Theorem unixpm 5303

Description: The double class union of an inhabited cross product is the union of its members. (Contributed by Jim Kingdon, 18-Dec-2018.)

Assertion

Ref	Expression
unixpm	⊢ (∃𝑥 𝑥 ∈ (𝐴 × 𝐵) → ∪ ∪ (𝐴 × 𝐵) = (𝐴 ∪ 𝐵))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem unixpm

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relxp 4864	. . 3 ⊢ Rel (𝐴 × 𝐵)
2		relfld 5296	. . 3 ⊢ (Rel (𝐴 × 𝐵) → ∪ ∪ (𝐴 × 𝐵) = (dom (𝐴 × 𝐵) ∪ ran (𝐴 × 𝐵)))
3	1, 2	ax-mp 5	. 2 ⊢ ∪ ∪ (𝐴 × 𝐵) = (dom (𝐴 × 𝐵) ∪ ran (𝐴 × 𝐵))
4		ancom 266	. . . 4 ⊢ ((∃𝑏 𝑏 ∈ 𝐵 ∧ ∃𝑎 𝑎 ∈ 𝐴) ↔ (∃𝑎 𝑎 ∈ 𝐴 ∧ ∃𝑏 𝑏 ∈ 𝐵))
5		xpm 5189	. . . 4 ⊢ ((∃𝑎 𝑎 ∈ 𝐴 ∧ ∃𝑏 𝑏 ∈ 𝐵) ↔ ∃𝑥 𝑥 ∈ (𝐴 × 𝐵))
6	4, 5	bitri 184	. . 3 ⊢ ((∃𝑏 𝑏 ∈ 𝐵 ∧ ∃𝑎 𝑎 ∈ 𝐴) ↔ ∃𝑥 𝑥 ∈ (𝐴 × 𝐵))
7		dmxpm 4982	. . . 4 ⊢ (∃𝑏 𝑏 ∈ 𝐵 → dom (𝐴 × 𝐵) = 𝐴)
8		rnxpm 5197	. . . 4 ⊢ (∃𝑎 𝑎 ∈ 𝐴 → ran (𝐴 × 𝐵) = 𝐵)
9		uneq12 3372	. . . 4 ⊢ ((dom (𝐴 × 𝐵) = 𝐴 ∧ ran (𝐴 × 𝐵) = 𝐵) → (dom (𝐴 × 𝐵) ∪ ran (𝐴 × 𝐵)) = (𝐴 ∪ 𝐵))
10	7, 8, 9	syl2an 289	. . 3 ⊢ ((∃𝑏 𝑏 ∈ 𝐵 ∧ ∃𝑎 𝑎 ∈ 𝐴) → (dom (𝐴 × 𝐵) ∪ ran (𝐴 × 𝐵)) = (𝐴 ∪ 𝐵))
11	6, 10	sylbir 135	. 2 ⊢ (∃𝑥 𝑥 ∈ (𝐴 × 𝐵) → (dom (𝐴 × 𝐵) ∪ ran (𝐴 × 𝐵)) = (𝐴 ∪ 𝐵))
12	3, 11	eqtrid 2279	1 ⊢ (∃𝑥 𝑥 ∈ (𝐴 × 𝐵) → ∪ ∪ (𝐴 × 𝐵) = (𝐴 ∪ 𝐵))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1398  ∃wex 1541   ∈ wcel 2205   ∪ cun 3212  ∪ cuni 3919   × cxp 4752  dom cdm 4754  ran crn 4755  Rel wrel 4759

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-xp 4760  df-rel 4761  df-cnv 4762  df-dm 4764  df-rn 4765

This theorem is referenced by: (None)