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Theorem uniinqs 8366
Description: Class union distributes over the intersection of two subclasses of a quotient space. Compare uniin 4850. (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 4850 . . 3 (𝐵𝐶) ⊆ ( 𝐵 𝐶)
21a1i 11 . 2 ((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) → (𝐵𝐶) ⊆ ( 𝐵 𝐶))
3 eluni2 4834 . . . . . 6 (𝑥 𝐵 ↔ ∃𝑏𝐵 𝑥𝑏)
4 eluni2 4834 . . . . . 6 (𝑥 𝐶 ↔ ∃𝑐𝐶 𝑥𝑐)
53, 4anbi12i 626 . . . . 5 ((𝑥 𝐵𝑥 𝐶) ↔ (∃𝑏𝐵 𝑥𝑏 ∧ ∃𝑐𝐶 𝑥𝑐))
6 elin 4166 . . . . 5 (𝑥 ∈ ( 𝐵 𝐶) ↔ (𝑥 𝐵𝑥 𝐶))
7 reeanv 3365 . . . . 5 (∃𝑏𝐵𝑐𝐶 (𝑥𝑏𝑥𝑐) ↔ (∃𝑏𝐵 𝑥𝑏 ∧ ∃𝑐𝐶 𝑥𝑐))
85, 6, 73bitr4i 304 . . . 4 (𝑥 ∈ ( 𝐵 𝐶) ↔ ∃𝑏𝐵𝑐𝐶 (𝑥𝑏𝑥𝑐))
9 simp3l 1193 . . . . . . 7 (((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) ∧ (𝑏𝐵𝑐𝐶) ∧ (𝑥𝑏𝑥𝑐)) → 𝑥𝑏)
10 simp2l 1191 . . . . . . . 8 (((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) ∧ (𝑏𝐵𝑐𝐶) ∧ (𝑥𝑏𝑥𝑐)) → 𝑏𝐵)
11 inelcm 4410 . . . . . . . . . . 11 ((𝑥𝑏𝑥𝑐) → (𝑏𝑐) ≠ ∅)
12113ad2ant3 1127 . . . . . . . . . 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 1189 . . . . . . . . . . . . . 14 (((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) ∧ (𝑏𝐵𝑐𝐶) ∧ (𝑥𝑏𝑥𝑐)) → 𝐵 ⊆ (𝐴 / 𝑅))
1615, 10sseldd 3965 . . . . . . . . . . . . 13 (((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) ∧ (𝑏𝐵𝑐𝐶) ∧ (𝑥𝑏𝑥𝑐)) → 𝑏 ∈ (𝐴 / 𝑅))
17 simp1r 1190 . . . . . . . . . . . . . 14 (((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) ∧ (𝑏𝐵𝑐𝐶) ∧ (𝑥𝑏𝑥𝑐)) → 𝐶 ⊆ (𝐴 / 𝑅))
18 simp2r 1192 . . . . . . . . . . . . . 14 (((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) ∧ (𝑏𝐵𝑐𝐶) ∧ (𝑥𝑏𝑥𝑐)) → 𝑐𝐶)
1917, 18sseldd 3965 . . . . . . . . . . . . 13 (((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) ∧ (𝑏𝐵𝑐𝐶) ∧ (𝑥𝑏𝑥𝑐)) → 𝑐 ∈ (𝐴 / 𝑅))
2014, 16, 19qsdisj 8363 . . . . . . . . . . . 12 (((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) ∧ (𝑏𝐵𝑐𝐶) ∧ (𝑥𝑏𝑥𝑐)) → (𝑏 = 𝑐 ∨ (𝑏𝑐) = ∅))
2120ord 858 . . . . . . . . . . 11 (((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) ∧ (𝑏𝐵𝑐𝐶) ∧ (𝑥𝑏𝑥𝑐)) → (¬ 𝑏 = 𝑐 → (𝑏𝑐) = ∅))
2221necon1ad 3030 . . . . . . . . . 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 2910 . . . . . . . 8 (((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) ∧ (𝑏𝐵𝑐𝐶) ∧ (𝑥𝑏𝑥𝑐)) → 𝑏𝐶)
2510, 24elind 4168 . . . . . . 7 (((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) ∧ (𝑏𝐵𝑐𝐶) ∧ (𝑥𝑏𝑥𝑐)) → 𝑏 ∈ (𝐵𝐶))
26 elunii 4835 . . . . . . 7 ((𝑥𝑏𝑏 ∈ (𝐵𝐶)) → 𝑥 (𝐵𝐶))
279, 25, 26syl2anc 584 . . . . . 6 (((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) ∧ (𝑏𝐵𝑐𝐶) ∧ (𝑥𝑏𝑥𝑐)) → 𝑥 (𝐵𝐶))
28273expia 1113 . . . . 5 (((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) ∧ (𝑏𝐵𝑐𝐶)) → ((𝑥𝑏𝑥𝑐) → 𝑥 (𝐵𝐶)))
2928rexlimdvva 3291 . . . 4 ((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) → (∃𝑏𝐵𝑐𝐶 (𝑥𝑏𝑥𝑐) → 𝑥 (𝐵𝐶)))
308, 29syl5bi 243 . . 3 ((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) → (𝑥 ∈ ( 𝐵 𝐶) → 𝑥 (𝐵𝐶)))
3130ssrdv 3970 . 2 ((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) → ( 𝐵 𝐶) ⊆ (𝐵𝐶))
322, 31eqssd 3981 1 ((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) → (𝐵𝐶) = ( 𝐵 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3a 1079   = wceq 1528  wcel 2105  wne 3013  wrex 3136  cin 3932  wss 3933  c0 4288   cuni 4830   Er wer 8275   / cqs 8277
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-er 8278  df-ec 8280  df-qs 8284
This theorem is referenced by: (None)
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