Theorem unixp 3523

Description: The double class union of a non-empty cross product is the union of it members.

Assertion

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

Proof of Theorem unixp

Step	Hyp	Ref	Expression
1		uneq12 2182	. . . 4 ⊢ ((dom ( A × B) = A ⋀ ran ( A × B) = B) → (dom ( A × B) ∪ ran ( A × B)) = (A ∪ B))
2		dmxp 3338	. . . 4 ⊢ (B ≠ ∅ → dom ( A × B) = A)
3		rnxp 3478	. . . 4 ⊢ (A ≠ ∅ → ran ( A × B) = B)
4	1, 2, 3	syl2an 456	. . 3 ⊢ ((B ≠ ∅ ⋀ A ≠ ∅) → (dom ( A × B) ∪ ran ( A × B)) = (A ∪ B))
5		xpeq2 3207	. . . . 5 ⊢ (B = ∅ → (A × B) = (A × ∅))
6		xp0 3471	. . . . 5 ⊢ (A × ∅) = ∅
7	5, 6	syl6eq 1526	. . . 4 ⊢ (B = ∅ → (A × B) = ∅)
8	7	necon3i 1608	. . 3 ⊢ ((A × B) ≠ ∅ → B ≠ ∅)
9		xpeq1 3206	. . . . 5 ⊢ (A = ∅ → (A × B) = (∅ × B))
10		xp0r 3245	. . . . 5 ⊢ (∅ × B) = ∅
11	9, 10	syl6eq 1526	. . . 4 ⊢ (A = ∅ → (A × B) = ∅)
12	11	necon3i 1608	. . 3 ⊢ ((A × B) ≠ ∅ → A ≠ ∅)
13	4, 8, 12	sylanc 473	. 2 ⊢ ((A × B) ≠ ∅ → (dom ( A × B) ∪ ran ( A × B)) = (A ∪ B))
14		relxp 3261	. . 3 ⊢ Rel (A × B)
15		relfld 3521	. . 3 ⊢ (Rel (A × B) → ∪∪(A × B) = (dom ( A × B) ∪ ran ( A × B)))
16	14, 15	ax-mp 7	. 2 ⊢ ∪∪(A × B) = (dom ( A × B) ∪ ran ( A × B))
17	13, 16	syl5eq 1522	1 ⊢ ((A × B) ≠ ∅ → ∪∪(A × B) = (A ∪ B))

Syntax hints:   → wi 3   = wceq 958   ≠ wne 1588   ∪ cun 2048  ∅c0 2283  ∪cuni 2507   × cxp 3174  dom cdm 3176  ran crn 3177  Rel wrel 3181

This theorem is referenced by:  rankxpl 4720  rankxplim2 4723  rankxplim3 4724  rankxpsuc 4725

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-pow 2748  ax-pr 2785

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-xp 3190  df-rel 3191  df-cnv 3192  df-dm 3194  df-rn 3195

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