Theorem xpexr2m 5178

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 5158	. 2 ⊢ ((∃𝑎 𝑎 ∈ 𝐴 ∧ ∃𝑏 𝑏 ∈ 𝐵) ↔ ∃𝑥 𝑥 ∈ (𝐴 × 𝐵))
2		dmxpm 4952	. . . . . 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 4996	. . . . . 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 2309	. . . 4 ⊢ (((𝐴 × 𝐵) ∈ 𝐶 ∧ ∃𝑏 𝑏 ∈ 𝐵) → 𝐴 ∈ V)
7		rnxpm 5166	. . . . . 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 4997	. . . . . 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 2309	. . . 4 ⊢ (((𝐴 × 𝐵) ∈ 𝐶 ∧ ∃𝑎 𝑎 ∈ 𝐴) → 𝐵 ∈ V)
12	6, 11	anim12dan 604	. . 3 ⊢ (((𝐴 × 𝐵) ∈ 𝐶 ∧ (∃𝑏 𝑏 ∈ 𝐵 ∧ ∃𝑎 𝑎 ∈ 𝐴)) → (𝐴 ∈ V ∧ 𝐵 ∈ V))
13	12	ancom2s 568	. 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 1397  ∃wex 1540   ∈ wcel 2202  Vcvv 2802   × cxp 4723  dom cdm 4725  ran crn 4726

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-pow 4264  ax-pr 4299  ax-un 4530

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-rex 2516  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-br 4089  df-opab 4151  df-xp 4731  df-rel 4732  df-cnv 4733  df-dm 4735  df-rn 4736

This theorem is referenced by: (None)