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Theorem undom 9010
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 5325. (Revised by BTernaryTau, 4-Dec-2024.)
Assertion
Ref Expression
undom (((𝐴𝐵𝐶𝐷) ∧ (𝐵𝐷) = ∅) → (𝐴𝐶) ≼ (𝐵𝐷))

Proof of Theorem undom
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 undif2 4441 . 2 (𝐴 ∪ (𝐶𝐴)) = (𝐴𝐶)
2 reldom 8896 . . . . . 6 Rel ≼
32brrelex2i 5694 . . . . 5 (𝐴𝐵𝐵 ∈ V)
42brrelex2i 5694 . . . . 5 (𝐶𝐷𝐷 ∈ V)
5 unexg 7688 . . . . 5 ((𝐵 ∈ V ∧ 𝐷 ∈ V) → (𝐵𝐷) ∈ V)
63, 4, 5syl2an 597 . . . 4 ((𝐴𝐵𝐶𝐷) → (𝐵𝐷) ∈ V)
76adantr 482 . . 3 (((𝐴𝐵𝐶𝐷) ∧ (𝐵𝐷) = ∅) → (𝐵𝐷) ∈ V)
8 brdomi 8905 . . . . 5 (𝐴𝐵 → ∃𝑥 𝑥:𝐴1-1𝐵)
9 brdomi 8905 . . . . 5 (𝐶𝐷 → ∃𝑦 𝑦:𝐶1-1𝐷)
10 exdistrv 1960 . . . . . 6 (∃𝑥𝑦(𝑥:𝐴1-1𝐵𝑦:𝐶1-1𝐷) ↔ (∃𝑥 𝑥:𝐴1-1𝐵 ∧ ∃𝑦 𝑦:𝐶1-1𝐷))
11 disjdif 4436 . . . . . . . . . 10 (𝐴 ∩ (𝐶𝐴)) = ∅
12 difss 4096 . . . . . . . . . . . 12 (𝐶𝐴) ⊆ 𝐶
13 f1ssres 6751 . . . . . . . . . . . 12 ((𝑦:𝐶1-1𝐷 ∧ (𝐶𝐴) ⊆ 𝐶) → (𝑦 ↾ (𝐶𝐴)):(𝐶𝐴)–1-1𝐷)
1412, 13mpan2 690 . . . . . . . . . . 11 (𝑦:𝐶1-1𝐷 → (𝑦 ↾ (𝐶𝐴)):(𝐶𝐴)–1-1𝐷)
15 f1un 6809 . . . . . . . . . . 11 (((𝑥:𝐴1-1𝐵 ∧ (𝑦 ↾ (𝐶𝐴)):(𝐶𝐴)–1-1𝐷) ∧ ((𝐴 ∩ (𝐶𝐴)) = ∅ ∧ (𝐵𝐷) = ∅)) → (𝑥 ∪ (𝑦 ↾ (𝐶𝐴))):(𝐴 ∪ (𝐶𝐴))–1-1→(𝐵𝐷))
1614, 15sylanl2 680 . . . . . . . . . 10 (((𝑥:𝐴1-1𝐵𝑦:𝐶1-1𝐷) ∧ ((𝐴 ∩ (𝐶𝐴)) = ∅ ∧ (𝐵𝐷) = ∅)) → (𝑥 ∪ (𝑦 ↾ (𝐶𝐴))):(𝐴 ∪ (𝐶𝐴))–1-1→(𝐵𝐷))
1711, 16mpanr1 702 . . . . . . . . 9 (((𝑥:𝐴1-1𝐵𝑦:𝐶1-1𝐷) ∧ (𝐵𝐷) = ∅) → (𝑥 ∪ (𝑦 ↾ (𝐶𝐴))):(𝐴 ∪ (𝐶𝐴))–1-1→(𝐵𝐷))
18 vex 3452 . . . . . . . . . . . 12 𝑥 ∈ V
19 vex 3452 . . . . . . . . . . . . 13 𝑦 ∈ V
2019resex 5990 . . . . . . . . . . . 12 (𝑦 ↾ (𝐶𝐴)) ∈ V
2118, 20unex 7685 . . . . . . . . . . 11 (𝑥 ∪ (𝑦 ↾ (𝐶𝐴))) ∈ V
22 f1dom3g 8914 . . . . . . . . . . 11 (((𝑥 ∪ (𝑦 ↾ (𝐶𝐴))) ∈ V ∧ (𝐵𝐷) ∈ V ∧ (𝑥 ∪ (𝑦 ↾ (𝐶𝐴))):(𝐴 ∪ (𝐶𝐴))–1-1→(𝐵𝐷)) → (𝐴 ∪ (𝐶𝐴)) ≼ (𝐵𝐷))
2321, 22mp3an1 1449 . . . . . . . . . 10 (((𝐵𝐷) ∈ V ∧ (𝑥 ∪ (𝑦 ↾ (𝐶𝐴))):(𝐴 ∪ (𝐶𝐴))–1-1→(𝐵𝐷)) → (𝐴 ∪ (𝐶𝐴)) ≼ (𝐵𝐷))
2423expcom 415 . . . . . . . . 9 ((𝑥 ∪ (𝑦 ↾ (𝐶𝐴))):(𝐴 ∪ (𝐶𝐴))–1-1→(𝐵𝐷) → ((𝐵𝐷) ∈ V → (𝐴 ∪ (𝐶𝐴)) ≼ (𝐵𝐷)))
2517, 24syl 17 . . . . . . . 8 (((𝑥:𝐴1-1𝐵𝑦:𝐶1-1𝐷) ∧ (𝐵𝐷) = ∅) → ((𝐵𝐷) ∈ V → (𝐴 ∪ (𝐶𝐴)) ≼ (𝐵𝐷)))
2625ex 414 . . . . . . 7 ((𝑥:𝐴1-1𝐵𝑦:𝐶1-1𝐷) → ((𝐵𝐷) = ∅ → ((𝐵𝐷) ∈ V → (𝐴 ∪ (𝐶𝐴)) ≼ (𝐵𝐷))))
2726exlimivv 1936 . . . . . 6 (∃𝑥𝑦(𝑥:𝐴1-1𝐵𝑦:𝐶1-1𝐷) → ((𝐵𝐷) = ∅ → ((𝐵𝐷) ∈ V → (𝐴 ∪ (𝐶𝐴)) ≼ (𝐵𝐷))))
2810, 27sylbir 234 . . . . 5 ((∃𝑥 𝑥:𝐴1-1𝐵 ∧ ∃𝑦 𝑦:𝐶1-1𝐷) → ((𝐵𝐷) = ∅ → ((𝐵𝐷) ∈ V → (𝐴 ∪ (𝐶𝐴)) ≼ (𝐵𝐷))))
298, 9, 28syl2an 597 . . . 4 ((𝐴𝐵𝐶𝐷) → ((𝐵𝐷) = ∅ → ((𝐵𝐷) ∈ V → (𝐴 ∪ (𝐶𝐴)) ≼ (𝐵𝐷))))
3029imp 408 . . 3 (((𝐴𝐵𝐶𝐷) ∧ (𝐵𝐷) = ∅) → ((𝐵𝐷) ∈ V → (𝐴 ∪ (𝐶𝐴)) ≼ (𝐵𝐷)))
317, 30mpd 15 . 2 (((𝐴𝐵𝐶𝐷) ∧ (𝐵𝐷) = ∅) → (𝐴 ∪ (𝐶𝐴)) ≼ (𝐵𝐷))
321, 31eqbrtrrid 5146 1 (((𝐴𝐵𝐶𝐷) ∧ (𝐵𝐷) = ∅) → (𝐴𝐶) ≼ (𝐵𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wex 1782  wcel 2107  Vcvv 3448  cdif 3912  cun 3913  cin 3914  wss 3915  c0 4287   class class class wbr 5110  cres 5640  1-1wf1 6498  cdom 8888
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-dom 8892
This theorem is referenced by:  domunsncan  9023  sucdom2OLD  9033  domunsn  9078  sucdom2  9157  unxpdom2  9205  sucxpdom  9206  fodomfi  9276  undjudom  10110  djudom1  10125
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