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Theorem uniinqs 7714
Description: Class union distributes over the intersection of two subclasses of a quotient space. Compare uniin 4393. (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 4393 . . 3 (𝐵𝐶) ⊆ ( 𝐵 𝐶)
21a1i 11 . 2 ((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) → (𝐵𝐶) ⊆ ( 𝐵 𝐶))
3 eluni2 4376 . . . . . 6 (𝑥 𝐵 ↔ ∃𝑏𝐵 𝑥𝑏)
4 eluni2 4376 . . . . . 6 (𝑥 𝐶 ↔ ∃𝑐𝐶 𝑥𝑐)
53, 4anbi12i 729 . . . . 5 ((𝑥 𝐵𝑥 𝐶) ↔ (∃𝑏𝐵 𝑥𝑏 ∧ ∃𝑐𝐶 𝑥𝑐))
6 elin 3758 . . . . 5 (𝑥 ∈ ( 𝐵 𝐶) ↔ (𝑥 𝐵𝑥 𝐶))
7 reeanv 3086 . . . . 5 (∃𝑏𝐵𝑐𝐶 (𝑥𝑏𝑥𝑐) ↔ (∃𝑏𝐵 𝑥𝑏 ∧ ∃𝑐𝐶 𝑥𝑐))
85, 6, 73bitr4i 291 . . . 4 (𝑥 ∈ ( 𝐵 𝐶) ↔ ∃𝑏𝐵𝑐𝐶 (𝑥𝑏𝑥𝑐))
9 simp3l 1082 . . . . . . 7 (((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) ∧ (𝑏𝐵𝑐𝐶) ∧ (𝑥𝑏𝑥𝑐)) → 𝑥𝑏)
10 simp2l 1080 . . . . . . . 8 (((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) ∧ (𝑏𝐵𝑐𝐶) ∧ (𝑥𝑏𝑥𝑐)) → 𝑏𝐵)
11 inelcm 3984 . . . . . . . . . . 11 ((𝑥𝑏𝑥𝑐) → (𝑏𝑐) ≠ ∅)
12113ad2ant3 1077 . . . . . . . . . 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 1078 . . . . . . . . . . . . . 14 (((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) ∧ (𝑏𝐵𝑐𝐶) ∧ (𝑥𝑏𝑥𝑐)) → 𝐵 ⊆ (𝐴 / 𝑅))
1615, 10sseldd 3569 . . . . . . . . . . . . 13 (((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) ∧ (𝑏𝐵𝑐𝐶) ∧ (𝑥𝑏𝑥𝑐)) → 𝑏 ∈ (𝐴 / 𝑅))
17 simp1r 1079 . . . . . . . . . . . . . 14 (((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) ∧ (𝑏𝐵𝑐𝐶) ∧ (𝑥𝑏𝑥𝑐)) → 𝐶 ⊆ (𝐴 / 𝑅))
18 simp2r 1081 . . . . . . . . . . . . . 14 (((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) ∧ (𝑏𝐵𝑐𝐶) ∧ (𝑥𝑏𝑥𝑐)) → 𝑐𝐶)
1917, 18sseldd 3569 . . . . . . . . . . . . 13 (((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) ∧ (𝑏𝐵𝑐𝐶) ∧ (𝑥𝑏𝑥𝑐)) → 𝑐 ∈ (𝐴 / 𝑅))
2014, 16, 19qsdisj 7711 . . . . . . . . . . . 12 (((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) ∧ (𝑏𝐵𝑐𝐶) ∧ (𝑥𝑏𝑥𝑐)) → (𝑏 = 𝑐 ∨ (𝑏𝑐) = ∅))
2120ord 391 . . . . . . . . . . 11 (((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) ∧ (𝑏𝐵𝑐𝐶) ∧ (𝑥𝑏𝑥𝑐)) → (¬ 𝑏 = 𝑐 → (𝑏𝑐) = ∅))
2221necon1ad 2799 . . . . . . . . . 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 2688 . . . . . . . 8 (((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) ∧ (𝑏𝐵𝑐𝐶) ∧ (𝑥𝑏𝑥𝑐)) → 𝑏𝐶)
2510, 24elind 3760 . . . . . . 7 (((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) ∧ (𝑏𝐵𝑐𝐶) ∧ (𝑥𝑏𝑥𝑐)) → 𝑏 ∈ (𝐵𝐶))
26 elunii 4377 . . . . . . 7 ((𝑥𝑏𝑏 ∈ (𝐵𝐶)) → 𝑥 (𝐵𝐶))
279, 25, 26syl2anc 691 . . . . . 6 (((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) ∧ (𝑏𝐵𝑐𝐶) ∧ (𝑥𝑏𝑥𝑐)) → 𝑥 (𝐵𝐶))
28273expia 1259 . . . . 5 (((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) ∧ (𝑏𝐵𝑐𝐶)) → ((𝑥𝑏𝑥𝑐) → 𝑥 (𝐵𝐶)))
2928rexlimdvva 3020 . . . 4 ((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) → (∃𝑏𝐵𝑐𝐶 (𝑥𝑏𝑥𝑐) → 𝑥 (𝐵𝐶)))
308, 29syl5bi 231 . . 3 ((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) → (𝑥 ∈ ( 𝐵 𝐶) → 𝑥 (𝐵𝐶)))
3130ssrdv 3574 . 2 ((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) → ( 𝐵 𝐶) ⊆ (𝐵𝐶))
322, 31eqssd 3585 1 ((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) → (𝐵𝐶) = ( 𝐵 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1031   = wceq 1475  wcel 1977  wne 2780  wrex 2897  cin 3539  wss 3540  c0 3874   cuni 4372   Er wer 7626   / cqs 7628
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-er 7629  df-ec 7631  df-qs 7635
This theorem is referenced by: (None)
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