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Theorem uniinqs 8737
Description: Class union distributes over the intersection of two subclasses of a quotient space. Compare uniin 4893. (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 4893 . . 3 (𝐵𝐶) ⊆ ( 𝐵 𝐶)
21a1i 11 . 2 ((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) → (𝐵𝐶) ⊆ ( 𝐵 𝐶))
3 eluni2 4870 . . . . . 6 (𝑥 𝐵 ↔ ∃𝑏𝐵 𝑥𝑏)
4 eluni2 4870 . . . . . 6 (𝑥 𝐶 ↔ ∃𝑐𝐶 𝑥𝑐)
53, 4anbi12i 628 . . . . 5 ((𝑥 𝐵𝑥 𝐶) ↔ (∃𝑏𝐵 𝑥𝑏 ∧ ∃𝑐𝐶 𝑥𝑐))
6 elin 3927 . . . . 5 (𝑥 ∈ ( 𝐵 𝐶) ↔ (𝑥 𝐵𝑥 𝐶))
7 reeanv 3218 . . . . 5 (∃𝑏𝐵𝑐𝐶 (𝑥𝑏𝑥𝑐) ↔ (∃𝑏𝐵 𝑥𝑏 ∧ ∃𝑐𝐶 𝑥𝑐))
85, 6, 73bitr4i 303 . . . 4 (𝑥 ∈ ( 𝐵 𝐶) ↔ ∃𝑏𝐵𝑐𝐶 (𝑥𝑏𝑥𝑐))
9 simp3l 1202 . . . . . . 7 (((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) ∧ (𝑏𝐵𝑐𝐶) ∧ (𝑥𝑏𝑥𝑐)) → 𝑥𝑏)
10 simp2l 1200 . . . . . . . 8 (((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) ∧ (𝑏𝐵𝑐𝐶) ∧ (𝑥𝑏𝑥𝑐)) → 𝑏𝐵)
11 inelcm 4425 . . . . . . . . . . 11 ((𝑥𝑏𝑥𝑐) → (𝑏𝑐) ≠ ∅)
12113ad2ant3 1136 . . . . . . . . . 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 1198 . . . . . . . . . . . . . 14 (((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) ∧ (𝑏𝐵𝑐𝐶) ∧ (𝑥𝑏𝑥𝑐)) → 𝐵 ⊆ (𝐴 / 𝑅))
1615, 10sseldd 3946 . . . . . . . . . . . . 13 (((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) ∧ (𝑏𝐵𝑐𝐶) ∧ (𝑥𝑏𝑥𝑐)) → 𝑏 ∈ (𝐴 / 𝑅))
17 simp1r 1199 . . . . . . . . . . . . . 14 (((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) ∧ (𝑏𝐵𝑐𝐶) ∧ (𝑥𝑏𝑥𝑐)) → 𝐶 ⊆ (𝐴 / 𝑅))
18 simp2r 1201 . . . . . . . . . . . . . 14 (((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) ∧ (𝑏𝐵𝑐𝐶) ∧ (𝑥𝑏𝑥𝑐)) → 𝑐𝐶)
1917, 18sseldd 3946 . . . . . . . . . . . . 13 (((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) ∧ (𝑏𝐵𝑐𝐶) ∧ (𝑥𝑏𝑥𝑐)) → 𝑐 ∈ (𝐴 / 𝑅))
2014, 16, 19qsdisj 8734 . . . . . . . . . . . 12 (((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) ∧ (𝑏𝐵𝑐𝐶) ∧ (𝑥𝑏𝑥𝑐)) → (𝑏 = 𝑐 ∨ (𝑏𝑐) = ∅))
2120ord 863 . . . . . . . . . . 11 (((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) ∧ (𝑏𝐵𝑐𝐶) ∧ (𝑥𝑏𝑥𝑐)) → (¬ 𝑏 = 𝑐 → (𝑏𝑐) = ∅))
2221necon1ad 2961 . . . . . . . . . 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 2838 . . . . . . . 8 (((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) ∧ (𝑏𝐵𝑐𝐶) ∧ (𝑥𝑏𝑥𝑐)) → 𝑏𝐶)
2510, 24elind 4155 . . . . . . 7 (((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) ∧ (𝑏𝐵𝑐𝐶) ∧ (𝑥𝑏𝑥𝑐)) → 𝑏 ∈ (𝐵𝐶))
26 elunii 4871 . . . . . . 7 ((𝑥𝑏𝑏 ∈ (𝐵𝐶)) → 𝑥 (𝐵𝐶))
279, 25, 26syl2anc 585 . . . . . 6 (((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) ∧ (𝑏𝐵𝑐𝐶) ∧ (𝑥𝑏𝑥𝑐)) → 𝑥 (𝐵𝐶))
28273expia 1122 . . . . 5 (((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) ∧ (𝑏𝐵𝑐𝐶)) → ((𝑥𝑏𝑥𝑐) → 𝑥 (𝐵𝐶)))
2928rexlimdvva 3206 . . . 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 3951 . 2 ((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) → ( 𝐵 𝐶) ⊆ (𝐵𝐶))
322, 31eqssd 3962 1 ((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) → (𝐵𝐶) = ( 𝐵 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088   = wceq 1542  wcel 2107  wne 2944  wrex 3074  cin 3910  wss 3911  c0 4283   cuni 4866   Er wer 8646   / cqs 8648
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-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385
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-ne 2945  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-er 8649  df-ec 8651  df-qs 8655
This theorem is referenced by: (None)
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