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 5307. (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 4417	. 2 ⊢ (𝐴 ∪ (𝐶 ∖ 𝐴)) = (𝐴 ∪ 𝐶)
2		reldom 8899	. . . . . 6 ⊢ Rel ≼
3	2	brrelex2i 5688	. . . . 5 ⊢ (𝐴 ≼ 𝐵 → 𝐵 ∈ V)
4	2	brrelex2i 5688	. . . . 5 ⊢ (𝐶 ≼ 𝐷 → 𝐷 ∈ V)
5		unexg 7697	. . . . 5 ⊢ ((𝐵 ∈ V ∧ 𝐷 ∈ V) → (𝐵 ∪ 𝐷) ∈ V)
6	3, 4, 5	syl2an 597	. . . 4 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐶 ≼ 𝐷) → (𝐵 ∪ 𝐷) ∈ V)
7	6	adantr 480	. . 3 ⊢ (((𝐴 ≼ 𝐵 ∧ 𝐶 ≼ 𝐷) ∧ (𝐵 ∩ 𝐷) = ∅) → (𝐵 ∪ 𝐷) ∈ V)
8		brdomi 8906	. . . . 5 ⊢ (𝐴 ≼ 𝐵 → ∃𝑥 𝑥:𝐴–1-1→𝐵)
9		brdomi 8906	. . . . 5 ⊢ (𝐶 ≼ 𝐷 → ∃𝑦 𝑦:𝐶–1-1→𝐷)
10		exdistrv 1957	. . . . . 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 4412	. . . . . . . . . 10 ⊢ (𝐴 ∩ (𝐶 ∖ 𝐴)) = ∅
12		difss 4076	. . . . . . . . . . . 12 ⊢ (𝐶 ∖ 𝐴) ⊆ 𝐶
13		f1ssres 6743	. . . . . . . . . . . 12 ⊢ ((𝑦:𝐶–1-1→𝐷 ∧ (𝐶 ∖ 𝐴) ⊆ 𝐶) → (𝑦 ↾ (𝐶 ∖ 𝐴)):(𝐶 ∖ 𝐴)–1-1→𝐷)
14	12, 13	mpan2 692	. . . . . . . . . . 11 ⊢ (𝑦:𝐶–1-1→𝐷 → (𝑦 ↾ (𝐶 ∖ 𝐴)):(𝐶 ∖ 𝐴)–1-1→𝐷)
15		f1un 6800	. . . . . . . . . . 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 682	. . . . . . . . . 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 704	. . . . . . . . 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 3433	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
19		vex 3433	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
20	19	resex 5994	. . . . . . . . . . . 12 ⊢ (𝑦 ↾ (𝐶 ∖ 𝐴)) ∈ V
21	18, 20	unex 7698	. . . . . . . . . . 11 ⊢ (𝑥 ∪ (𝑦 ↾ (𝐶 ∖ 𝐴))) ∈ V
22		f1dom3g 8914	. . . . . . . . . . 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 1451	. . . . . . . . . 10 ⊢ (((𝐵 ∪ 𝐷) ∈ V ∧ (𝑥 ∪ (𝑦 ↾ (𝐶 ∖ 𝐴))):(𝐴 ∪ (𝐶 ∖ 𝐴))–1-1→(𝐵 ∪ 𝐷)) → (𝐴 ∪ (𝐶 ∖ 𝐴)) ≼ (𝐵 ∪ 𝐷))
24	23	expcom 413	. . . . . . . . 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 412	. . . . . . 7 ⊢ ((𝑥:𝐴–1-1→𝐵 ∧ 𝑦:𝐶–1-1→𝐷) → ((𝐵 ∩ 𝐷) = ∅ → ((𝐵 ∪ 𝐷) ∈ V → (𝐴 ∪ (𝐶 ∖ 𝐴)) ≼ (𝐵 ∪ 𝐷))))
27	26	exlimivv 1934	. . . . . 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 235	. . . . 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 597	. . . 4 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐶 ≼ 𝐷) → ((𝐵 ∩ 𝐷) = ∅ → ((𝐵 ∪ 𝐷) ∈ V → (𝐴 ∪ (𝐶 ∖ 𝐴)) ≼ (𝐵 ∪ 𝐷))))
30	29	imp 406	. . 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 5121	1 ⊢ (((𝐴 ≼ 𝐵 ∧ 𝐶 ≼ 𝐷) ∧ (𝐵 ∩ 𝐷) = ∅) → (𝐴 ∪ 𝐶) ≼ (𝐵 ∪ 𝐷))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542  ∃wex 1781   ∈ wcel 2114  Vcvv 3429   ∖ cdif 3886   ∪ cun 3887   ∩ cin 3888   ⊆ wss 3889  ∅c0 4273   class class class wbr 5085   ↾ cres 5633  –1-1→wf1 6495   ≼ cdom 8891

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-12 2185  ax-ext 2708  ax-sep 5231  ax-pr 5375  ax-un 7689

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 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4851  df-br 5086  df-opab 5148  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-dom 8895

This theorem is referenced by:  domunsncan  9015  domunsn  9065  sucdom2  9137  unxpdom2  9170  sucxpdom  9171  fodomfi  9222  fodomfiOLD  9240  undjudom  10090  djudom1  10105