Theorem xpexr2m 4705

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 4688	. 2 ⊢ ((∃𝑎 𝑎 ∈ A ∧ ∃𝑏 𝑏 ∈ B) ↔ ∃x x ∈ (A × B))
2		dmxpm 4498	. . . . . 6 ⊢ (∃𝑏 𝑏 ∈ B → dom (A × B) = A)
3	2	adantl 262	. . . . 5 ⊢ (((A × B) ∈ 𝐶 ∧ ∃𝑏 𝑏 ∈ B) → dom (A × B) = A)
4		dmexg 4539	. . . . . 6 ⊢ ((A × B) ∈ 𝐶 → dom (A × B) ∈ V)
5	4	adantr 261	. . . . 5 ⊢ (((A × B) ∈ 𝐶 ∧ ∃𝑏 𝑏 ∈ B) → dom (A × B) ∈ V)
6	3, 5	eqeltrrd 2112	. . . 4 ⊢ (((A × B) ∈ 𝐶 ∧ ∃𝑏 𝑏 ∈ B) → A ∈ V)
7		rnxpm 4695	. . . . . 6 ⊢ (∃𝑎 𝑎 ∈ A → ran (A × B) = B)
8	7	adantl 262	. . . . 5 ⊢ (((A × B) ∈ 𝐶 ∧ ∃𝑎 𝑎 ∈ A) → ran (A × B) = B)
9		rnexg 4540	. . . . . 6 ⊢ ((A × B) ∈ 𝐶 → ran (A × B) ∈ V)
10	9	adantr 261	. . . . 5 ⊢ (((A × B) ∈ 𝐶 ∧ ∃𝑎 𝑎 ∈ A) → ran (A × B) ∈ V)
11	8, 10	eqeltrrd 2112	. . . 4 ⊢ (((A × B) ∈ 𝐶 ∧ ∃𝑎 𝑎 ∈ A) → B ∈ V)
12	6, 11	anim12dan 532	. . 3 ⊢ (((A × B) ∈ 𝐶 ∧ (∃𝑏 𝑏 ∈ B ∧ ∃𝑎 𝑎 ∈ A)) → (A ∈ V ∧ B ∈ V))
13	12	ancom2s 500	. 2 ⊢ (((A × B) ∈ 𝐶 ∧ (∃𝑎 𝑎 ∈ A ∧ ∃𝑏 𝑏 ∈ B)) → (A ∈ V ∧ B ∈ V))
14	1, 13	sylan2br 272	1 ⊢ (((A × B) ∈ 𝐶 ∧ ∃x x ∈ (A × B)) → (A ∈ V ∧ B ∈ V))

Syntax hints:   → wi 4   ∧ wa 97   = wceq 1242  ∃wex 1378   ∈ wcel 1390  Vcvv 2551   × cxp 4286  dom cdm 4288  ran crn 4289

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296  df-dm 4298  df-rn 4299

This theorem is referenced by: (None)