Theorem suplubti 7102

Description: A supremum is the least upper bound. See also supclti 7100 and supubti 7101. (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 4047	. . . . . . . 8 ⊢ (𝑦 = 𝑤 → (𝑦𝑅𝑥 ↔ 𝑤𝑅𝑥))
3		breq1 4047	. . . . . . . . 9 ⊢ (𝑦 = 𝑤 → (𝑦𝑅𝑧 ↔ 𝑤𝑅𝑧))
4	3	rexbidv 2507	. . . . . . . 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 2738	. . . . . 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 3275	. . 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 7097	. . . 4 ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) = (℩𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))))
13	10, 11	supeuti 7096	. . . . 5 ⊢ (𝜑 → ∃!𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧)))
14		riotacl2 5913	. . . . 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 2282	. . 3 ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ {𝑥 ∈ 𝐴 ∣ (∀𝑦 ∈ 𝐵 ¬ 𝑥𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦𝑅𝑧))})
17	9, 16	sselid 3191	. 2 ⊢ (𝜑 → sup(𝐵, 𝐴, 𝑅) ∈ {𝑥 ∈ 𝐴 ∣ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑤𝑅𝑧)})
18		breq2 4048	. . . . . 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 2506	. . . 4 ⊢ (𝑥 = sup(𝐵, 𝐴, 𝑅) → (∀𝑤 ∈ 𝐴 (𝑤𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑤𝑅𝑧) ↔ ∀𝑤 ∈ 𝐴 (𝑤𝑅sup(𝐵, 𝐴, 𝑅) → ∃𝑧 ∈ 𝐵 𝑤𝑅𝑧)))
21	20	elrab 2929	. . 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 4047	. . . . 5 ⊢ (𝑤 = 𝐶 → (𝑤𝑅sup(𝐵, 𝐴, 𝑅) ↔ 𝐶𝑅sup(𝐵, 𝐴, 𝑅)))
24		breq1 4047	. . . . . 6 ⊢ (𝑤 = 𝐶 → (𝑤𝑅𝑧 ↔ 𝐶𝑅𝑧))
25	24	rexbidv 2507	. . . . 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 2874	. . 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 1373   ∈ wcel 2176  ∀wral 2484  ∃wrex 2485  ∃!wreu 2486  {crab 2488   class class class wbr 4044  ℩crio 5898  supcsup 7084

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 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1484  df-sb 1786  df-eu 2057  df-mo 2058  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-ral 2489  df-rex 2490  df-reu 2491  df-rmo 2492  df-rab 2493  df-v 2774  df-sbc 2999  df-un 3170  df-in 3172  df-ss 3179  df-sn 3639  df-pr 3640  df-op 3642  df-uni 3851  df-br 4045  df-iota 5232  df-riota 5899  df-sup 7086

This theorem is referenced by:  suplub2ti  7103  supisoti  7112  infglbti  7127  sup3exmid  9030  maxleast  11524