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Theorem undom 8846
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 5288. (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 4410 . 2 (𝐴 ∪ (𝐶𝐴)) = (𝐴𝐶)
2 reldom 8739 . . . . . 6 Rel ≼
32brrelex2i 5644 . . . . 5 (𝐴𝐵𝐵 ∈ V)
42brrelex2i 5644 . . . . 5 (𝐶𝐷𝐷 ∈ V)
5 unexg 7599 . . . . 5 ((𝐵 ∈ V ∧ 𝐷 ∈ V) → (𝐵𝐷) ∈ V)
63, 4, 5syl2an 596 . . . 4 ((𝐴𝐵𝐶𝐷) → (𝐵𝐷) ∈ V)
76adantr 481 . . 3 (((𝐴𝐵𝐶𝐷) ∧ (𝐵𝐷) = ∅) → (𝐵𝐷) ∈ V)
8 brdomi 8748 . . . . 5 (𝐴𝐵 → ∃𝑥 𝑥:𝐴1-1𝐵)
9 brdomi 8748 . . . . 5 (𝐶𝐷 → ∃𝑦 𝑦:𝐶1-1𝐷)
10 exdistrv 1959 . . . . . 6 (∃𝑥𝑦(𝑥:𝐴1-1𝐵𝑦:𝐶1-1𝐷) ↔ (∃𝑥 𝑥:𝐴1-1𝐵 ∧ ∃𝑦 𝑦:𝐶1-1𝐷))
11 disjdif 4405 . . . . . . . . . 10 (𝐴 ∩ (𝐶𝐴)) = ∅
12 difss 4066 . . . . . . . . . . . 12 (𝐶𝐴) ⊆ 𝐶
13 f1ssres 6678 . . . . . . . . . . . 12 ((𝑦:𝐶1-1𝐷 ∧ (𝐶𝐴) ⊆ 𝐶) → (𝑦 ↾ (𝐶𝐴)):(𝐶𝐴)–1-1𝐷)
1412, 13mpan2 688 . . . . . . . . . . 11 (𝑦:𝐶1-1𝐷 → (𝑦 ↾ (𝐶𝐴)):(𝐶𝐴)–1-1𝐷)
15 f1un 6736 . . . . . . . . . . 11 (((𝑥:𝐴1-1𝐵 ∧ (𝑦 ↾ (𝐶𝐴)):(𝐶𝐴)–1-1𝐷) ∧ ((𝐴 ∩ (𝐶𝐴)) = ∅ ∧ (𝐵𝐷) = ∅)) → (𝑥 ∪ (𝑦 ↾ (𝐶𝐴))):(𝐴 ∪ (𝐶𝐴))–1-1→(𝐵𝐷))
1614, 15sylanl2 678 . . . . . . . . . 10 (((𝑥:𝐴1-1𝐵𝑦:𝐶1-1𝐷) ∧ ((𝐴 ∩ (𝐶𝐴)) = ∅ ∧ (𝐵𝐷) = ∅)) → (𝑥 ∪ (𝑦 ↾ (𝐶𝐴))):(𝐴 ∪ (𝐶𝐴))–1-1→(𝐵𝐷))
1711, 16mpanr1 700 . . . . . . . . 9 (((𝑥:𝐴1-1𝐵𝑦:𝐶1-1𝐷) ∧ (𝐵𝐷) = ∅) → (𝑥 ∪ (𝑦 ↾ (𝐶𝐴))):(𝐴 ∪ (𝐶𝐴))–1-1→(𝐵𝐷))
18 vex 3436 . . . . . . . . . . . 12 𝑥 ∈ V
19 vex 3436 . . . . . . . . . . . . 13 𝑦 ∈ V
2019resex 5939 . . . . . . . . . . . 12 (𝑦 ↾ (𝐶𝐴)) ∈ V
2118, 20unex 7596 . . . . . . . . . . 11 (𝑥 ∪ (𝑦 ↾ (𝐶𝐴))) ∈ V
22 f1dom3g 8755 . . . . . . . . . . 11 (((𝑥 ∪ (𝑦 ↾ (𝐶𝐴))) ∈ V ∧ (𝐵𝐷) ∈ V ∧ (𝑥 ∪ (𝑦 ↾ (𝐶𝐴))):(𝐴 ∪ (𝐶𝐴))–1-1→(𝐵𝐷)) → (𝐴 ∪ (𝐶𝐴)) ≼ (𝐵𝐷))
2321, 22mp3an1 1447 . . . . . . . . . 10 (((𝐵𝐷) ∈ V ∧ (𝑥 ∪ (𝑦 ↾ (𝐶𝐴))):(𝐴 ∪ (𝐶𝐴))–1-1→(𝐵𝐷)) → (𝐴 ∪ (𝐶𝐴)) ≼ (𝐵𝐷))
2423expcom 414 . . . . . . . . 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 413 . . . . . . 7 ((𝑥:𝐴1-1𝐵𝑦:𝐶1-1𝐷) → ((𝐵𝐷) = ∅ → ((𝐵𝐷) ∈ V → (𝐴 ∪ (𝐶𝐴)) ≼ (𝐵𝐷))))
2726exlimivv 1935 . . . . . 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 596 . . . 4 ((𝐴𝐵𝐶𝐷) → ((𝐵𝐷) = ∅ → ((𝐵𝐷) ∈ V → (𝐴 ∪ (𝐶𝐴)) ≼ (𝐵𝐷))))
3029imp 407 . . 3 (((𝐴𝐵𝐶𝐷) ∧ (𝐵𝐷) = ∅) → ((𝐵𝐷) ∈ V → (𝐴 ∪ (𝐶𝐴)) ≼ (𝐵𝐷)))
317, 30mpd 15 . 2 (((𝐴𝐵𝐶𝐷) ∧ (𝐵𝐷) = ∅) → (𝐴 ∪ (𝐶𝐴)) ≼ (𝐵𝐷))
321, 31eqbrtrrid 5110 1 (((𝐴𝐵𝐶𝐷) ∧ (𝐵𝐷) = ∅) → (𝐴𝐶) ≼ (𝐵𝐷))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1539  wex 1782  wcel 2106  Vcvv 3432  cdif 3884  cun 3885  cin 3886  wss 3887  c0 4256   class class class wbr 5074  cres 5591  1-1wf1 6430  cdom 8731
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-dom 8735
This theorem is referenced by:  domunsncan  8859  sucdom2OLD  8869  domunsn  8914  sucdom2  8989  unxpdom2  9031  sucxpdom  9032  fodomfi  9092  undjudom  9923  djudom1  9938
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