 𝑎 ∈ 𝐴 ∧ ∃𝑏 𝑏 ∈ 𝐵) ↔ ∃𝑥 𝑥 ∈ (𝐴 × 𝐵))
2		dmxpm 4824	. . . . . 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 4868	. . . . . 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 2244	. . . 4 ⊢ (((𝐴 × 𝐵) ∈ 𝐶 ∧ ∃𝑏 𝑏 ∈ 𝐵) → 𝐴 ∈ V)
7		rnxpm 5033	. . . . . 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 4869	. . . . . 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 2244	. . . 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 1343  ∃wex 1480   ∈ wcel 2136  Vcvv 2726   × cxp 4602  dom cdm 4604  ran crn 4605

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 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-xp 4610  df-rel 4611  df-cnv 4612  df-dm 4614  df-rn 4615

This theorem is referenced by: (None)