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Mirrors > Home > MPE Home > Th. List > infglbb | Structured version Visualization version GIF version |
Description: Bidirectional form of infglb 9249. (Contributed by AV, 3-Sep-2020.) |
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) |
Ref | Expression |
---|---|
infglbb | ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → (inf(𝐵, 𝐴, 𝑅)𝑅𝐶 ↔ ∃𝑧 ∈ 𝐵 𝑧𝑅𝐶)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-inf 9202 | . . 3 ⊢ inf(𝐵, 𝐴, 𝑅) = sup(𝐵, 𝐴, ◡𝑅) | |
2 | 1 | breq1i 5081 | . 2 ⊢ (inf(𝐵, 𝐴, 𝑅)𝑅𝐶 ↔ sup(𝐵, 𝐴, ◡𝑅)𝑅𝐶) |
3 | simpr 485 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → 𝐶 ∈ 𝐴) | |
4 | infcl.1 | . . . . . . 7 ⊢ (𝜑 → 𝑅 Or 𝐴) | |
5 | cnvso 6191 | . . . . . . 7 ⊢ (𝑅 Or 𝐴 ↔ ◡𝑅 Or 𝐴) | |
6 | 4, 5 | sylib 217 | . . . . . 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 9246 | . . . . . 6 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐵 ¬ 𝑥◡𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦◡𝑅𝑥 → ∃𝑧 ∈ 𝐵 𝑦◡𝑅𝑧))) |
9 | 6, 8 | supcl 9217 | . . . . 5 ⊢ (𝜑 → sup(𝐵, 𝐴, ◡𝑅) ∈ 𝐴) |
10 | 9 | adantr 481 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → sup(𝐵, 𝐴, ◡𝑅) ∈ 𝐴) |
11 | brcnvg 5788 | . . . . 5 ⊢ ((𝐶 ∈ 𝐴 ∧ sup(𝐵, 𝐴, ◡𝑅) ∈ 𝐴) → (𝐶◡𝑅sup(𝐵, 𝐴, ◡𝑅) ↔ sup(𝐵, 𝐴, ◡𝑅)𝑅𝐶)) | |
12 | 11 | bicomd 222 | . . . 4 ⊢ ((𝐶 ∈ 𝐴 ∧ sup(𝐵, 𝐴, ◡𝑅) ∈ 𝐴) → (sup(𝐵, 𝐴, ◡𝑅)𝑅𝐶 ↔ 𝐶◡𝑅sup(𝐵, 𝐴, ◡𝑅))) |
13 | 3, 10, 12 | syl2anc 584 | . . 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 9220 | . . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → (𝐶◡𝑅sup(𝐵, 𝐴, ◡𝑅) ↔ ∃𝑧 ∈ 𝐵 𝐶◡𝑅𝑧)) |
16 | vex 3436 | . . . . 5 ⊢ 𝑧 ∈ V | |
17 | brcnvg 5788 | . . . . 5 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝑧 ∈ V) → (𝐶◡𝑅𝑧 ↔ 𝑧𝑅𝐶)) | |
18 | 3, 16, 17 | sylancl 586 | . . . 4 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → (𝐶◡𝑅𝑧 ↔ 𝑧𝑅𝐶)) |
19 | 18 | rexbidv 3226 | . . 3 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → (∃𝑧 ∈ 𝐵 𝐶◡𝑅𝑧 ↔ ∃𝑧 ∈ 𝐵 𝑧𝑅𝐶)) |
20 | 13, 15, 19 | 3bitrd 305 | . 2 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → (sup(𝐵, 𝐴, ◡𝑅)𝑅𝐶 ↔ ∃𝑧 ∈ 𝐵 𝑧𝑅𝐶)) |
21 | 2, 20 | bitrid 282 | 1 ⊢ ((𝜑 ∧ 𝐶 ∈ 𝐴) → (inf(𝐵, 𝐴, 𝑅)𝑅𝐶 ↔ ∃𝑧 ∈ 𝐵 𝑧𝑅𝐶)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 ∈ wcel 2106 ∀wral 3064 ∃wrex 3065 Vcvv 3432 ⊆ wss 3887 class class class wbr 5074 Or wor 5502 ◡ccnv 5588 supcsup 9199 infcinf 9200 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-rmo 3071 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-po 5503 df-so 5504 df-cnv 5597 df-iota 6391 df-riota 7232 df-sup 9201 df-inf 9202 |
This theorem is referenced by: infregelb 11959 infxrgelb 13069 infxrge0glb 31088 infxrglb 42879 infrglb 43131 |
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