MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  uniinqs Structured version   Visualization version   GIF version

Theorem uniinqs 8743
Description: Class union distributes over the intersection of two subclasses of a quotient space. Compare uniin 4897. (Contributed by FL, 25-May-2007.) (Proof shortened by Mario Carneiro, 11-Jul-2014.)
Hypothesis
Ref Expression
uniinqs.1 𝑅 Er 𝑋
Assertion
Ref Expression
uniinqs ((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) → (𝐵𝐶) = ( 𝐵 𝐶))

Proof of Theorem uniinqs
Dummy variables 𝑏 𝑐 𝑥 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 uniin 4897 . . 3 (𝐵𝐶) ⊆ ( 𝐵 𝐶)
21a1i 11 . 2 ((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) → (𝐵𝐶) ⊆ ( 𝐵 𝐶))
3 eluni2 4874 . . . . . 6 (𝑥 𝐵 ↔ ∃𝑏𝐵 𝑥𝑏)
4 eluni2 4874 . . . . . 6 (𝑥 𝐶 ↔ ∃𝑐𝐶 𝑥𝑐)
53, 4anbi12i 627 . . . . 5 ((𝑥 𝐵𝑥 𝐶) ↔ (∃𝑏𝐵 𝑥𝑏 ∧ ∃𝑐𝐶 𝑥𝑐))
6 elin 3929 . . . . 5 (𝑥 ∈ ( 𝐵 𝐶) ↔ (𝑥 𝐵𝑥 𝐶))
7 reeanv 3215 . . . . 5 (∃𝑏𝐵𝑐𝐶 (𝑥𝑏𝑥𝑐) ↔ (∃𝑏𝐵 𝑥𝑏 ∧ ∃𝑐𝐶 𝑥𝑐))
85, 6, 73bitr4i 302 . . . 4 (𝑥 ∈ ( 𝐵 𝐶) ↔ ∃𝑏𝐵𝑐𝐶 (𝑥𝑏𝑥𝑐))
9 simp3l 1201 . . . . . . 7 (((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) ∧ (𝑏𝐵𝑐𝐶) ∧ (𝑥𝑏𝑥𝑐)) → 𝑥𝑏)
10 simp2l 1199 . . . . . . . 8 (((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) ∧ (𝑏𝐵𝑐𝐶) ∧ (𝑥𝑏𝑥𝑐)) → 𝑏𝐵)
11 inelcm 4429 . . . . . . . . . . 11 ((𝑥𝑏𝑥𝑐) → (𝑏𝑐) ≠ ∅)
12113ad2ant3 1135 . . . . . . . . . 10 (((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) ∧ (𝑏𝐵𝑐𝐶) ∧ (𝑥𝑏𝑥𝑐)) → (𝑏𝑐) ≠ ∅)
13 uniinqs.1 . . . . . . . . . . . . . 14 𝑅 Er 𝑋
1413a1i 11 . . . . . . . . . . . . 13 (((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) ∧ (𝑏𝐵𝑐𝐶) ∧ (𝑥𝑏𝑥𝑐)) → 𝑅 Er 𝑋)
15 simp1l 1197 . . . . . . . . . . . . . 14 (((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) ∧ (𝑏𝐵𝑐𝐶) ∧ (𝑥𝑏𝑥𝑐)) → 𝐵 ⊆ (𝐴 / 𝑅))
1615, 10sseldd 3948 . . . . . . . . . . . . 13 (((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) ∧ (𝑏𝐵𝑐𝐶) ∧ (𝑥𝑏𝑥𝑐)) → 𝑏 ∈ (𝐴 / 𝑅))
17 simp1r 1198 . . . . . . . . . . . . . 14 (((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) ∧ (𝑏𝐵𝑐𝐶) ∧ (𝑥𝑏𝑥𝑐)) → 𝐶 ⊆ (𝐴 / 𝑅))
18 simp2r 1200 . . . . . . . . . . . . . 14 (((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) ∧ (𝑏𝐵𝑐𝐶) ∧ (𝑥𝑏𝑥𝑐)) → 𝑐𝐶)
1917, 18sseldd 3948 . . . . . . . . . . . . 13 (((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) ∧ (𝑏𝐵𝑐𝐶) ∧ (𝑥𝑏𝑥𝑐)) → 𝑐 ∈ (𝐴 / 𝑅))
2014, 16, 19qsdisj 8740 . . . . . . . . . . . 12 (((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) ∧ (𝑏𝐵𝑐𝐶) ∧ (𝑥𝑏𝑥𝑐)) → (𝑏 = 𝑐 ∨ (𝑏𝑐) = ∅))
2120ord 862 . . . . . . . . . . 11 (((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) ∧ (𝑏𝐵𝑐𝐶) ∧ (𝑥𝑏𝑥𝑐)) → (¬ 𝑏 = 𝑐 → (𝑏𝑐) = ∅))
2221necon1ad 2956 . . . . . . . . . 10 (((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) ∧ (𝑏𝐵𝑐𝐶) ∧ (𝑥𝑏𝑥𝑐)) → ((𝑏𝑐) ≠ ∅ → 𝑏 = 𝑐))
2312, 22mpd 15 . . . . . . . . 9 (((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) ∧ (𝑏𝐵𝑐𝐶) ∧ (𝑥𝑏𝑥𝑐)) → 𝑏 = 𝑐)
2423, 18eqeltrd 2832 . . . . . . . 8 (((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) ∧ (𝑏𝐵𝑐𝐶) ∧ (𝑥𝑏𝑥𝑐)) → 𝑏𝐶)
2510, 24elind 4159 . . . . . . 7 (((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) ∧ (𝑏𝐵𝑐𝐶) ∧ (𝑥𝑏𝑥𝑐)) → 𝑏 ∈ (𝐵𝐶))
26 elunii 4875 . . . . . . 7 ((𝑥𝑏𝑏 ∈ (𝐵𝐶)) → 𝑥 (𝐵𝐶))
279, 25, 26syl2anc 584 . . . . . 6 (((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) ∧ (𝑏𝐵𝑐𝐶) ∧ (𝑥𝑏𝑥𝑐)) → 𝑥 (𝐵𝐶))
28273expia 1121 . . . . 5 (((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) ∧ (𝑏𝐵𝑐𝐶)) → ((𝑥𝑏𝑥𝑐) → 𝑥 (𝐵𝐶)))
2928rexlimdvva 3201 . . . 4 ((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) → (∃𝑏𝐵𝑐𝐶 (𝑥𝑏𝑥𝑐) → 𝑥 (𝐵𝐶)))
308, 29biimtrid 241 . . 3 ((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) → (𝑥 ∈ ( 𝐵 𝐶) → 𝑥 (𝐵𝐶)))
3130ssrdv 3953 . 2 ((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) → ( 𝐵 𝐶) ⊆ (𝐵𝐶))
322, 31eqssd 3964 1 ((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) → (𝐵𝐶) = ( 𝐵 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1087   = wceq 1541  wcel 2106  wne 2939  wrex 3069  cin 3912  wss 3913  c0 4287   cuni 4870   Er wer 8652   / cqs 8654
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-11 2154  ax-12 2171  ax-ext 2702  ax-sep 5261  ax-nul 5268  ax-pr 5389
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 2709  df-cleq 2723  df-clel 2809  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3406  df-v 3448  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  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-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-er 8655  df-ec 8657  df-qs 8661
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator