Theorem unixp 6307

Description: The double class union of a nonempty Cartesian product is the union of it members. (Contributed by NM, 17-Sep-2006.)

Assertion

Ref	Expression
unixp	⊢ ((𝐴 × 𝐵) ≠ ∅ → ∪ ∪ (𝐴 × 𝐵) = (𝐴 ∪ 𝐵))

Proof of Theorem unixp

Step	Hyp	Ref	Expression
1		relxp 5708	. . 3 ⊢ Rel (𝐴 × 𝐵)
2		relfld 6300	. . 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		xpeq2 5711	. . . . 5 ⊢ (𝐵 = ∅ → (𝐴 × 𝐵) = (𝐴 × ∅))
5		xp0 6183	. . . . 5 ⊢ (𝐴 × ∅) = ∅
6	4, 5	eqtrdi 2792	. . . 4 ⊢ (𝐵 = ∅ → (𝐴 × 𝐵) = ∅)
7	6	necon3i 2972	. . 3 ⊢ ((𝐴 × 𝐵) ≠ ∅ → 𝐵 ≠ ∅)
8		xpeq1 5704	. . . . 5 ⊢ (𝐴 = ∅ → (𝐴 × 𝐵) = (∅ × 𝐵))
9		0xp 5788	. . . . 5 ⊢ (∅ × 𝐵) = ∅
10	8, 9	eqtrdi 2792	. . . 4 ⊢ (𝐴 = ∅ → (𝐴 × 𝐵) = ∅)
11	10	necon3i 2972	. . 3 ⊢ ((𝐴 × 𝐵) ≠ ∅ → 𝐴 ≠ ∅)
12		dmxp 5943	. . . 4 ⊢ (𝐵 ≠ ∅ → dom (𝐴 × 𝐵) = 𝐴)
13		rnxp 6195	. . . 4 ⊢ (𝐴 ≠ ∅ → ran (𝐴 × 𝐵) = 𝐵)
14		uneq12 4174	. . . 4 ⊢ ((dom (𝐴 × 𝐵) = 𝐴 ∧ ran (𝐴 × 𝐵) = 𝐵) → (dom (𝐴 × 𝐵) ∪ ran (𝐴 × 𝐵)) = (𝐴 ∪ 𝐵))
15	12, 13, 14	syl2an 596	. . 3 ⊢ ((𝐵 ≠ ∅ ∧ 𝐴 ≠ ∅) → (dom (𝐴 × 𝐵) ∪ ran (𝐴 × 𝐵)) = (𝐴 ∪ 𝐵))
16	7, 11, 15	syl2anc 584	. 2 ⊢ ((𝐴 × 𝐵) ≠ ∅ → (dom (𝐴 × 𝐵) ∪ ran (𝐴 × 𝐵)) = (𝐴 ∪ 𝐵))
17	3, 16	eqtrid 2788	1 ⊢ ((𝐴 × 𝐵) ≠ ∅ → ∪ ∪ (𝐴 × 𝐵) = (𝐴 ∪ 𝐵))

Syntax hints:   → wi 4   = wceq 1538   ≠ wne 2939   ∪ cun 3962  ∅c0 4340  ∪ cuni 4913   × cxp 5688  dom cdm 5690  ran crn 5691  Rel wrel 5695

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-11 2156  ax-12 2176  ax-ext 2707  ax-sep 5303  ax-nul 5313  ax-pr 5439

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1541  df-fal 1551  df-ex 1778  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3435  df-v 3481  df-dif 3967  df-un 3969  df-ss 3981  df-nul 4341  df-if 4533  df-pw 4608  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4914  df-br 5150  df-opab 5212  df-xp 5696  df-rel 5697  df-cnv 5698  df-dm 5700  df-rn 5701

This theorem is referenced by:  unixpid  6309  rankxpl  9919  rankxplim2  9924  rankxplim3  9925  rankxpsuc  9926