Theorem infval 9448

Description: Alternate expression for the infimum. (Contributed by AV, 2-Sep-2020.)

Hypothesis

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

Assertion

Ref	Expression
infval	⊢ (𝜑 → inf(𝐵, 𝐴, 𝑅) = (℩𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦))))

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

Proof of Theorem infval

Step	Hyp	Ref	Expression
1		df-inf 9404	. 2 ⊢ inf(𝐵, 𝐴, 𝑅) = sup(𝐵, 𝐴, ◡𝑅)
2		infexd.1	. . . . 5 ⊢ (𝜑 → 𝑅 Or 𝐴)
3		cnvso 6291	. . . . 5 ⊢ (𝑅 Or 𝐴 ↔ ◡𝑅 Or 𝐴)
4	2, 3	sylib 221	. . . 4 ⊢ (𝜑 → ◡𝑅 Or 𝐴)
5	4	supval2 9416	. . 3 ⊢ (𝜑 → sup(𝐵, 𝐴, ◡𝑅) = (℩𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥◡𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦◡𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦◡𝑅𝑧))))
6		vex 3459	. . . . . . . . 9 ⊢ 𝑥 ∈ V
7		vex 3459	. . . . . . . . 9 ⊢ 𝑦 ∈ V
8	6, 7	brcnv 5870	. . . . . . . 8 ⊢ (𝑥◡𝑅𝑦 ↔ 𝑦𝑅𝑥)
9	8	a1i 11	. . . . . . 7 ⊢ (𝜑 → (𝑥◡𝑅𝑦 ↔ 𝑦𝑅𝑥))
10	9	notbid 321	. . . . . 6 ⊢ (𝜑 → (¬ 𝑥◡𝑅𝑦 ↔ ¬ 𝑦𝑅𝑥))
11	10	ralbidv 3188	. . . . 5 ⊢ (𝜑 → (∀𝑦 ∈ 𝐵 ¬ 𝑥◡𝑅𝑦 ↔ ∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥))
12	7, 6	brcnv 5870	. . . . . . . 8 ⊢ (𝑦◡𝑅𝑥 ↔ 𝑥𝑅𝑦)
13	12	a1i 11	. . . . . . 7 ⊢ (𝜑 → (𝑦◡𝑅𝑥 ↔ 𝑥𝑅𝑦))
14		vex 3459	. . . . . . . . . 10 ⊢ 𝑧 ∈ V
15	7, 14	brcnv 5870	. . . . . . . . 9 ⊢ (𝑦◡𝑅𝑧 ↔ 𝑧𝑅𝑦)
16	15	a1i 11	. . . . . . . 8 ⊢ (𝜑 → (𝑦◡𝑅𝑧 ↔ 𝑧𝑅𝑦))
17	16	rexbidv 3189	. . . . . . 7 ⊢ (𝜑 → (∃𝑧 ∈ 𝐵 𝑦◡𝑅𝑧 ↔ ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦))
18	13, 17	imbi12d 347	. . . . . 6 ⊢ (𝜑 → ((𝑦◡𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦◡𝑅𝑧) ↔ (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)))
19	18	ralbidv 3188	. . . . 5 ⊢ (𝜑 → (∀𝑦 ∈ 𝐴 (𝑦◡𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦◡𝑅𝑧) ↔ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)))
20	11, 19	anbi12d 643	. . . 4 ⊢ (𝜑 → ((∀𝑦 ∈ 𝐵 ¬ 𝑥◡𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦◡𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦◡𝑅𝑧)) ↔ (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦))))
21	20	riotabidv 7371	. . 3 ⊢ (𝜑 → (℩𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥◡𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦◡𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦◡𝑅𝑧))) = (℩𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦))))
22	5, 21	eqtrd 2798	. 2 ⊢ (𝜑 → sup(𝐵, 𝐴, ◡𝑅) = (℩𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦))))
23	1, 22	eqtrid 2810	1 ⊢ (𝜑 → inf(𝐵, 𝐴, 𝑅) = (℩𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦))))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1570  ∀wral 3079  ∃wrex 3089   class class class wbr 5110   Or wor 5570  ^◡ccnv 5662  ℩crio 7368  supcsup 9401  infcinf 9402

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3369  df-reu 3370  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-po 5571  df-so 5572  df-cnv 5671  df-iota 6494  df-riota 7369  df-sup 9403  df-inf 9404

This theorem is referenced by: (None)