Theorem unixp 6250

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 5652	. . 3 ⊢ Rel (𝐴 × 𝐵)
2		relfld 6243	. . 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 5655	. . . . 5 ⊢ (𝐵 = ∅ → (𝐴 × 𝐵) = (𝐴 × ∅))
5		xp0 5734	. . . . 5 ⊢ (𝐴 × ∅) = ∅
6	4, 5	eqtrdi 2788	. . . 4 ⊢ (𝐵 = ∅ → (𝐴 × 𝐵) = ∅)
7	6	necon3i 2965	. . 3 ⊢ ((𝐴 × 𝐵) ≠ ∅ → 𝐵 ≠ ∅)
8		xpeq1 5648	. . . . 5 ⊢ (𝐴 = ∅ → (𝐴 × 𝐵) = (∅ × 𝐵))
9		0xp 5733	. . . . 5 ⊢ (∅ × 𝐵) = ∅
10	8, 9	eqtrdi 2788	. . . 4 ⊢ (𝐴 = ∅ → (𝐴 × 𝐵) = ∅)
11	10	necon3i 2965	. . 3 ⊢ ((𝐴 × 𝐵) ≠ ∅ → 𝐴 ≠ ∅)
12		dmxp 5888	. . . 4 ⊢ (𝐵 ≠ ∅ → dom (𝐴 × 𝐵) = 𝐴)
13		rnxp 6138	. . . 4 ⊢ (𝐴 ≠ ∅ → ran (𝐴 × 𝐵) = 𝐵)
14		uneq12 4117	. . . 4 ⊢ ((dom (𝐴 × 𝐵) = 𝐴 ∧ ran (𝐴 × 𝐵) = 𝐵) → (dom (𝐴 × 𝐵) ∪ ran (𝐴 × 𝐵)) = (𝐴 ∪ 𝐵))
15	12, 13, 14	syl2an 597	. . 3 ⊢ ((𝐵 ≠ ∅ ∧ 𝐴 ≠ ∅) → (dom (𝐴 × 𝐵) ∪ ran (𝐴 × 𝐵)) = (𝐴 ∪ 𝐵))
16	7, 11, 15	syl2anc 585	. 2 ⊢ ((𝐴 × 𝐵) ≠ ∅ → (dom (𝐴 × 𝐵) ∪ ran (𝐴 × 𝐵)) = (𝐴 ∪ 𝐵))
17	3, 16	eqtrid 2784	1 ⊢ ((𝐴 × 𝐵) ≠ ∅ → ∪ ∪ (𝐴 × 𝐵) = (𝐴 ∪ 𝐵))

Syntax hints:   → wi 4   = wceq 1542   ≠ wne 2933   ∪ cun 3901  ∅c0 4287  ∪ cuni 4865   × cxp 5632  dom cdm 5634  ran crn 5635  Rel wrel 5639

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-11 2163  ax-ext 2709  ax-sep 5245  ax-pr 5381

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-br 5101  df-opab 5163  df-xp 5640  df-rel 5641  df-cnv 5642  df-dm 5644  df-rn 5645

This theorem is referenced by:  unixpid  6252  rankxpl  9801  rankxplim2  9806  rankxplim3  9807  rankxpsuc  9808