Theorem undom 9120

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 5332. (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 4434	. 2 ⊢ (𝐴 ∪ (𝐶 ∖ 𝐴)) = (𝐴 ∪ 𝐶)
2		reldom 9015	. . . . . 6 ⊢ Rel ≼
3	2	brrelex2i 5713	. . . . 5 ⊢ (𝐴 ≼ 𝐵 → 𝐵 ∈ V)
4	2	brrelex2i 5713	. . . . 5 ⊢ (𝐶 ≼ 𝐷 → 𝐷 ∈ V)
5		unexg 7790	. . . . 5 ⊢ ((𝐵 ∈ V ∧ 𝐷 ∈ V) → (𝐵 ∪ 𝐷) ∈ V)
6	3, 4, 5	syl2an 608	. . . 4 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐶 ≼ 𝐷) → (𝐵 ∪ 𝐷) ∈ V)
7	6	adantr 486	. . 3 ⊢ (((𝐴 ≼ 𝐵 ∧ 𝐶 ≼ 𝐷) ∧ (𝐵 ∩ 𝐷) = ∅) → (𝐵 ∪ 𝐷) ∈ V)
8		brdomi 9022	. . . . 5 ⊢ (𝐴 ≼ 𝐵 → ∃𝑥 𝑥:𝐴⟶1-1𝐵)
9		brdomi 9022	. . . . 5 ⊢ (𝐶 ≼ 𝐷 → ∃𝑦 𝑦:𝐶⟶1-1𝐷)
10		exdistrv 1990	. . . . . 6 ⊢ (∃𝑥∃𝑦(𝑥:𝐴⟶1-1𝐵 ∧ 𝑦:𝐶⟶1-1𝐷) ↔ (∃𝑥 𝑥:𝐴⟶1-1𝐵 ∧ ∃𝑦 𝑦:𝐶⟶1-1𝐷))
11		disjdif 4429	. . . . . . . . . 10 ⊢ (𝐴 ∩ (𝐶 ∖ 𝐴)) = ∅
12		difss 4075	. . . . . . . . . . . 12 ⊢ (𝐶 ∖ 𝐴) ⊆ 𝐶
13		f1ssres 6815	. . . . . . . . . . . 12 ⊢ ((𝑦:𝐶⟶1-1𝐷 ∧ (𝐶 ∖ 𝐴) ⊆ 𝐶) → (𝑦 ↾ (𝐶 ∖ 𝐴)):(𝐶 ∖ 𝐴)⟶1-1𝐷)
14	12, 13	mpan2 704	. . . . . . . . . . 11 ⊢ (𝑦:𝐶⟶1-1𝐷 → (𝑦 ↾ (𝐶 ∖ 𝐴)):(𝐶 ∖ 𝐴)⟶1-1𝐷)
15		f1un 6873	. . . . . . . . . . 11 ⊢ (((𝑥:𝐴⟶1-1𝐵 ∧ (𝑦 ↾ (𝐶 ∖ 𝐴)):(𝐶 ∖ 𝐴)⟶1-1𝐷) ∧ ((𝐴 ∩ (𝐶 ∖ 𝐴)) = ∅ ∧ (𝐵 ∩ 𝐷) = ∅)) → (𝑥 ∪ (𝑦 ↾ (𝐶 ∖ 𝐴))):(𝐴 ∪ (𝐶 ∖ 𝐴))⟶1-1(𝐵 ∪ 𝐷))
16	14, 15	sylanl2 694	. . . . . . . . . 10 ⊢ (((𝑥:𝐴⟶1-1𝐵 ∧ 𝑦:𝐶⟶1-1𝐷) ∧ ((𝐴 ∩ (𝐶 ∖ 𝐴)) = ∅ ∧ (𝐵 ∩ 𝐷) = ∅)) → (𝑥 ∪ (𝑦 ↾ (𝐶 ∖ 𝐴))):(𝐴 ∪ (𝐶 ∖ 𝐴))⟶1-1(𝐵 ∪ 𝐷))
17	11, 16	mpanr1 716	. . . . . . . . 9 ⊢ (((𝑥:𝐴⟶1-1𝐵 ∧ 𝑦:𝐶⟶1-1𝐷) ∧ (𝐵 ∩ 𝐷) = ∅) → (𝑥 ∪ (𝑦 ↾ (𝐶 ∖ 𝐴))):(𝐴 ∪ (𝐶 ∖ 𝐴))⟶1-1(𝐵 ∪ 𝐷))
18		vex 3458	. . . . . . . . . . . 12 ⊢ 𝑥 ∈ V
19		vex 3458	. . . . . . . . . . . . 13 ⊢ 𝑦 ∈ V
20	19	resex 6034	. . . . . . . . . . . 12 ⊢ (𝑦 ↾ (𝐶 ∖ 𝐴)) ∈ V
21	18, 20	unex 7791	. . . . . . . . . . 11 ⊢ (𝑥 ∪ (𝑦 ↾ (𝐶 ∖ 𝐴))) ∈ V
22		f1dom3g 9030	. . . . . . . . . . 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 1478	. . . . . . . . . 10 ⊢ (((𝐵 ∪ 𝐷) ∈ V ∧ (𝑥 ∪ (𝑦 ↾ (𝐶 ∖ 𝐴))):(𝐴 ∪ (𝐶 ∖ 𝐴))⟶1-1(𝐵 ∪ 𝐷)) → (𝐴 ∪ (𝐶 ∖ 𝐴)) ≼ (𝐵 ∪ 𝐷))
24	23	expcom 419	. . . . . . . . 9 ⊢ ((𝑥 ∪ (𝑦 ↾ (𝐶 ∖ 𝐴))):(𝐴 ∪ (𝐶 ∖ 𝐴))⟶1-1(𝐵 ∪ 𝐷) → ((𝐵 ∪ 𝐷) ∈ V → (𝐴 ∪ (𝐶 ∖ 𝐴)) ≼ (𝐵 ∪ 𝐷)))
25	17, 24	syl 18	. . . . . . . 8 ⊢ (((𝑥:𝐴⟶1-1𝐵 ∧ 𝑦:𝐶⟶1-1𝐷) ∧ (𝐵 ∩ 𝐷) = ∅) → ((𝐵 ∪ 𝐷) ∈ V → (𝐴 ∪ (𝐶 ∖ 𝐴)) ≼ (𝐵 ∪ 𝐷)))
26	25	ex 418	. . . . . . 7 ⊢ ((𝑥:𝐴⟶1-1𝐵 ∧ 𝑦:𝐶⟶1-1𝐷) → ((𝐵 ∩ 𝐷) = ∅ → ((𝐵 ∪ 𝐷) ∈ V → (𝐴 ∪ (𝐶 ∖ 𝐴)) ≼ (𝐵 ∪ 𝐷))))
27	26	exlimivv 1967	. . . . . 6 ⊢ (∃𝑥∃𝑦(𝑥:𝐴⟶1-1𝐵 ∧ 𝑦:𝐶⟶1-1𝐷) → ((𝐵 ∩ 𝐷) = ∅ → ((𝐵 ∪ 𝐷) ∈ V → (𝐴 ∪ (𝐶 ∖ 𝐴)) ≼ (𝐵 ∪ 𝐷))))
28	10, 27	sylbir 238	. . . . 5 ⊢ ((∃𝑥 𝑥:𝐴⟶1-1𝐵 ∧ ∃𝑦 𝑦:𝐶⟶1-1𝐷) → ((𝐵 ∩ 𝐷) = ∅ → ((𝐵 ∪ 𝐷) ∈ V → (𝐴 ∪ (𝐶 ∖ 𝐴)) ≼ (𝐵 ∪ 𝐷))))
29	8, 9, 28	syl2an 608	. . . 4 ⊢ ((𝐴 ≼ 𝐵 ∧ 𝐶 ≼ 𝐷) → ((𝐵 ∩ 𝐷) = ∅ → ((𝐵 ∪ 𝐷) ∈ V → (𝐴 ∪ (𝐶 ∖ 𝐴)) ≼ (𝐵 ∪ 𝐷))))
30	29	imp 412	. . 3 ⊢ (((𝐴 ≼ 𝐵 ∧ 𝐶 ≼ 𝐷) ∧ (𝐵 ∩ 𝐷) = ∅) → ((𝐵 ∪ 𝐷) ∈ V → (𝐴 ∪ (𝐶 ∖ 𝐴)) ≼ (𝐵 ∪ 𝐷)))
31	7, 30	mpd 16	. 2 ⊢ (((𝐴 ≼ 𝐵 ∧ 𝐶 ≼ 𝐷) ∧ (𝐵 ∩ 𝐷) = ∅) → (𝐴 ∪ (𝐶 ∖ 𝐴)) ≼ (𝐵 ∪ 𝐷))
32	1, 31	eqbrtrrid 5145	1 ⊢ (((𝐴 ≼ 𝐵 ∧ 𝐶 ≼ 𝐷) ∧ (𝐵 ∩ 𝐷) = ∅) → (𝐴 ∪ 𝐶) ≼ (𝐵 ∪ 𝐷))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1570  ∃wex 1812   ∈ wcel 2147  Vcvv 3454   ⊆ wss 3900   ∖ cdif 4048   ∩ cin 4092   ∪ cun 4152  ∅c0 4281   class class class wbr 5106   ↾ cres 5658  ⟶1-1wf1 6561   ≼ cdom 9007

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-12 2216  ax-ext 2735  ax-sep 5253  ax-pr 5396  ax-un 7781

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 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5172  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-res 5668  df-fun 6566  df-fn 6567  df-f 6568  df-f1 6569  df-fo 6570  df-f1o 6571  df-dom 9011

This theorem is used by:  domunsncan  9132  domunsn  9182  sucdom2  9254  unxpdom2  9287  sucxpdom  9288  fodomfi  9340  undjudom  10283  djudom1  10298