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Mirrors > Home > MPE Home > Th. List > isorel | Structured version Visualization version GIF version |
Description: An isomorphism connects binary relations via its function values. (Contributed by NM, 27-Apr-2004.) |
Ref | Expression |
---|---|
isorel | ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝐶𝑅𝐷 ↔ (𝐻‘𝐶)𝑆(𝐻‘𝐷))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-isom 6333 | . . 3 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐻:𝐴–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)))) | |
2 | 1 | simprbi 500 | . 2 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦))) |
3 | breq1 5033 | . . . 4 ⊢ (𝑥 = 𝐶 → (𝑥𝑅𝑦 ↔ 𝐶𝑅𝑦)) | |
4 | fveq2 6645 | . . . . 5 ⊢ (𝑥 = 𝐶 → (𝐻‘𝑥) = (𝐻‘𝐶)) | |
5 | 4 | breq1d 5040 | . . . 4 ⊢ (𝑥 = 𝐶 → ((𝐻‘𝑥)𝑆(𝐻‘𝑦) ↔ (𝐻‘𝐶)𝑆(𝐻‘𝑦))) |
6 | 3, 5 | bibi12d 349 | . . 3 ⊢ (𝑥 = 𝐶 → ((𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)) ↔ (𝐶𝑅𝑦 ↔ (𝐻‘𝐶)𝑆(𝐻‘𝑦)))) |
7 | breq2 5034 | . . . 4 ⊢ (𝑦 = 𝐷 → (𝐶𝑅𝑦 ↔ 𝐶𝑅𝐷)) | |
8 | fveq2 6645 | . . . . 5 ⊢ (𝑦 = 𝐷 → (𝐻‘𝑦) = (𝐻‘𝐷)) | |
9 | 8 | breq2d 5042 | . . . 4 ⊢ (𝑦 = 𝐷 → ((𝐻‘𝐶)𝑆(𝐻‘𝑦) ↔ (𝐻‘𝐶)𝑆(𝐻‘𝐷))) |
10 | 7, 9 | bibi12d 349 | . . 3 ⊢ (𝑦 = 𝐷 → ((𝐶𝑅𝑦 ↔ (𝐻‘𝐶)𝑆(𝐻‘𝑦)) ↔ (𝐶𝑅𝐷 ↔ (𝐻‘𝐶)𝑆(𝐻‘𝐷)))) |
11 | 6, 10 | rspc2v 3581 | . 2 ⊢ ((𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴) → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)) → (𝐶𝑅𝐷 ↔ (𝐻‘𝐶)𝑆(𝐻‘𝐷)))) |
12 | 2, 11 | mpan9 510 | 1 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → (𝐶𝑅𝐷 ↔ (𝐻‘𝐶)𝑆(𝐻‘𝐷))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∀wral 3106 class class class wbr 5030 –1-1-onto→wf1o 6323 ‘cfv 6324 Isom wiso 6325 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-ext 2770 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-ex 1782 df-sb 2070 df-clab 2777 df-cleq 2791 df-clel 2870 df-ral 3111 df-v 3443 df-un 3886 df-in 3888 df-ss 3898 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-iota 6283 df-fv 6332 df-isom 6333 |
This theorem is referenced by: soisores 7059 isomin 7069 isoini 7070 isopolem 7077 isosolem 7079 weniso 7086 smoiso 7982 supisolem 8921 ordiso2 8963 cantnflt 9119 cantnfp1lem3 9127 cantnflem1b 9133 cantnflem1 9136 wemapwe 9144 cnfcomlem 9146 cnfcom 9147 cnfcom3lem 9150 fpwwe2lem6 10046 fpwwe2lem7 10047 fpwwe2lem9 10049 leisorel 13814 seqcoll 13818 seqcoll2 13819 isercoll 15016 ordthmeolem 22406 iccpnfhmeo 23550 xrhmeo 23551 dvcnvrelem1 24620 dvcvx 24623 isoun 30461 erdszelem8 32558 erdsze2lem2 32564 fourierdlem20 42769 fourierdlem46 42794 fourierdlem50 42798 fourierdlem63 42811 fourierdlem64 42812 fourierdlem65 42813 fourierdlem76 42824 fourierdlem79 42827 fourierdlem102 42850 fourierdlem103 42851 fourierdlem104 42852 fourierdlem114 42862 |
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