Theorem xpexr2m 5166

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 5146	. 2 ⊢ ((∃𝑎 𝑎 ∈ 𝐴 ∧ ∃𝑏 𝑏 ∈ 𝐵) ↔ ∃𝑥 𝑥 ∈ (𝐴 × 𝐵))
2		dmxpm 4940	. . . . . 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 4984	. . . . . 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 2307	. . . 4 ⊢ (((𝐴 × 𝐵) ∈ 𝐶 ∧ ∃𝑏 𝑏 ∈ 𝐵) → 𝐴 ∈ V)
7		rnxpm 5154	. . . . . 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 4985	. . . . . 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 2307	. . . 4 ⊢ (((𝐴 × 𝐵) ∈ 𝐶 ∧ ∃𝑎 𝑎 ∈ 𝐴) → 𝐵 ∈ V)
12	6, 11	anim12dan 602	. . 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 1395  ∃wex 1538   ∈ wcel 2200  Vcvv 2799   × cxp 4714  dom cdm 4716  ran crn 4717

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4521

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-br 4083  df-opab 4145  df-xp 4722  df-rel 4723  df-cnv 4724  df-dm 4726  df-rn 4727

This theorem is referenced by: (None)