Theorem suplub 8918

Description: A supremum is the least upper bound. See also supcl 8916 and supub 8917. (Contributed by NM, 13-Oct-2004.) (Revised by Mario Carneiro, 24-Dec-2016.)

Hypotheses

Ref	Expression
supmo.1	⊢ (𝜑 → 𝑅 Or 𝐴)
supcl.2	⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)))

Assertion

Ref	Expression
suplub	⊢ (𝜑 → ((𝐶 ∈ 𝐴 ∧ 𝐶𝑅sup(𝐵, 𝐴, 𝑅)) → ∃𝑧 ∈ 𝐵 𝐶𝑅𝑧))

Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝑅,𝑦,𝑧   𝑥,𝐵,𝑦,𝑧   𝑧,𝐶

Allowed substitution hints:   𝜑(𝑥,𝑦,𝑧)   𝐶(𝑥,𝑦)

Proof of Theorem suplub

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 487	. . . . . . 7 ⊢ ((∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)) → ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))
2		breq1 5061	. . . . . . . . 9 ⊢ (𝑦 = 𝑤 → (𝑦𝑅𝑥 ↔ 𝑤𝑅𝑥))
3		breq1 5061	. . . . . . . . . 10 ⊢ (𝑦 = 𝑤 → (𝑦𝑅𝑧 ↔ 𝑤𝑅𝑧))
4	3	rexbidv 3297	. . . . . . . . 9 ⊢ (𝑦 = 𝑤 → (∃𝑧 ∈ 𝐵 𝑦𝑅𝑧 ↔ ∃𝑧 ∈ 𝐵 𝑤𝑅𝑧))
5	2, 4	imbi12d 347	. . . . . . . 8 ⊢ (𝑦 = 𝑤 → ((𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧) ↔ (𝑤𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑤𝑅𝑧)))
6	5	cbvralvw 3449	. . . . . . 7 ⊢ (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧) ↔ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑤𝑅𝑧))
7	1, 6	sylib 220	. . . . . 6 ⊢ ((∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)) → ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑤𝑅𝑧))
8	7	a1i 11	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → ((∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)) → ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑤𝑅𝑧)))
9	8	ss2rabi 4052	. . . 4 ⊢ {𝑥 ∈ 𝐴 ∣ (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))} ⊆ {𝑥 ∈ 𝐴 ∣ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑤𝑅𝑧)}
10		supmo.1	. . . . . 6 ⊢ (𝜑 → 𝑅 Or 𝐴)
11	10	supval2 8913	. . . . 5 ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) = (℩𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))))
12		supcl.2	. . . . . . 7 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)))
13	10, 12	supeu 8912	. . . . . 6 ⊢ (𝜑 → ∃!𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)))
14		riotacl2 7124	. . . . . 6 ⊢ (∃!𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)) → (℩𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))) ∈ {𝑥 ∈ 𝐴 ∣ (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))})
15	13, 14	syl 17	. . . . 5 ⊢ (𝜑 → (℩𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))) ∈ {𝑥 ∈ 𝐴 ∣ (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))})
16	11, 15	eqeltrd 2913	. . . 4 ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ {𝑥 ∈ 𝐴 ∣ (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))})
17	9, 16	sseldi 3964	. . 3 ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ {𝑥 ∈ 𝐴 ∣ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑤𝑅𝑧)})
18		breq2 5062	. . . . . . 7 ⊢ (𝑥 = sup(𝐵, 𝐴, 𝑅) → (𝑤𝑅𝑥 ↔ 𝑤𝑅sup(𝐵, 𝐴, 𝑅)))
19	18	imbi1d 344	. . . . . 6 ⊢ (𝑥 = sup(𝐵, 𝐴, 𝑅) → ((𝑤𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑤𝑅𝑧) ↔ (𝑤𝑅sup(𝐵, 𝐴, 𝑅) → ∃𝑧 ∈ 𝐵 𝑤𝑅𝑧)))
20	19	ralbidv 3197	. . . . 5 ⊢ (𝑥 = sup(𝐵, 𝐴, 𝑅) → (∀𝑤 ∈ 𝐴 (𝑤𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑤𝑅𝑧) ↔ ∀𝑤 ∈ 𝐴 (𝑤𝑅sup(𝐵, 𝐴, 𝑅) → ∃𝑧 ∈ 𝐵 𝑤𝑅𝑧)))
21	20	elrab 3679	. . . 4 ⊢ (sup(𝐵, 𝐴, 𝑅) ∈ {𝑥 ∈ 𝐴 ∣ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑤𝑅𝑧)} ↔ (sup(𝐵, 𝐴, 𝑅) ∈ 𝐴 ∧ ∀𝑤 ∈ 𝐴 (𝑤𝑅sup(𝐵, 𝐴, 𝑅) → ∃𝑧 ∈ 𝐵 𝑤𝑅𝑧)))
22	21	simprbi 499	. . 3 ⊢ (sup(𝐵, 𝐴, 𝑅) ∈ {𝑥 ∈ 𝐴 ∣ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑤𝑅𝑧)} → ∀𝑤 ∈ 𝐴 (𝑤𝑅sup(𝐵, 𝐴, 𝑅) → ∃𝑧 ∈ 𝐵 𝑤𝑅𝑧))
23	17, 22	syl 17	. 2 ⊢ (𝜑 → ∀𝑤 ∈ 𝐴 (𝑤𝑅sup(𝐵, 𝐴, 𝑅) → ∃𝑧 ∈ 𝐵 𝑤𝑅𝑧))
24		breq1 5061	. . . . 5 ⊢ (𝑤 = 𝐶 → (𝑤𝑅sup(𝐵, 𝐴, 𝑅) ↔ 𝐶𝑅sup(𝐵, 𝐴, 𝑅)))
25		breq1 5061	. . . . . 6 ⊢ (𝑤 = 𝐶 → (𝑤𝑅𝑧 ↔ 𝐶𝑅𝑧))
26	25	rexbidv 3297	. . . . 5 ⊢ (𝑤 = 𝐶 → (∃𝑧 ∈ 𝐵 𝑤𝑅𝑧 ↔ ∃𝑧 ∈ 𝐵 𝐶𝑅𝑧))
27	24, 26	imbi12d 347	. . . 4 ⊢ (𝑤 = 𝐶 → ((𝑤𝑅sup(𝐵, 𝐴, 𝑅) → ∃𝑧 ∈ 𝐵 𝑤𝑅𝑧) ↔ (𝐶𝑅sup(𝐵, 𝐴, 𝑅) → ∃𝑧 ∈ 𝐵 𝐶𝑅𝑧)))
28	27	rspccv 3619	. . 3 ⊢ (∀𝑤 ∈ 𝐴 (𝑤𝑅sup(𝐵, 𝐴, 𝑅) → ∃𝑧 ∈ 𝐵 𝑤𝑅𝑧) → (𝐶 ∈ 𝐴 → (𝐶𝑅sup(𝐵, 𝐴, 𝑅) → ∃𝑧 ∈ 𝐵 𝐶𝑅𝑧)))
29	28	impd 413	. 2 ⊢ (∀𝑤 ∈ 𝐴 (𝑤𝑅sup(𝐵, 𝐴, 𝑅) → ∃𝑧 ∈ 𝐵 𝑤𝑅𝑧) → ((𝐶 ∈ 𝐴 ∧ 𝐶𝑅sup(𝐵, 𝐴, 𝑅)) → ∃𝑧 ∈ 𝐵 𝐶𝑅𝑧))
30	23, 29	syl 17	1 ⊢ (𝜑 → ((𝐶 ∈ 𝐴 ∧ 𝐶𝑅sup(𝐵, 𝐴, 𝑅)) → ∃𝑧 ∈ 𝐵 𝐶𝑅𝑧))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 398   = wceq 1533   ∈ wcel 2110  ∀wral 3138  ∃wrex 3139  ∃!wreu 3140  {crab 3142   class class class wbr 5058   Or wor 5467  ℩crio 7107  supcsup 8898

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-reu 3145  df-rmo 3146  df-rab 3147  df-v 3496  df-sbc 3772  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-po 5468  df-so 5469  df-iota 6308  df-riota 7108  df-sup 8900

This theorem is referenced by:  suplub2  8919  supnub  8920  supiso  8933  infglb  8948  supxrun  12703  supxrunb1  12706  supxrunb2  12707  esum2d  31347