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Theorem uniinqs 7772
Description: Class union distributes over the intersection of two subclasses of a quotient space. Compare uniin 4423. (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 4423 . . 3 (𝐵𝐶) ⊆ ( 𝐵 𝐶)
21a1i 11 . 2 ((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) → (𝐵𝐶) ⊆ ( 𝐵 𝐶))
3 eluni2 4406 . . . . . 6 (𝑥 𝐵 ↔ ∃𝑏𝐵 𝑥𝑏)
4 eluni2 4406 . . . . . 6 (𝑥 𝐶 ↔ ∃𝑐𝐶 𝑥𝑐)
53, 4anbi12i 732 . . . . 5 ((𝑥 𝐵𝑥 𝐶) ↔ (∃𝑏𝐵 𝑥𝑏 ∧ ∃𝑐𝐶 𝑥𝑐))
6 elin 3774 . . . . 5 (𝑥 ∈ ( 𝐵 𝐶) ↔ (𝑥 𝐵𝑥 𝐶))
7 reeanv 3097 . . . . 5 (∃𝑏𝐵𝑐𝐶 (𝑥𝑏𝑥𝑐) ↔ (∃𝑏𝐵 𝑥𝑏 ∧ ∃𝑐𝐶 𝑥𝑐))
85, 6, 73bitr4i 292 . . . 4 (𝑥 ∈ ( 𝐵 𝐶) ↔ ∃𝑏𝐵𝑐𝐶 (𝑥𝑏𝑥𝑐))
9 simp3l 1087 . . . . . . 7 (((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) ∧ (𝑏𝐵𝑐𝐶) ∧ (𝑥𝑏𝑥𝑐)) → 𝑥𝑏)
10 simp2l 1085 . . . . . . . 8 (((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) ∧ (𝑏𝐵𝑐𝐶) ∧ (𝑥𝑏𝑥𝑐)) → 𝑏𝐵)
11 inelcm 4004 . . . . . . . . . . 11 ((𝑥𝑏𝑥𝑐) → (𝑏𝑐) ≠ ∅)
12113ad2ant3 1082 . . . . . . . . . 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 1083 . . . . . . . . . . . . . 14 (((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) ∧ (𝑏𝐵𝑐𝐶) ∧ (𝑥𝑏𝑥𝑐)) → 𝐵 ⊆ (𝐴 / 𝑅))
1615, 10sseldd 3584 . . . . . . . . . . . . 13 (((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) ∧ (𝑏𝐵𝑐𝐶) ∧ (𝑥𝑏𝑥𝑐)) → 𝑏 ∈ (𝐴 / 𝑅))
17 simp1r 1084 . . . . . . . . . . . . . 14 (((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) ∧ (𝑏𝐵𝑐𝐶) ∧ (𝑥𝑏𝑥𝑐)) → 𝐶 ⊆ (𝐴 / 𝑅))
18 simp2r 1086 . . . . . . . . . . . . . 14 (((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) ∧ (𝑏𝐵𝑐𝐶) ∧ (𝑥𝑏𝑥𝑐)) → 𝑐𝐶)
1917, 18sseldd 3584 . . . . . . . . . . . . 13 (((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) ∧ (𝑏𝐵𝑐𝐶) ∧ (𝑥𝑏𝑥𝑐)) → 𝑐 ∈ (𝐴 / 𝑅))
2014, 16, 19qsdisj 7769 . . . . . . . . . . . 12 (((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) ∧ (𝑏𝐵𝑐𝐶) ∧ (𝑥𝑏𝑥𝑐)) → (𝑏 = 𝑐 ∨ (𝑏𝑐) = ∅))
2120ord 392 . . . . . . . . . . 11 (((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) ∧ (𝑏𝐵𝑐𝐶) ∧ (𝑥𝑏𝑥𝑐)) → (¬ 𝑏 = 𝑐 → (𝑏𝑐) = ∅))
2221necon1ad 2807 . . . . . . . . . 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 2698 . . . . . . . 8 (((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) ∧ (𝑏𝐵𝑐𝐶) ∧ (𝑥𝑏𝑥𝑐)) → 𝑏𝐶)
2510, 24elind 3776 . . . . . . 7 (((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) ∧ (𝑏𝐵𝑐𝐶) ∧ (𝑥𝑏𝑥𝑐)) → 𝑏 ∈ (𝐵𝐶))
26 elunii 4407 . . . . . . 7 ((𝑥𝑏𝑏 ∈ (𝐵𝐶)) → 𝑥 (𝐵𝐶))
279, 25, 26syl2anc 692 . . . . . 6 (((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) ∧ (𝑏𝐵𝑐𝐶) ∧ (𝑥𝑏𝑥𝑐)) → 𝑥 (𝐵𝐶))
28273expia 1264 . . . . 5 (((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) ∧ (𝑏𝐵𝑐𝐶)) → ((𝑥𝑏𝑥𝑐) → 𝑥 (𝐵𝐶)))
2928rexlimdvva 3031 . . . 4 ((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) → (∃𝑏𝐵𝑐𝐶 (𝑥𝑏𝑥𝑐) → 𝑥 (𝐵𝐶)))
308, 29syl5bi 232 . . 3 ((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) → (𝑥 ∈ ( 𝐵 𝐶) → 𝑥 (𝐵𝐶)))
3130ssrdv 3589 . 2 ((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) → ( 𝐵 𝐶) ⊆ (𝐵𝐶))
322, 31eqssd 3600 1 ((𝐵 ⊆ (𝐴 / 𝑅) ∧ 𝐶 ⊆ (𝐴 / 𝑅)) → (𝐵𝐶) = ( 𝐵 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  w3a 1036   = wceq 1480  wcel 1987  wne 2790  wrex 2908  cin 3554  wss 3555  c0 3891   cuni 4402   Er wer 7684   / cqs 7686
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-er 7687  df-ec 7689  df-qs 7693
This theorem is referenced by: (None)
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