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Mirrors > Home > MPE Home > Th. List > infiso | Structured version Visualization version GIF version |
Description: Image of an infimum under an isomorphism. (Contributed by AV, 4-Sep-2020.) |
Ref | Expression |
---|---|
infiso.1 | ⊢ (𝜑 → 𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵)) |
infiso.2 | ⊢ (𝜑 → 𝐶 ⊆ 𝐴) |
infiso.3 | ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐶 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐶 𝑧𝑅𝑦))) |
infiso.4 | ⊢ (𝜑 → 𝑅 Or 𝐴) |
Ref | Expression |
---|---|
infiso | ⊢ (𝜑 → inf((𝐹 “ 𝐶), 𝐵, 𝑆) = (𝐹‘inf(𝐶, 𝐴, 𝑅))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | infiso.1 | . . . 4 ⊢ (𝜑 → 𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵)) | |
2 | isocnv2 6481 | . . . 4 ⊢ (𝐹 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ 𝐹 Isom ◡𝑅, ◡𝑆(𝐴, 𝐵)) | |
3 | 1, 2 | sylib 207 | . . 3 ⊢ (𝜑 → 𝐹 Isom ◡𝑅, ◡𝑆(𝐴, 𝐵)) |
4 | infiso.2 | . . 3 ⊢ (𝜑 → 𝐶 ⊆ 𝐴) | |
5 | infiso.4 | . . . 4 ⊢ (𝜑 → 𝑅 Or 𝐴) | |
6 | infiso.3 | . . . 4 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐶 ¬ 𝑦𝑅𝑥 ∧ ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 → ∃𝑧 ∈ 𝐶 𝑧𝑅𝑦))) | |
7 | 5, 6 | infcllem 8276 | . . 3 ⊢ (𝜑 → ∃𝑥 ∈ 𝐴 (∀𝑦 ∈ 𝐶 ¬ 𝑥◡𝑅𝑦 ∧ ∀𝑦 ∈ 𝐴 (𝑦◡𝑅𝑥 → ∃𝑧 ∈ 𝐶 𝑦◡𝑅𝑧))) |
8 | cnvso 5591 | . . . 4 ⊢ (𝑅 Or 𝐴 ↔ ◡𝑅 Or 𝐴) | |
9 | 5, 8 | sylib 207 | . . 3 ⊢ (𝜑 → ◡𝑅 Or 𝐴) |
10 | 3, 4, 7, 9 | supiso 8264 | . 2 ⊢ (𝜑 → sup((𝐹 “ 𝐶), 𝐵, ◡𝑆) = (𝐹‘sup(𝐶, 𝐴, ◡𝑅))) |
11 | df-inf 8232 | . 2 ⊢ inf((𝐹 “ 𝐶), 𝐵, 𝑆) = sup((𝐹 “ 𝐶), 𝐵, ◡𝑆) | |
12 | df-inf 8232 | . . 3 ⊢ inf(𝐶, 𝐴, 𝑅) = sup(𝐶, 𝐴, ◡𝑅) | |
13 | 12 | fveq2i 6106 | . 2 ⊢ (𝐹‘inf(𝐶, 𝐴, 𝑅)) = (𝐹‘sup(𝐶, 𝐴, ◡𝑅)) |
14 | 10, 11, 13 | 3eqtr4g 2669 | 1 ⊢ (𝜑 → inf((𝐹 “ 𝐶), 𝐵, 𝑆) = (𝐹‘inf(𝐶, 𝐴, 𝑅))) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 383 = wceq 1475 ∀wral 2896 ∃wrex 2897 ⊆ wss 3540 class class class wbr 4583 Or wor 4958 ◡ccnv 5037 “ cima 5041 ‘cfv 5804 Isom wiso 5805 supcsup 8229 infcinf 8230 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-isom 5813 df-riota 6511 df-sup 8231 df-inf 8232 |
This theorem is referenced by: (None) |
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