Theorem unixp 6282

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 5673	. . 3 ⊢ Rel (𝐴 × 𝐵)
2		relfld 6274	. . 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 5676	. . . . 5 ⊢ (𝐵 = ∅ → (𝐴 × 𝐵) = (𝐴 × ∅))
5		xp0 5755	. . . . 5 ⊢ (𝐴 × ∅) = ∅
6	4, 5	eqtrdi 2811	. . . 4 ⊢ (𝐵 = ∅ → (𝐴 × 𝐵) = ∅)
7	6	necon3i 2987	. . 3 ⊢ ((𝐴 × 𝐵) ≠ ∅ → 𝐵 ≠ ∅)
8		xpeq1 5669	. . . . 5 ⊢ (𝐴 = ∅ → (𝐴 × 𝐵) = (∅ × 𝐵))
9		0xp 5754	. . . . 5 ⊢ (∅ × 𝐵) = ∅
10	8, 9	eqtrdi 2811	. . . 4 ⊢ (𝐴 = ∅ → (𝐴 × 𝐵) = ∅)
11	10	necon3i 2987	. . 3 ⊢ ((𝐴 × 𝐵) ≠ ∅ → 𝐴 ≠ ∅)
12		dmxp 5915	. . . 4 ⊢ (𝐵 ≠ ∅ → dom (𝐴 × 𝐵) = 𝐴)
13		rnxp 6165	. . . 4 ⊢ (𝐴 ≠ ∅ → ran (𝐴 × 𝐵) = 𝐵)
14		uneq12 4110	. . . 4 ⊢ ((dom (𝐴 × 𝐵) = 𝐴 ∧ ran (𝐴 × 𝐵) = 𝐵) → (dom (𝐴 × 𝐵) ∪ ran (𝐴 × 𝐵)) = (𝐴 ∪ 𝐵))
15	12, 13, 14	syl2an 608	. . 3 ⊢ ((𝐵 ≠ ∅ ∧ 𝐴 ≠ ∅) → (dom (𝐴 × 𝐵) ∪ ran (𝐴 × 𝐵)) = (𝐴 ∪ 𝐵))
16	7, 11, 15	syl2anc 596	. 2 ⊢ ((𝐴 × 𝐵) ≠ ∅ → (dom (𝐴 × 𝐵) ∪ ran (𝐴 × 𝐵)) = (𝐴 ∪ 𝐵))
17	3, 16	eqtrid 2807	1 ⊢ ((𝐴 × 𝐵) ≠ ∅ → ∪ ∪ (𝐴 × 𝐵) = (𝐴 ∪ 𝐵))

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ≠ wne 2955   ∪ cun 3897  ∅c0 4279  ∪ cuni 4867   × cxp 5653  dom cdm 5655  ran crn 5656  Rel wrel 5660

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732  ax-sep 5251  ax-pr 5398

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-xp 5661  df-rel 5662  df-cnv 5663  df-dm 5665  df-rn 5666

This theorem is used by:  unixpid  6284  rankxpl  9868  rankxplim2  9873  rankxplim3  9874  rankxpsuc  9875