Theorem infglbb 8672

Description: Bidirectional form of infglb 8671. (Contributed by AV, 3-Sep-2020.)

Hypotheses

Ref	Expression
infcl.1	⊢ (𝜑 → 𝑅 Or 𝐴)
infcl.2	⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)))
infglbb.3	⊢ (𝜑 → 𝐵 ⊆ 𝐴)

Assertion

Ref	Expression
infglbb	⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → (inf(𝐵, 𝐴, 𝑅)𝑅𝐶 ↔ ∃𝑧 ∈ 𝐵 𝑧𝑅𝐶))

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

Allowed substitution hints:   𝜑(𝑥,𝑦)   𝐶(𝑥,𝑦)

Proof of Theorem infglbb

Step	Hyp	Ref	Expression
1		df-inf 8624	. . 3 ⊢ inf(𝐵, 𝐴, 𝑅) = sup(𝐵, 𝐴, ◡𝑅)
2	1	breq1i 4882	. 2 ⊢ (inf(𝐵, 𝐴, 𝑅)𝑅𝐶 ↔ sup(𝐵, 𝐴, ◡𝑅)𝑅𝐶)
3		simpr 479	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → 𝐶 ∈ 𝐴)
4		infcl.1	. . . . . . 7 ⊢ (𝜑 → 𝑅 Or 𝐴)
5		cnvso 5919	. . . . . . 7 ⊢ (𝑅 Or 𝐴 ↔ ◡𝑅 Or 𝐴)
6	4, 5	sylib 210	. . . . . 6 ⊢ (𝜑 → ◡𝑅 Or 𝐴)
7		infcl.2	. . . . . . 7 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐵 𝑧𝑅𝑦)))
8	4, 7	infcllem 8668	. . . . . 6 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥◡𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦◡𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦◡𝑅𝑧)))
9	6, 8	supcl 8639	. . . . 5 ⊢ (𝜑 → sup(𝐵, 𝐴, ◡𝑅) ∈ 𝐴)
10	9	adantr 474	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → sup(𝐵, 𝐴, ◡𝑅) ∈ 𝐴)
11		brcnvg 5538	. . . . 5 ⊢ ((𝐶 ∈ 𝐴 ∧ sup(𝐵, 𝐴, ◡𝑅) ∈ 𝐴) → (𝐶◡𝑅sup(𝐵, 𝐴, ◡𝑅) ↔ sup(𝐵, 𝐴, ◡𝑅)𝑅𝐶))
12	11	bicomd 215	. . . 4 ⊢ ((𝐶 ∈ 𝐴 ∧ sup(𝐵, 𝐴, ◡𝑅) ∈ 𝐴) → (sup(𝐵, 𝐴, ◡𝑅)𝑅𝐶 ↔ 𝐶◡𝑅sup(𝐵, 𝐴, ◡𝑅)))
13	3, 10, 12	syl2anc 579	. . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → (sup(𝐵, 𝐴, ◡𝑅)𝑅𝐶 ↔ 𝐶◡𝑅sup(𝐵, 𝐴, ◡𝑅)))
14		infglbb.3	. . . 4 ⊢ (𝜑 → 𝐵 ⊆ 𝐴)
15	6, 8, 14	suplub2 8642	. . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → (𝐶◡𝑅sup(𝐵, 𝐴, ◡𝑅) ↔ ∃𝑧 ∈ 𝐵 𝐶◡𝑅𝑧))
16		vex 3417	. . . . 5 ⊢ 𝑧 ∈ V
17		brcnvg 5538	. . . . 5 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝑧 ∈ V) → (𝐶◡𝑅𝑧 ↔ 𝑧𝑅𝐶))
18	3, 16, 17	sylancl 580	. . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → (𝐶◡𝑅𝑧 ↔ 𝑧𝑅𝐶))
19	18	rexbidv 3262	. . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → (∃𝑧 ∈ 𝐵 𝐶◡𝑅𝑧 ↔ ∃𝑧 ∈ 𝐵 𝑧𝑅𝐶))
20	13, 15, 19	3bitrd 297	. 2 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → (sup(𝐵, 𝐴, ◡𝑅)𝑅𝐶 ↔ ∃𝑧 ∈ 𝐵 𝑧𝑅𝐶))
21	2, 20	syl5bb 275	1 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → (inf(𝐵, 𝐴, 𝑅)𝑅𝐶 ↔ ∃𝑧 ∈ 𝐵 𝑧𝑅𝐶))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 386   ∈ wcel 2164  ∀wral 3117  ∃wrex 3118  Vcvv 3414   ⊆ wss 3798   class class class wbr 4875   Or wor 5264  ^◡ccnv 5345  supcsup 8621  infcinf 8622

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1894  ax-4 1908  ax-5 2009  ax-6 2075  ax-7 2112  ax-9 2173  ax-10 2192  ax-11 2207  ax-12 2220  ax-13 2389  ax-ext 2803  ax-sep 5007  ax-nul 5015  ax-pr 5129

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 879  df-3or 1112  df-3an 1113  df-tru 1660  df-ex 1879  df-nf 1883  df-sb 2068  df-mo 2605  df-eu 2640  df-clab 2812  df-cleq 2818  df-clel 2821  df-nfc 2958  df-ne 3000  df-ral 3122  df-rex 3123  df-reu 3124  df-rmo 3125  df-rab 3126  df-v 3416  df-sbc 3663  df-dif 3801  df-un 3803  df-in 3805  df-ss 3812  df-nul 4147  df-if 4309  df-sn 4400  df-pr 4402  df-op 4406  df-uni 4661  df-br 4876  df-opab 4938  df-po 5265  df-so 5266  df-cnv 5354  df-iota 6090  df-riota 6871  df-sup 8623  df-inf 8624

This theorem is referenced by:  infregelb  11344  infxrgelb  12460  infxrge0glb  30073  infxrglb  40351  infrglb  40615