Theorem dfsup2 8892

Description: Quantifier-free definition of supremum. (Contributed by Scott Fenton, 19-Feb-2013.)

Assertion

Ref	Expression
dfsup2	⊢ sup(𝐵, 𝐴, 𝑅) = ∪ (𝐴 ∖ ((◡𝑅 “ 𝐵) ∪ (𝑅 “ (𝐴 ∖ (◡𝑅 “ 𝐵)))))

Proof of Theorem dfsup2

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-sup 8890	. 2 ⊢ sup(𝐵, 𝐴, 𝑅) = ∪ {𝑥 ∈ 𝐴 ∣ (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))}
2		dfrab3 4230	. . . 4 ⊢ {𝑥 ∈ 𝐴 ∣ (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))} = (𝐴 ∩ {𝑥 ∣ (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))})
3		abeq1 2923	. . . . . . 7 ⊢ ({𝑥 ∣ (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))} = (V ∖ ((◡𝑅 “ 𝐵) ∪ (𝑅 “ (𝐴 ∖ (◡𝑅 “ 𝐵))))) ↔ ∀𝑥((∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)) ↔ 𝑥 ∈ (V ∖ ((◡𝑅 “ 𝐵) ∪ (𝑅 “ (𝐴 ∖ (◡𝑅 “ 𝐵)))))))
4		vex 3444	. . . . . . . . 9 ⊢ 𝑥 ∈ V
5		eldif 3891	. . . . . . . . 9 ⊢ (𝑥 ∈ (V ∖ ((◡𝑅 “ 𝐵) ∪ (𝑅 “ (𝐴 ∖ (◡𝑅 “ 𝐵))))) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ ((◡𝑅 “ 𝐵) ∪ (𝑅 “ (𝐴 ∖ (◡𝑅 “ 𝐵))))))
6	4, 5	mpbiran 708	. . . . . . . 8 ⊢ (𝑥 ∈ (V ∖ ((◡𝑅 “ 𝐵) ∪ (𝑅 “ (𝐴 ∖ (◡𝑅 “ 𝐵))))) ↔ ¬ 𝑥 ∈ ((◡𝑅 “ 𝐵) ∪ (𝑅 “ (𝐴 ∖ (◡𝑅 “ 𝐵)))))
7	4	elima 5901	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (◡𝑅 “ 𝐵) ↔ ∃𝑦 ∈ 𝐵 𝑦◡𝑅𝑥)
8		dfrex2 3202	. . . . . . . . . . . 12 ⊢ (∃𝑦 ∈ 𝐵 𝑦◡𝑅𝑥 ↔ ¬ ∀𝑦 ∈ 𝐵 ¬ 𝑦◡𝑅𝑥)
9	7, 8	bitri 278	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (◡𝑅 “ 𝐵) ↔ ¬ ∀𝑦 ∈ 𝐵 ¬ 𝑦◡𝑅𝑥)
10	4	elima 5901	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝑅 “ (𝐴 ∖ (◡𝑅 “ 𝐵))) ↔ ∃𝑦 ∈ (𝐴 ∖ (◡𝑅 “ 𝐵))𝑦𝑅𝑥)
11		dfrex2 3202	. . . . . . . . . . . 12 ⊢ (∃𝑦 ∈ (𝐴 ∖ (◡𝑅 “ 𝐵))𝑦𝑅𝑥 ↔ ¬ ∀𝑦 ∈ (𝐴 ∖ (◡𝑅 “ 𝐵)) ¬ 𝑦𝑅𝑥)
12	10, 11	bitri 278	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝑅 “ (𝐴 ∖ (◡𝑅 “ 𝐵))) ↔ ¬ ∀𝑦 ∈ (𝐴 ∖ (◡𝑅 “ 𝐵)) ¬ 𝑦𝑅𝑥)
13	9, 12	orbi12i 912	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (◡𝑅 “ 𝐵) ∨ 𝑥 ∈ (𝑅 “ (𝐴 ∖ (◡𝑅 “ 𝐵)))) ↔ (¬ ∀𝑦 ∈ 𝐵 ¬ 𝑦◡𝑅𝑥 ∨ ¬ ∀𝑦 ∈ (𝐴 ∖ (◡𝑅 “ 𝐵)) ¬ 𝑦𝑅𝑥))
14		elun 4076	. . . . . . . . . 10 ⊢ (𝑥 ∈ ((◡𝑅 “ 𝐵) ∪ (𝑅 “ (𝐴 ∖ (◡𝑅 “ 𝐵)))) ↔ (𝑥 ∈ (◡𝑅 “ 𝐵) ∨ 𝑥 ∈ (𝑅 “ (𝐴 ∖ (◡𝑅 “ 𝐵)))))
15		ianor 979	. . . . . . . . . 10 ⊢ (¬ (∀𝑦 ∈ 𝐵 ¬ 𝑦◡𝑅𝑥 ∧ ∀𝑦 ∈ (𝐴 ∖ (◡𝑅 “ 𝐵)) ¬ 𝑦𝑅𝑥) ↔ (¬ ∀𝑦 ∈ 𝐵 ¬ 𝑦◡𝑅𝑥 ∨ ¬ ∀𝑦 ∈ (𝐴 ∖ (◡𝑅 “ 𝐵)) ¬ 𝑦𝑅𝑥))
16	13, 14, 15	3bitr4i 306	. . . . . . . . 9 ⊢ (𝑥 ∈ ((◡𝑅 “ 𝐵) ∪ (𝑅 “ (𝐴 ∖ (◡𝑅 “ 𝐵)))) ↔ ¬ (∀𝑦 ∈ 𝐵 ¬ 𝑦◡𝑅𝑥 ∧ ∀𝑦 ∈ (𝐴 ∖ (◡𝑅 “ 𝐵)) ¬ 𝑦𝑅𝑥))
17	16	con2bii 361	. . . . . . . 8 ⊢ ((∀𝑦 ∈ 𝐵 ¬ 𝑦◡𝑅𝑥 ∧ ∀𝑦 ∈ (𝐴 ∖ (◡𝑅 “ 𝐵)) ¬ 𝑦𝑅𝑥) ↔ ¬ 𝑥 ∈ ((◡𝑅 “ 𝐵) ∪ (𝑅 “ (𝐴 ∖ (◡𝑅 “ 𝐵)))))
18		vex 3444	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
19	18, 4	brcnv 5717	. . . . . . . . . . 11 ⊢ (𝑦◡𝑅𝑥 ↔ 𝑥𝑅𝑦)
20	19	notbii 323	. . . . . . . . . 10 ⊢ (¬ 𝑦◡𝑅𝑥 ↔ ¬ 𝑥𝑅𝑦)
21	20	ralbii 3133	. . . . . . . . 9 ⊢ (∀𝑦 ∈ 𝐵 ¬ 𝑦◡𝑅𝑥 ↔ ∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦)
22		impexp 454	. . . . . . . . . . 11 ⊢ (((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ (◡𝑅 “ 𝐵)) → ¬ 𝑦𝑅𝑥) ↔ (𝑦 ∈ 𝐴 → (¬ 𝑦 ∈ (◡𝑅 “ 𝐵) → ¬ 𝑦𝑅𝑥)))
23		eldif 3891	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (𝐴 ∖ (◡𝑅 “ 𝐵)) ↔ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ (◡𝑅 “ 𝐵)))
24	23	imbi1i 353	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ (𝐴 ∖ (◡𝑅 “ 𝐵)) → ¬ 𝑦𝑅𝑥) ↔ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ (◡𝑅 “ 𝐵)) → ¬ 𝑦𝑅𝑥))
25	18	elima 5901	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (◡𝑅 “ 𝐵) ↔ ∃𝑧 ∈ 𝐵 𝑧◡𝑅𝑦)
26		vex 3444	. . . . . . . . . . . . . . . . 17 ⊢ 𝑧 ∈ V
27	26, 18	brcnv 5717	. . . . . . . . . . . . . . . 16 ⊢ (𝑧◡𝑅𝑦 ↔ 𝑦𝑅𝑧)
28	27	rexbii 3210	. . . . . . . . . . . . . . 15 ⊢ (∃𝑧 ∈ 𝐵 𝑧◡𝑅𝑦 ↔ ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)
29	25, 28	bitri 278	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (◡𝑅 “ 𝐵) ↔ ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)
30	29	imbi2i 339	. . . . . . . . . . . . 13 ⊢ ((𝑦𝑅𝑥 → 𝑦 ∈ (◡𝑅 “ 𝐵)) ↔ (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))
31		con34b 319	. . . . . . . . . . . . 13 ⊢ ((𝑦𝑅𝑥 → 𝑦 ∈ (◡𝑅 “ 𝐵)) ↔ (¬ 𝑦 ∈ (◡𝑅 “ 𝐵) → ¬ 𝑦𝑅𝑥))
32	30, 31	bitr3i 280	. . . . . . . . . . . 12 ⊢ ((𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧) ↔ (¬ 𝑦 ∈ (◡𝑅 “ 𝐵) → ¬ 𝑦𝑅𝑥))
33	32	imbi2i 339	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ 𝐴 → (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)) ↔ (𝑦 ∈ 𝐴 → (¬ 𝑦 ∈ (◡𝑅 “ 𝐵) → ¬ 𝑦𝑅𝑥)))
34	22, 24, 33	3bitr4i 306	. . . . . . . . . 10 ⊢ ((𝑦 ∈ (𝐴 ∖ (◡𝑅 “ 𝐵)) → ¬ 𝑦𝑅𝑥) ↔ (𝑦 ∈ 𝐴 → (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)))
35	34	ralbii2 3131	. . . . . . . . 9 ⊢ (∀𝑦 ∈ (𝐴 ∖ (◡𝑅 “ 𝐵)) ¬ 𝑦𝑅𝑥 ↔ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))
36	21, 35	anbi12i 629	. . . . . . . 8 ⊢ ((∀𝑦 ∈ 𝐵 ¬ 𝑦◡𝑅𝑥 ∧ ∀𝑦 ∈ (𝐴 ∖ (◡𝑅 “ 𝐵)) ¬ 𝑦𝑅𝑥) ↔ (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)))
37	6, 17, 36	3bitr2ri 303	. . . . . . 7 ⊢ ((∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)) ↔ 𝑥 ∈ (V ∖ ((◡𝑅 “ 𝐵) ∪ (𝑅 “ (𝐴 ∖ (◡𝑅 “ 𝐵))))))
38	3, 37	mpgbir 1801	. . . . . 6 ⊢ {𝑥 ∣ (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))} = (V ∖ ((◡𝑅 “ 𝐵) ∪ (𝑅 “ (𝐴 ∖ (◡𝑅 “ 𝐵)))))
39	38	ineq2i 4136	. . . . 5 ⊢ (𝐴 ∩ {𝑥 ∣ (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))}) = (𝐴 ∩ (V ∖ ((◡𝑅 “ 𝐵) ∪ (𝑅 “ (𝐴 ∖ (◡𝑅 “ 𝐵))))))
40		invdif 4195	. . . . 5 ⊢ (𝐴 ∩ (V ∖ ((◡𝑅 “ 𝐵) ∪ (𝑅 “ (𝐴 ∖ (◡𝑅 “ 𝐵)))))) = (𝐴 ∖ ((◡𝑅 “ 𝐵) ∪ (𝑅 “ (𝐴 ∖ (◡𝑅 “ 𝐵)))))
41	39, 40	eqtri 2821	. . . 4 ⊢ (𝐴 ∩ {𝑥 ∣ (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))}) = (𝐴 ∖ ((◡𝑅 “ 𝐵) ∪ (𝑅 “ (𝐴 ∖ (◡𝑅 “ 𝐵)))))
42	2, 41	eqtri 2821	. . 3 ⊢ {𝑥 ∈ 𝐴 ∣ (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))} = (𝐴 ∖ ((◡𝑅 “ 𝐵) ∪ (𝑅 “ (𝐴 ∖ (◡𝑅 “ 𝐵)))))
43	42	unieqi 4813	. 2 ⊢ ∪ {𝑥 ∈ 𝐴 ∣ (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))} = ∪ (𝐴 ∖ ((◡𝑅 “ 𝐵) ∪ (𝑅 “ (𝐴 ∖ (◡𝑅 “ 𝐵)))))
44	1, 43	eqtri 2821	1 ⊢ sup(𝐵, 𝐴, 𝑅) = ∪ (𝐴 ∖ ((◡𝑅 “ 𝐵) ∪ (𝑅 “ (𝐴 ∖ (◡𝑅 “ 𝐵)))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   = wceq 1538   ∈ wcel 2111  {cab 2776  ∀wral 3106  ∃wrex 3107  {crab 3110  Vcvv 3441   ∖ cdif 3878   ∪ cun 3879   ∩ cin 3880  ∪ cuni 4800   class class class wbr 5030  ^◡ccnv 5518   “ cima 5522  supcsup 8888

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-xp 5525  df-cnv 5527  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-sup 8890

This theorem is referenced by:  nfsup  8899