Theorem unixp 6272

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 5685	. . 3 ⊢ Rel (𝐴 × 𝐵)
2		relfld 6265	. . 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 5688	. . . . 5 ⊢ (𝐵 = ∅ → (𝐴 × 𝐵) = (𝐴 × ∅))
5		xp0 6148	. . . . 5 ⊢ (𝐴 × ∅) = ∅
6	4, 5	eqtrdi 2780	. . . 4 ⊢ (𝐵 = ∅ → (𝐴 × 𝐵) = ∅)
7	6	necon3i 2965	. . 3 ⊢ ((𝐴 × 𝐵) ≠ ∅ → 𝐵 ≠ ∅)
8		xpeq1 5681	. . . . 5 ⊢ (𝐴 = ∅ → (𝐴 × 𝐵) = (∅ × 𝐵))
9		0xp 5765	. . . . 5 ⊢ (∅ × 𝐵) = ∅
10	8, 9	eqtrdi 2780	. . . 4 ⊢ (𝐴 = ∅ → (𝐴 × 𝐵) = ∅)
11	10	necon3i 2965	. . 3 ⊢ ((𝐴 × 𝐵) ≠ ∅ → 𝐴 ≠ ∅)
12		dmxp 5919	. . . 4 ⊢ (𝐵 ≠ ∅ → dom (𝐴 × 𝐵) = 𝐴)
13		rnxp 6160	. . . 4 ⊢ (𝐴 ≠ ∅ → ran (𝐴 × 𝐵) = 𝐵)
14		uneq12 4151	. . . 4 ⊢ ((dom (𝐴 × 𝐵) = 𝐴 ∧ ran (𝐴 × 𝐵) = 𝐵) → (dom (𝐴 × 𝐵) ∪ ran (𝐴 × 𝐵)) = (𝐴 ∪ 𝐵))
15	12, 13, 14	syl2an 595	. . 3 ⊢ ((𝐵 ≠ ∅ ∧ 𝐴 ≠ ∅) → (dom (𝐴 × 𝐵) ∪ ran (𝐴 × 𝐵)) = (𝐴 ∪ 𝐵))
16	7, 11, 15	syl2anc 583	. 2 ⊢ ((𝐴 × 𝐵) ≠ ∅ → (dom (𝐴 × 𝐵) ∪ ran (𝐴 × 𝐵)) = (𝐴 ∪ 𝐵))
17	3, 16	eqtrid 2776	1 ⊢ ((𝐴 × 𝐵) ≠ ∅ → ∪ ∪ (𝐴 × 𝐵) = (𝐴 ∪ 𝐵))

Syntax hints:   → wi 4   = wceq 1533   ≠ wne 2932   ∪ cun 3939  ∅c0 4315  ∪ cuni 4900   × cxp 5665  dom cdm 5667  ran crn 5668  Rel wrel 5672

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-pw 4597  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-xp 5673  df-rel 5674  df-cnv 5675  df-dm 5677  df-rn 5678

This theorem is referenced by:  unixpid  6274  rankxpl  9867  rankxplim2  9872  rankxplim3  9873  rankxpsuc  9874