Theorem suplubti 7242

Description: A supremum is the least upper bound. See also supclti 7240 and supubti 7241. (Contributed by Jim Kingdon, 24-Nov-2021.)

Hypotheses

Ref	Expression
supmoti.ti	⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))
supclti.2	⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)))

Assertion

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

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

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

Proof of Theorem suplubti

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		simpr 110	. . . . . 6 ⊢ ((∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)) → ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))
2		breq1 4096	. . . . . . . 8 ⊢ (𝑦 = 𝑤 → (𝑦𝑅𝑥 ↔ 𝑤𝑅𝑥))
3		breq1 4096	. . . . . . . . 9 ⊢ (𝑦 = 𝑤 → (𝑦𝑅𝑧 ↔ 𝑤𝑅𝑧))
4	3	rexbidv 2534	. . . . . . . 8 ⊢ (𝑦 = 𝑤 → (∃𝑧 ∈ 𝐵 𝑦𝑅𝑧 ↔ ∃𝑧 ∈ 𝐵 𝑤𝑅𝑧))
5	2, 4	imbi12d 234	. . . . . . 7 ⊢ (𝑦 = 𝑤 → ((𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧) ↔ (𝑤𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑤𝑅𝑧)))
6	5	cbvralv 2768	. . . . . 6 ⊢ (∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧) ↔ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑤𝑅𝑧))
7	1, 6	sylib 122	. . . . 5 ⊢ ((∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)) → ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑤𝑅𝑧))
8	7	a1i 9	. . . 4 ⊢ (𝑥 ∈ 𝐴 → ((∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)) → ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑤𝑅𝑧)))
9	8	ss2rabi 3310	. . 3 ⊢ {𝑥 ∈ 𝐴 ∣ (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))} ⊆ {𝑥 ∈ 𝐴 ∣ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑤𝑅𝑧)}
10		supmoti.ti	. . . . 5 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))
11		supclti.2	. . . . 5 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)))
12	10, 11	supval2ti 7237	. . . 4 ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) = (℩𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))))
13	10, 11	supeuti 7236	. . . . 5 ⊢ (𝜑 → ∃!𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)))
14		riotacl2 5996	. . . . 5 ⊢ (∃!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 14	. . . 4 ⊢ (𝜑 → (℩𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))) ∈ {𝑥 ∈ 𝐴 ∣ (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))})
16	12, 15	eqeltrd 2308	. . 3 ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ {𝑥 ∈ 𝐴 ∣ (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))})
17	9, 16	sselid 3226	. 2 ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ {𝑥 ∈ 𝐴 ∣ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑤𝑅𝑧)})
18		breq2 4097	. . . . . 6 ⊢ (𝑥 = sup(𝐵, 𝐴, 𝑅) → (𝑤𝑅𝑥 ↔ 𝑤𝑅sup(𝐵, 𝐴, 𝑅)))
19	18	imbi1d 231	. . . . 5 ⊢ (𝑥 = sup(𝐵, 𝐴, 𝑅) → ((𝑤𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑤𝑅𝑧) ↔ (𝑤𝑅sup(𝐵, 𝐴, 𝑅) → ∃𝑧 ∈ 𝐵 𝑤𝑅𝑧)))
20	19	ralbidv 2533	. . . 4 ⊢ (𝑥 = sup(𝐵, 𝐴, 𝑅) → (∀𝑤 ∈ 𝐴 (𝑤𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑤𝑅𝑧) ↔ ∀𝑤 ∈ 𝐴 (𝑤𝑅sup(𝐵, 𝐴, 𝑅) → ∃𝑧 ∈ 𝐵 𝑤𝑅𝑧)))
21	20	elrab 2963	. . 3 ⊢ (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 275	. 2 ⊢ (sup(𝐵, 𝐴, 𝑅) ∈ {𝑥 ∈ 𝐴 ∣ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑤𝑅𝑧)} → ∀𝑤 ∈ 𝐴 (𝑤𝑅sup(𝐵, 𝐴, 𝑅) → ∃𝑧 ∈ 𝐵 𝑤𝑅𝑧))
23		breq1 4096	. . . . 5 ⊢ (𝑤 = 𝐶 → (𝑤𝑅sup(𝐵, 𝐴, 𝑅) ↔ 𝐶𝑅sup(𝐵, 𝐴, 𝑅)))
24		breq1 4096	. . . . . 6 ⊢ (𝑤 = 𝐶 → (𝑤𝑅𝑧 ↔ 𝐶𝑅𝑧))
25	24	rexbidv 2534	. . . . 5 ⊢ (𝑤 = 𝐶 → (∃𝑧 ∈ 𝐵 𝑤𝑅𝑧 ↔ ∃𝑧 ∈ 𝐵 𝐶𝑅𝑧))
26	23, 25	imbi12d 234	. . . 4 ⊢ (𝑤 = 𝐶 → ((𝑤𝑅sup(𝐵, 𝐴, 𝑅) → ∃𝑧 ∈ 𝐵 𝑤𝑅𝑧) ↔ (𝐶𝑅sup(𝐵, 𝐴, 𝑅) → ∃𝑧 ∈ 𝐵 𝐶𝑅𝑧)))
27	26	rspccv 2908	. . 3 ⊢ (∀𝑤 ∈ 𝐴 (𝑤𝑅sup(𝐵, 𝐴, 𝑅) → ∃𝑧 ∈ 𝐵 𝑤𝑅𝑧) → (𝐶 ∈ 𝐴 → (𝐶𝑅sup(𝐵, 𝐴, 𝑅) → ∃𝑧 ∈ 𝐵 𝐶𝑅𝑧)))
28	27	impd 254	. 2 ⊢ (∀𝑤 ∈ 𝐴 (𝑤𝑅sup(𝐵, 𝐴, 𝑅) → ∃𝑧 ∈ 𝐵 𝑤𝑅𝑧) → ((𝐶 ∈ 𝐴 ∧ 𝐶𝑅sup(𝐵, 𝐴, 𝑅)) → ∃𝑧 ∈ 𝐵 𝐶𝑅𝑧))
29	17, 22, 28	3syl 17	1 ⊢ (𝜑 → ((𝐶 ∈ 𝐴 ∧ 𝐶𝑅sup(𝐵, 𝐴, 𝑅)) → ∃𝑧 ∈ 𝐵 𝐶𝑅𝑧))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398   ∈ wcel 2202  ∀wral 2511  ∃wrex 2512  ∃!wreu 2513  {crab 2515   class class class wbr 4093  ℩crio 5980  supcsup 7224

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-reu 2518  df-rmo 2519  df-rab 2520  df-v 2805  df-sbc 3033  df-un 3205  df-in 3207  df-ss 3214  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-iota 5293  df-riota 5981  df-sup 7226

This theorem is referenced by:  suplub2ti  7243  supisoti  7252  infglbti  7267  sup3exmid  9179  maxleast  11836