Theorem eqinf 9444

Description: Sufficient condition for an element to be equal to the infimum. (Contributed by AV, 2-Sep-2020.)

Hypothesis

Ref	Expression
infexd.1	⊢ (𝜑 → 𝑅 Or 𝐴)

Assertion

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

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

Allowed substitution hints:   𝜑(𝑦,𝑧)

Proof of Theorem eqinf

Step	Hyp	Ref	Expression
1		df-inf 9402	. . 3 ⊢ inf(𝐵, 𝐴, 𝑅) = sup(𝐵, 𝐴, ◡𝑅)
2		infexd.1	. . . . . 6 ⊢ (𝜑 → 𝑅 Or 𝐴)
3		cnvso 6289	. . . . . 6 ⊢ (𝑅 Or 𝐴 ↔ ◡𝑅 Or 𝐴)
4	2, 3	sylib 221	. . . . 5 ⊢ (𝜑 → ◡𝑅 Or 𝐴)
5	4	eqsup 9415	. . . 4 ⊢ (𝜑 → ((𝐶 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝐶◡𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦◡𝑅𝐶 → ∃𝑧 ∈ 𝐵 𝑦◡𝑅𝑧)) → sup(𝐵, 𝐴, ◡𝑅) = 𝐶))
6		brcnvg 5865	. . . . . . . . . . . 12 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝑦 ∈ V) → (𝐶◡𝑅𝑦 ↔ 𝑦𝑅𝐶))
7	6	bicomd 226	. . . . . . . . . . 11 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝑦 ∈ V) → (𝑦𝑅𝐶 ↔ 𝐶◡𝑅𝑦))
8	7	elvd 3459	. . . . . . . . . 10 ⊢ (𝐶 ∈ 𝐴 → (𝑦𝑅𝐶 ↔ 𝐶◡𝑅𝑦))
9	8	notbid 321	. . . . . . . . 9 ⊢ (𝐶 ∈ 𝐴 → (¬ 𝑦𝑅𝐶 ↔ ¬ 𝐶◡𝑅𝑦))
10	9	ralbidv 3186	. . . . . . . 8 ⊢ (𝐶 ∈ 𝐴 → (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝐶 ↔ ∀𝑦 ∈ 𝐵 ¬ 𝐶◡𝑅𝑦))
11		vex 3457	. . . . . . . . . . . 12 ⊢ 𝑦 ∈ V
12		brcnvg 5865	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ V ∧ 𝐶 ∈ 𝐴) → (𝑦◡𝑅𝐶 ↔ 𝐶𝑅𝑦))
13	11, 12	mpan 702	. . . . . . . . . . 11 ⊢ (𝐶 ∈ 𝐴 → (𝑦◡𝑅𝐶 ↔ 𝐶𝑅𝑦))
14	13	bicomd 226	. . . . . . . . . 10 ⊢ (𝐶 ∈ 𝐴 → (𝐶𝑅𝑦 ↔ 𝑦◡𝑅𝐶))
15		vex 3457	. . . . . . . . . . . . . 14 ⊢ 𝑧 ∈ V
16	11, 15	brcnv 5868	. . . . . . . . . . . . 13 ⊢ (𝑦◡𝑅𝑧 ↔ 𝑧𝑅𝑦)
17	16	a1i 11	. . . . . . . . . . . 12 ⊢ (𝐶 ∈ 𝐴 → (𝑦◡𝑅𝑧 ↔ 𝑧𝑅𝑦))
18	17	bicomd 226	. . . . . . . . . . 11 ⊢ (𝐶 ∈ 𝐴 → (𝑧𝑅𝑦 ↔ 𝑦◡𝑅𝑧))
19	18	rexbidv 3187	. . . . . . . . . 10 ⊢ (𝐶 ∈ 𝐴 → (∃𝑧 ∈ 𝐵 𝑧𝑅𝑦 ↔ ∃𝑧 ∈ 𝐵 𝑦◡𝑅𝑧))
20	14, 19	imbi12d 347	. . . . . . . . 9 ⊢ (𝐶 ∈ 𝐴 → ((𝐶𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦) ↔ (𝑦◡𝑅𝐶 → ∃𝑧 ∈ 𝐵 𝑦◡𝑅𝑧)))
21	20	ralbidv 3186	. . . . . . . 8 ⊢ (𝐶 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 (𝐶𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦) ↔ ∀𝑦 ∈ 𝐴 (𝑦◡𝑅𝐶 → ∃𝑧 ∈ 𝐵 𝑦◡𝑅𝑧)))
22	10, 21	anbi12d 643	. . . . . . 7 ⊢ (𝐶 ∈ 𝐴 → ((∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝐶 ∧ ∀𝑦 ∈ 𝐴 (𝐶𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)) ↔ (∀𝑦 ∈ 𝐵 ¬ 𝐶◡𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦◡𝑅𝐶 → ∃𝑧 ∈ 𝐵 𝑦◡𝑅𝑧))))
23	22	pm5.32i 584	. . . . . 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))))
24		3anass 1109	. . . . . 6 ⊢ ((𝐶 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝐶 ∧ ∀𝑦 ∈ 𝐴 (𝐶𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)) ↔ (𝐶 ∈ 𝐴 ∧ (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝐶 ∧ ∀𝑦 ∈ 𝐴 (𝐶𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦))))
25		3anass 1109	. . . . . 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))))
26	23, 24, 25	3bitr4i 306	. . . . 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)))
27	26	biimpi 219	. . . 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)))
28	5, 27	impel 514	. . 3 ⊢ ((𝜑 ∧ (𝐶 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝐶 ∧ ∀𝑦 ∈ 𝐴 (𝐶𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦))) → sup(𝐵, 𝐴, ◡𝑅) = 𝐶)
29	1, 28	eqtrid 2808	. 2 ⊢ ((𝜑 ∧ (𝐶 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝐶 ∧ ∀𝑦 ∈ 𝐴 (𝐶𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦))) → inf(𝐵, 𝐴, 𝑅) = 𝐶)
30	29	ex 417	1 ⊢ (𝜑 → ((𝐶 ∈ 𝐴 ∧ ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝐶 ∧ ∀𝑦 ∈ 𝐴 (𝐶𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)) → inf(𝐵, 𝐴, 𝑅) = 𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1101   = wceq 1568   ∈ wcel 2141  ∀wral 3077  ∃wrex 3087  Vcvv 3453   class class class wbr 5108   Or wor 5568  ^◡ccnv 5660  supcsup 9399  infcinf 9400

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-10 2174  ax-11 2190  ax-12 2211  ax-ext 2733  ax-sep 5256  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1102  df-3an 1103  df-tru 1571  df-fal 1581  df-ex 1808  df-nf 1812  df-sb 2095  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rmo 3367  df-reu 3368  df-rab 3415  df-v 3455  df-dif 3907  df-un 3909  df-in 3911  df-ss 3921  df-nul 4286  df-if 4487  df-sn 4589  df-pr 4591  df-op 4595  df-uni 4872  df-br 5109  df-opab 5173  df-po 5569  df-so 5570  df-cnv 5669  df-iota 6492  df-riota 7367  df-sup 9401  df-inf 9402

This theorem is referenced by:  eqinfd  9445  infxr  46052