Theorem eqinfti 7262

Description: Sufficient condition for an element to be equal to the infimum. (Contributed by Jim Kingdon, 16-Dec-2021.)

Hypothesis

Ref	Expression
eqinfti.ti	⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))

Assertion

Ref	Expression
eqinfti	⊢ (𝜑 → ((𝐶 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝐶 ∧ ∀𝑦 ∈ 𝐴 (𝐶𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)) → inf(𝐵, 𝐴, 𝑅) = 𝐶))

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

Allowed substitution hints:   𝜑(𝑦,𝑧)

Proof of Theorem eqinfti

Step	Hyp	Ref	Expression
1		df-inf 7227	. . 3 ⊢ inf(𝐵, 𝐴, 𝑅) = sup(𝐵, 𝐴, ◡𝑅)
2		eqinfti.ti	. . . . . 6 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢𝑅𝑣 ∧ ¬ 𝑣𝑅𝑢)))
3	2	cnvti 7261	. . . . 5 ⊢ ((𝜑 ∧ (𝑢 ∈ 𝐴 ∧ 𝑣 ∈ 𝐴)) → (𝑢 = 𝑣 ↔ (¬ 𝑢◡𝑅𝑣 ∧ ¬ 𝑣◡𝑅𝑢)))
4	3	eqsupti 7238	. . . 4 ⊢ (𝜑 → ((𝐶 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝐶◡𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦◡𝑅𝐶 → ∃𝑧 ∈ 𝐵 𝑦◡𝑅𝑧)) → sup(𝐵, 𝐴, ◡𝑅) = 𝐶))
5		vex 2806	. . . . . . . . . . 11 ⊢ 𝑦 ∈ V
6		brcnvg 4917	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝑦 ∈ V) → (𝐶◡𝑅𝑦 ↔ 𝑦𝑅𝐶))
7	6	bicomd 141	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝑦 ∈ V) → (𝑦𝑅𝐶 ↔ 𝐶◡𝑅𝑦))
8	5, 7	mpan2 425	. . . . . . . . . 10 ⊢ (𝐶 ∈ 𝐴 → (𝑦𝑅𝐶 ↔ 𝐶◡𝑅𝑦))
9	8	notbid 673	. . . . . . . . 9 ⊢ (𝐶 ∈ 𝐴 → (¬ 𝑦𝑅𝐶 ↔ ¬ 𝐶◡𝑅𝑦))
10	9	ralbidv 2533	. . . . . . . 8 ⊢ (𝐶 ∈ 𝐴 → (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝐶 ↔ ∀𝑦 ∈ 𝐵 ¬ 𝐶◡𝑅𝑦))
11		brcnvg 4917	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ V ∧ 𝐶 ∈ 𝐴) → (𝑦◡𝑅𝐶 ↔ 𝐶𝑅𝑦))
12	5, 11	mpan 424	. . . . . . . . . . 11 ⊢ (𝐶 ∈ 𝐴 → (𝑦◡𝑅𝐶 ↔ 𝐶𝑅𝑦))
13	12	bicomd 141	. . . . . . . . . 10 ⊢ (𝐶 ∈ 𝐴 → (𝐶𝑅𝑦 ↔ 𝑦◡𝑅𝐶))
14		vex 2806	. . . . . . . . . . . . . 14 ⊢ 𝑧 ∈ V
15	5, 14	brcnv 4919	. . . . . . . . . . . . 13 ⊢ (𝑦◡𝑅𝑧 ↔ 𝑧𝑅𝑦)
16	15	a1i 9	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ 𝐴 → (𝑦◡𝑅𝑧 ↔ 𝑧𝑅𝑦))
17	16	bicomd 141	. . . . . . . . . . 11 ⊢ (𝐶 ∈ 𝐴 → (𝑧𝑅𝑦 ↔ 𝑦◡𝑅𝑧))
18	17	rexbidv 2534	. . . . . . . . . 10 ⊢ (𝐶 ∈ 𝐴 → (∃𝑧 ∈ 𝐵 𝑧𝑅𝑦 ↔ ∃𝑧 ∈ 𝐵 𝑦◡𝑅𝑧))
19	13, 18	imbi12d 234	. . . . . . . . 9 ⊢ (𝐶 ∈ 𝐴 → ((𝐶𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦) ↔ (𝑦◡𝑅𝐶 → ∃𝑧 ∈ 𝐵 𝑦◡𝑅𝑧)))
20	19	ralbidv 2533	. . . . . . . 8 ⊢ (𝐶 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 (𝐶𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦) ↔ ∀𝑦 ∈ 𝐴 (𝑦◡𝑅𝐶 → ∃𝑧 ∈ 𝐵 𝑦◡𝑅𝑧)))
21	10, 20	anbi12d 473	. . . . . . 7 ⊢ (𝐶 ∈ 𝐴 → ((∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝐶 ∧ ∀𝑦 ∈ 𝐴 (𝐶𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)) ↔ (∀𝑦 ∈ 𝐵 ¬ 𝐶◡𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦◡𝑅𝐶 → ∃𝑧 ∈ 𝐵 𝑦◡𝑅𝑧))))
22	21	pm5.32i 454	. . . . . 6 ⊢ ((𝐶 ∈ 𝐴 ∧ (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝐶 ∧ ∀𝑦 ∈ 𝐴 (𝐶𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦))) ↔ (𝐶 ∈ 𝐴 ∧ (∀𝑦 ∈ 𝐵 ¬ 𝐶◡𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦◡𝑅𝐶 → ∃𝑧 ∈ 𝐵 𝑦◡𝑅𝑧))))
23		3anass 1009	. . . . . 6 ⊢ ((𝐶 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝐶 ∧ ∀𝑦 ∈ 𝐴 (𝐶𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)) ↔ (𝐶 ∈ 𝐴 ∧ (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝐶 ∧ ∀𝑦 ∈ 𝐴 (𝐶𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦))))
24		3anass 1009	. . . . . 6 ⊢ ((𝐶 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝐶◡𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦◡𝑅𝐶 → ∃𝑧 ∈ 𝐵 𝑦◡𝑅𝑧)) ↔ (𝐶 ∈ 𝐴 ∧ (∀𝑦 ∈ 𝐵 ¬ 𝐶◡𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦◡𝑅𝐶 → ∃𝑧 ∈ 𝐵 𝑦◡𝑅𝑧))))
25	22, 23, 24	3bitr4i 212	. . . . 5 ⊢ ((𝐶 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝐶 ∧ ∀𝑦 ∈ 𝐴 (𝐶𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)) ↔ (𝐶 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝐶◡𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦◡𝑅𝐶 → ∃𝑧 ∈ 𝐵 𝑦◡𝑅𝑧)))
26	25	biimpi 120	. . . 4 ⊢ ((𝐶 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝐶 ∧ ∀𝑦 ∈ 𝐴 (𝐶𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)) → (𝐶 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝐶◡𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦◡𝑅𝐶 → ∃𝑧 ∈ 𝐵 𝑦◡𝑅𝑧)))
27	4, 26	impel 280	. . 3 ⊢ ((𝜑 ∧ (𝐶 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝐶 ∧ ∀𝑦 ∈ 𝐴 (𝐶𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦))) → sup(𝐵, 𝐴, ◡𝑅) = 𝐶)
28	1, 27	eqtrid 2276	. 2 ⊢ ((𝜑 ∧ (𝐶 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝐶 ∧ ∀𝑦 ∈ 𝐴 (𝐶𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦))) → inf(𝐵, 𝐴, 𝑅) = 𝐶)
29	28	ex 115	1 ⊢ (𝜑 → ((𝐶 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝐶 ∧ ∀𝑦 ∈ 𝐴 (𝐶𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)) → inf(𝐵, 𝐴, 𝑅) = 𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1005   = wceq 1398   ∈ wcel 2202  ∀wral 2511  ∃wrex 2512  Vcvv 2803   class class class wbr 4093  ^◡ccnv 4730  supcsup 7224  infcinf 7225

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-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

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-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-cnv 4739  df-iota 5293  df-riota 5981  df-sup 7226  df-inf 7227

This theorem is referenced by:  eqinftid  7263