Theorem undom 9003

Description: Dominance law for union. Proposition 4.24(a) of [Mendelson] p. 257. (Contributed by NM, 3-Sep-2004.) (Revised by Mario Carneiro, 26-Apr-2015.) Avoid ax-pow 5320. (Revised by BTernaryTau, 4-Dec-2024.)

Assertion

Ref	Expression
undom	⊢ (((𝐴 ≼ 𝐵 ∧ 𝐶 ≼ 𝐷) ∧ (𝐵 ∩ 𝐷) = ∅) → (𝐴 ∪ 𝐶) ≼ (𝐵 ∪ 𝐷))

Proof of Theorem undom

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		undif2 4436	. 2 ⊢ (𝐴 ∪ (𝐶 ∖ 𝐴)) = (𝐴 ∪ 𝐶)
2		reldom 8889	. . . . . 6 ⊢ Rel ≼
3	2	brrelex2i 5689	. . . . 5 ⊢ (𝐴 ≼ 𝐵 → 𝐵 ∈ V)
4	2	brrelex2i 5689	. . . . 5 ⊢ (𝐶 ≼ 𝐷 → 𝐷 ∈ V)
5		unexg 7683	. . . . 5 ⊢ ((𝐵 ∈ V ∧ 𝐷 ∈ V) → (𝐵 ∪ 𝐷) ∈ V)
6	3, 4, 5	syl2an 596	. . . 4 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐶 ≼ 𝐷) → (𝐵 ∪ 𝐷) ∈ V)
7	6	adantr 481	. . 3 ⊢ (((𝐴 ≼ 𝐵 ∧ 𝐶 ≼ 𝐷) ∧ (𝐵 ∩ 𝐷) = ∅) → (𝐵 ∪ 𝐷) ∈ V)
8		brdomi 8898	. . . . 5 ⊢ (𝐴 ≼ 𝐵 → ∃𝑥 𝑥:𝐴–1-1→𝐵)
9		brdomi 8898	. . . . 5 ⊢ (𝐶 ≼ 𝐷 → ∃𝑦 𝑦:𝐶–1-1→𝐷)
10		exdistrv 1959	. . . . . 6 ⊢ (∃𝑥∃𝑦(𝑥:𝐴–1-1→𝐵 ∧ 𝑦:𝐶–1-1→𝐷) ↔ (∃𝑥 𝑥:𝐴–1-1→𝐵 ∧ ∃𝑦 𝑦:𝐶–1-1→𝐷))
11		disjdif 4431	. . . . . . . . . 10 ⊢ (𝐴 ∩ (𝐶 ∖ 𝐴)) = ∅
12		difss 4091	. . . . . . . . . . . 12 ⊢ (𝐶 ∖ 𝐴) ⊆ 𝐶
13		f1ssres 6746	. . . . . . . . . . . 12 ⊢ ((𝑦:𝐶–1-1→𝐷 ∧ (𝐶 ∖ 𝐴) ⊆ 𝐶) → (𝑦 ↾ (𝐶 ∖ 𝐴)):(𝐶 ∖ 𝐴)–1-1→𝐷)
14	12, 13	mpan2 689	. . . . . . . . . . 11 ⊢ (𝑦:𝐶–1-1→𝐷 → (𝑦 ↾ (𝐶 ∖ 𝐴)):(𝐶 ∖ 𝐴)–1-1→𝐷)
15		f1un 6804	. . . . . . . . . . 11 ⊢ (((𝑥:𝐴–1-1→𝐵 ∧ (𝑦 ↾ (𝐶 ∖ 𝐴)):(𝐶 ∖ 𝐴)–1-1→𝐷) ∧ ((𝐴 ∩ (𝐶 ∖ 𝐴)) = ∅ ∧ (𝐵 ∩ 𝐷) = ∅)) → (𝑥 ∪ (𝑦 ↾ (𝐶 ∖ 𝐴))):(𝐴 ∪ (𝐶 ∖ 𝐴))–1-1→(𝐵 ∪ 𝐷))
16	14, 15	sylanl2 679	. . . . . . . . . 10 ⊢ (((𝑥:𝐴–1-1→𝐵 ∧ 𝑦:𝐶–1-1→𝐷) ∧ ((𝐴 ∩ (𝐶 ∖ 𝐴)) = ∅ ∧ (𝐵 ∩ 𝐷) = ∅)) → (𝑥 ∪ (𝑦 ↾ (𝐶 ∖ 𝐴))):(𝐴 ∪ (𝐶 ∖ 𝐴))–1-1→(𝐵 ∪ 𝐷))
17	11, 16	mpanr1 701	. . . . . . . . 9 ⊢ (((𝑥:𝐴–1-1→𝐵 ∧ 𝑦:𝐶–1-1→𝐷) ∧ (𝐵 ∩ 𝐷) = ∅) → (𝑥 ∪ (𝑦 ↾ (𝐶 ∖ 𝐴))):(𝐴 ∪ (𝐶 ∖ 𝐴))–1-1→(𝐵 ∪ 𝐷))
18		vex 3449	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
19		vex 3449	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
20	19	resex 5985	. . . . . . . . . . . 12 ⊢ (𝑦 ↾ (𝐶 ∖ 𝐴)) ∈ V
21	18, 20	unex 7680	. . . . . . . . . . 11 ⊢ (𝑥 ∪ (𝑦 ↾ (𝐶 ∖ 𝐴))) ∈ V
22		f1dom3g 8907	. . . . . . . . . . 11 ⊢ (((𝑥 ∪ (𝑦 ↾ (𝐶 ∖ 𝐴))) ∈ V ∧ (𝐵 ∪ 𝐷) ∈ V ∧ (𝑥 ∪ (𝑦 ↾ (𝐶 ∖ 𝐴))):(𝐴 ∪ (𝐶 ∖ 𝐴))–1-1→(𝐵 ∪ 𝐷)) → (𝐴 ∪ (𝐶 ∖ 𝐴)) ≼ (𝐵 ∪ 𝐷))
23	21, 22	mp3an1 1448	. . . . . . . . . 10 ⊢ (((𝐵 ∪ 𝐷) ∈ V ∧ (𝑥 ∪ (𝑦 ↾ (𝐶 ∖ 𝐴))):(𝐴 ∪ (𝐶 ∖ 𝐴))–1-1→(𝐵 ∪ 𝐷)) → (𝐴 ∪ (𝐶 ∖ 𝐴)) ≼ (𝐵 ∪ 𝐷))
24	23	expcom 414	. . . . . . . . 9 ⊢ ((𝑥 ∪ (𝑦 ↾ (𝐶 ∖ 𝐴))):(𝐴 ∪ (𝐶 ∖ 𝐴))–1-1→(𝐵 ∪ 𝐷) → ((𝐵 ∪ 𝐷) ∈ V → (𝐴 ∪ (𝐶 ∖ 𝐴)) ≼ (𝐵 ∪ 𝐷)))
25	17, 24	syl 17	. . . . . . . 8 ⊢ (((𝑥:𝐴–1-1→𝐵 ∧ 𝑦:𝐶–1-1→𝐷) ∧ (𝐵 ∩ 𝐷) = ∅) → ((𝐵 ∪ 𝐷) ∈ V → (𝐴 ∪ (𝐶 ∖ 𝐴)) ≼ (𝐵 ∪ 𝐷)))
26	25	ex 413	. . . . . . 7 ⊢ ((𝑥:𝐴–1-1→𝐵 ∧ 𝑦:𝐶–1-1→𝐷) → ((𝐵 ∩ 𝐷) = ∅ → ((𝐵 ∪ 𝐷) ∈ V → (𝐴 ∪ (𝐶 ∖ 𝐴)) ≼ (𝐵 ∪ 𝐷))))
27	26	exlimivv 1935	. . . . . 6 ⊢ (∃𝑥∃𝑦(𝑥:𝐴–1-1→𝐵 ∧ 𝑦:𝐶–1-1→𝐷) → ((𝐵 ∩ 𝐷) = ∅ → ((𝐵 ∪ 𝐷) ∈ V → (𝐴 ∪ (𝐶 ∖ 𝐴)) ≼ (𝐵 ∪ 𝐷))))
28	10, 27	sylbir 234	. . . . 5 ⊢ ((∃𝑥 𝑥:𝐴–1-1→𝐵 ∧ ∃𝑦 𝑦:𝐶–1-1→𝐷) → ((𝐵 ∩ 𝐷) = ∅ → ((𝐵 ∪ 𝐷) ∈ V → (𝐴 ∪ (𝐶 ∖ 𝐴)) ≼ (𝐵 ∪ 𝐷))))
29	8, 9, 28	syl2an 596	. . . 4 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐶 ≼ 𝐷) → ((𝐵 ∩ 𝐷) = ∅ → ((𝐵 ∪ 𝐷) ∈ V → (𝐴 ∪ (𝐶 ∖ 𝐴)) ≼ (𝐵 ∪ 𝐷))))
30	29	imp 407	. . 3 ⊢ (((𝐴 ≼ 𝐵 ∧ 𝐶 ≼ 𝐷) ∧ (𝐵 ∩ 𝐷) = ∅) → ((𝐵 ∪ 𝐷) ∈ V → (𝐴 ∪ (𝐶 ∖ 𝐴)) ≼ (𝐵 ∪ 𝐷)))
31	7, 30	mpd 15	. 2 ⊢ (((𝐴 ≼ 𝐵 ∧ 𝐶 ≼ 𝐷) ∧ (𝐵 ∩ 𝐷) = ∅) → (𝐴 ∪ (𝐶 ∖ 𝐴)) ≼ (𝐵 ∪ 𝐷))
32	1, 31	eqbrtrrid 5141	1 ⊢ (((𝐴 ≼ 𝐵 ∧ 𝐶 ≼ 𝐷) ∧ (𝐵 ∩ 𝐷) = ∅) → (𝐴 ∪ 𝐶) ≼ (𝐵 ∪ 𝐷))

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1541  ∃wex 1781   ∈ wcel 2106  Vcvv 3445   ∖ cdif 3907   ∪ cun 3908   ∩ cin 3909   ⊆ wss 3910  ∅c0 4282   class class class wbr 5105   ↾ cres 5635  –1-1→wf1 6493   ≼ cdom 8881

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384  ax-un 7672

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-uni 4866  df-br 5106  df-opab 5168  df-id 5531  df-xp 5639  df-rel 5640  df-cnv 5641  df-co 5642  df-dm 5643  df-rn 5644  df-res 5645  df-fun 6498  df-fn 6499  df-f 6500  df-f1 6501  df-fo 6502  df-f1o 6503  df-dom 8885

This theorem is referenced by:  domunsncan  9016  sucdom2OLD  9026  domunsn  9071  sucdom2  9150  unxpdom2  9198  sucxpdom  9199  fodomfi  9269  undjudom  10103  djudom1  10118