Theorem dfsup2 9395

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 9393	. 2 ⊢ sup(𝐵, 𝐴, 𝑅) = ∪ {𝑥 ∈ 𝐴 ∣ (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))}
2		dfrab3 4282	. . . 4 ⊢ {𝑥 ∈ 𝐴 ∣ (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))} = (𝐴 ∩ {𝑥 ∣ (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))})
3		eqabcb 2869	. . . . . . 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 3451	. . . . . . . . 9 ⊢ 𝑥 ∈ V
5		eldif 3924	. . . . . . . . 9 ⊢ (𝑥 ∈ (V ∖ ((◡𝑅 “ 𝐵) ∪ (𝑅 “ (𝐴 ∖ (◡𝑅 “ 𝐵))))) ↔ (𝑥 ∈ V ∧ ¬ 𝑥 ∈ ((◡𝑅 “ 𝐵) ∪ (𝑅 “ (𝐴 ∖ (◡𝑅 “ 𝐵))))))
6	4, 5	mpbiran 709	. . . . . . . 8 ⊢ (𝑥 ∈ (V ∖ ((◡𝑅 “ 𝐵) ∪ (𝑅 “ (𝐴 ∖ (◡𝑅 “ 𝐵))))) ↔ ¬ 𝑥 ∈ ((◡𝑅 “ 𝐵) ∪ (𝑅 “ (𝐴 ∖ (◡𝑅 “ 𝐵)))))
7	4	elima 6036	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (◡𝑅 “ 𝐵) ↔ ∃𝑦 ∈ 𝐵 𝑦◡𝑅𝑥)
8		dfrex2 3056	. . . . . . . . . . . 12 ⊢ (∃𝑦 ∈ 𝐵 𝑦◡𝑅𝑥 ↔ ¬ ∀𝑦 ∈ 𝐵 ¬ 𝑦◡𝑅𝑥)
9	7, 8	bitri 275	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (◡𝑅 “ 𝐵) ↔ ¬ ∀𝑦 ∈ 𝐵 ¬ 𝑦◡𝑅𝑥)
10	4	elima 6036	. . . . . . . . . . . 12 ⊢ (𝑥 ∈ (𝑅 “ (𝐴 ∖ (◡𝑅 “ 𝐵))) ↔ ∃𝑦 ∈ (𝐴 ∖ (◡𝑅 “ 𝐵))𝑦𝑅𝑥)
11		dfrex2 3056	. . . . . . . . . . . 12 ⊢ (∃𝑦 ∈ (𝐴 ∖ (◡𝑅 “ 𝐵))𝑦𝑅𝑥 ↔ ¬ ∀𝑦 ∈ (𝐴 ∖ (◡𝑅 “ 𝐵)) ¬ 𝑦𝑅𝑥)
12	10, 11	bitri 275	. . . . . . . . . . 11 ⊢ (𝑥 ∈ (𝑅 “ (𝐴 ∖ (◡𝑅 “ 𝐵))) ↔ ¬ ∀𝑦 ∈ (𝐴 ∖ (◡𝑅 “ 𝐵)) ¬ 𝑦𝑅𝑥)
13	9, 12	orbi12i 914	. . . . . . . . . 10 ⊢ ((𝑥 ∈ (◡𝑅 “ 𝐵) ∨ 𝑥 ∈ (𝑅 “ (𝐴 ∖ (◡𝑅 “ 𝐵)))) ↔ (¬ ∀𝑦 ∈ 𝐵 ¬ 𝑦◡𝑅𝑥 ∨ ¬ ∀𝑦 ∈ (𝐴 ∖ (◡𝑅 “ 𝐵)) ¬ 𝑦𝑅𝑥))
14		elun 4116	. . . . . . . . . 10 ⊢ (𝑥 ∈ ((◡𝑅 “ 𝐵) ∪ (𝑅 “ (𝐴 ∖ (◡𝑅 “ 𝐵)))) ↔ (𝑥 ∈ (◡𝑅 “ 𝐵) ∨ 𝑥 ∈ (𝑅 “ (𝐴 ∖ (◡𝑅 “ 𝐵)))))
15		ianor 983	. . . . . . . . . 10 ⊢ (¬ (∀𝑦 ∈ 𝐵 ¬ 𝑦◡𝑅𝑥 ∧ ∀𝑦 ∈ (𝐴 ∖ (◡𝑅 “ 𝐵)) ¬ 𝑦𝑅𝑥) ↔ (¬ ∀𝑦 ∈ 𝐵 ¬ 𝑦◡𝑅𝑥 ∨ ¬ ∀𝑦 ∈ (𝐴 ∖ (◡𝑅 “ 𝐵)) ¬ 𝑦𝑅𝑥))
16	13, 14, 15	3bitr4i 303	. . . . . . . . 9 ⊢ (𝑥 ∈ ((◡𝑅 “ 𝐵) ∪ (𝑅 “ (𝐴 ∖ (◡𝑅 “ 𝐵)))) ↔ ¬ (∀𝑦 ∈ 𝐵 ¬ 𝑦◡𝑅𝑥 ∧ ∀𝑦 ∈ (𝐴 ∖ (◡𝑅 “ 𝐵)) ¬ 𝑦𝑅𝑥))
17	16	con2bii 357	. . . . . . . 8 ⊢ ((∀𝑦 ∈ 𝐵 ¬ 𝑦◡𝑅𝑥 ∧ ∀𝑦 ∈ (𝐴 ∖ (◡𝑅 “ 𝐵)) ¬ 𝑦𝑅𝑥) ↔ ¬ 𝑥 ∈ ((◡𝑅 “ 𝐵) ∪ (𝑅 “ (𝐴 ∖ (◡𝑅 “ 𝐵)))))
18		vex 3451	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
19	18, 4	brcnv 5846	. . . . . . . . . . 11 ⊢ (𝑦◡𝑅𝑥 ↔ 𝑥𝑅𝑦)
20	19	notbii 320	. . . . . . . . . 10 ⊢ (¬ 𝑦◡𝑅𝑥 ↔ ¬ 𝑥𝑅𝑦)
21	20	ralbii 3075	. . . . . . . . 9 ⊢ (∀𝑦 ∈ 𝐵 ¬ 𝑦◡𝑅𝑥 ↔ ∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦)
22		impexp 450	. . . . . . . . . . 11 ⊢ (((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ (◡𝑅 “ 𝐵)) → ¬ 𝑦𝑅𝑥) ↔ (𝑦 ∈ 𝐴 → (¬ 𝑦 ∈ (◡𝑅 “ 𝐵) → ¬ 𝑦𝑅𝑥)))
23		eldif 3924	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (𝐴 ∖ (◡𝑅 “ 𝐵)) ↔ (𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ (◡𝑅 “ 𝐵)))
24	23	imbi1i 349	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ (𝐴 ∖ (◡𝑅 “ 𝐵)) → ¬ 𝑦𝑅𝑥) ↔ ((𝑦 ∈ 𝐴 ∧ ¬ 𝑦 ∈ (◡𝑅 “ 𝐵)) → ¬ 𝑦𝑅𝑥))
25	18	elima 6036	. . . . . . . . . . . . . . 15 ⊢ (𝑦 ∈ (◡𝑅 “ 𝐵) ↔ ∃𝑧 ∈ 𝐵 𝑧◡𝑅𝑦)
26		vex 3451	. . . . . . . . . . . . . . . . 17 ⊢ 𝑧 ∈ V
27	26, 18	brcnv 5846	. . . . . . . . . . . . . . . 16 ⊢ (𝑧◡𝑅𝑦 ↔ 𝑦𝑅𝑧)
28	27	rexbii 3076	. . . . . . . . . . . . . . 15 ⊢ (∃𝑧 ∈ 𝐵 𝑧◡𝑅𝑦 ↔ ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)
29	25, 28	bitri 275	. . . . . . . . . . . . . 14 ⊢ (𝑦 ∈ (◡𝑅 “ 𝐵) ↔ ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)
30	29	imbi2i 336	. . . . . . . . . . . . 13 ⊢ ((𝑦𝑅𝑥 → 𝑦 ∈ (◡𝑅 “ 𝐵)) ↔ (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))
31		con34b 316	. . . . . . . . . . . . 13 ⊢ ((𝑦𝑅𝑥 → 𝑦 ∈ (◡𝑅 “ 𝐵)) ↔ (¬ 𝑦 ∈ (◡𝑅 “ 𝐵) → ¬ 𝑦𝑅𝑥))
32	30, 31	bitr3i 277	. . . . . . . . . . . 12 ⊢ ((𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧) ↔ (¬ 𝑦 ∈ (◡𝑅 “ 𝐵) → ¬ 𝑦𝑅𝑥))
33	32	imbi2i 336	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ 𝐴 → (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)) ↔ (𝑦 ∈ 𝐴 → (¬ 𝑦 ∈ (◡𝑅 “ 𝐵) → ¬ 𝑦𝑅𝑥)))
34	22, 24, 33	3bitr4i 303	. . . . . . . . . 10 ⊢ ((𝑦 ∈ (𝐴 ∖ (◡𝑅 “ 𝐵)) → ¬ 𝑦𝑅𝑥) ↔ (𝑦 ∈ 𝐴 → (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)))
35	34	ralbii2 3071	. . . . . . . . 9 ⊢ (∀𝑦 ∈ (𝐴 ∖ (◡𝑅 “ 𝐵)) ¬ 𝑦𝑅𝑥 ↔ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))
36	21, 35	anbi12i 628	. . . . . . . 8 ⊢ ((∀𝑦 ∈ 𝐵 ¬ 𝑦◡𝑅𝑥 ∧ ∀𝑦 ∈ (𝐴 ∖ (◡𝑅 “ 𝐵)) ¬ 𝑦𝑅𝑥) ↔ (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)))
37	6, 17, 36	3bitr2ri 300	. . . . . . 7 ⊢ ((∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)) ↔ 𝑥 ∈ (V ∖ ((◡𝑅 “ 𝐵) ∪ (𝑅 “ (𝐴 ∖ (◡𝑅 “ 𝐵))))))
38	3, 37	mpgbir 1799	. . . . . 6 ⊢ {𝑥 ∣ (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))} = (V ∖ ((◡𝑅 “ 𝐵) ∪ (𝑅 “ (𝐴 ∖ (◡𝑅 “ 𝐵)))))
39	38	ineq2i 4180	. . . . 5 ⊢ (𝐴 ∩ {𝑥 ∣ (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))}) = (𝐴 ∩ (V ∖ ((◡𝑅 “ 𝐵) ∪ (𝑅 “ (𝐴 ∖ (◡𝑅 “ 𝐵))))))
40		invdif 4242	. . . . 5 ⊢ (𝐴 ∩ (V ∖ ((◡𝑅 “ 𝐵) ∪ (𝑅 “ (𝐴 ∖ (◡𝑅 “ 𝐵)))))) = (𝐴 ∖ ((◡𝑅 “ 𝐵) ∪ (𝑅 “ (𝐴 ∖ (◡𝑅 “ 𝐵)))))
41	39, 40	eqtri 2752	. . . 4 ⊢ (𝐴 ∩ {𝑥 ∣ (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))}) = (𝐴 ∖ ((◡𝑅 “ 𝐵) ∪ (𝑅 “ (𝐴 ∖ (◡𝑅 “ 𝐵)))))
42	2, 41	eqtri 2752	. . 3 ⊢ {𝑥 ∈ 𝐴 ∣ (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))} = (𝐴 ∖ ((◡𝑅 “ 𝐵) ∪ (𝑅 “ (𝐴 ∖ (◡𝑅 “ 𝐵)))))
43	42	unieqi 4883	. 2 ⊢ ∪ {𝑥 ∈ 𝐴 ∣ (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))} = ∪ (𝐴 ∖ ((◡𝑅 “ 𝐵) ∪ (𝑅 “ (𝐴 ∖ (◡𝑅 “ 𝐵)))))
44	1, 43	eqtri 2752	1 ⊢ sup(𝐵, 𝐴, 𝑅) = ∪ (𝐴 ∖ ((◡𝑅 “ 𝐵) ∪ (𝑅 “ (𝐴 ∖ (◡𝑅 “ 𝐵)))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   = wceq 1540   ∈ wcel 2109  {cab 2707  ∀wral 3044  ∃wrex 3053  {crab 3405  Vcvv 3447   ∖ cdif 3911   ∪ cun 3912   ∩ cin 3913  ∪ cuni 4871   class class class wbr 5107  ^◡ccnv 5637   “ cima 5641  supcsup 9391

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-xp 5644  df-cnv 5646  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-sup 9393

This theorem is referenced by:  nfsup  9402