![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 6510 | . . 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 498 | . 2 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦))) |
3 | breq1 5113 | . . . 4 ⊢ (𝑥 = 𝐶 → (𝑥𝑅𝑦 ↔ 𝐶𝑅𝑦)) | |
4 | fveq2 6847 | . . . . 5 ⊢ (𝑥 = 𝐶 → (𝐻‘𝑥) = (𝐻‘𝐶)) | |
5 | 4 | breq1d 5120 | . . . 4 ⊢ (𝑥 = 𝐶 → ((𝐻‘𝑥)𝑆(𝐻‘𝑦) ↔ (𝐻‘𝐶)𝑆(𝐻‘𝑦))) |
6 | 3, 5 | bibi12d 346 | . . 3 ⊢ (𝑥 = 𝐶 → ((𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)) ↔ (𝐶𝑅𝑦 ↔ (𝐻‘𝐶)𝑆(𝐻‘𝑦)))) |
7 | breq2 5114 | . . . 4 ⊢ (𝑦 = 𝐷 → (𝐶𝑅𝑦 ↔ 𝐶𝑅𝐷)) | |
8 | fveq2 6847 | . . . . 5 ⊢ (𝑦 = 𝐷 → (𝐻‘𝑦) = (𝐻‘𝐷)) | |
9 | 8 | breq2d 5122 | . . . 4 ⊢ (𝑦 = 𝐷 → ((𝐻‘𝐶)𝑆(𝐻‘𝑦) ↔ (𝐻‘𝐶)𝑆(𝐻‘𝐷))) |
10 | 7, 9 | bibi12d 346 | . . 3 ⊢ (𝑦 = 𝐷 → ((𝐶𝑅𝑦 ↔ (𝐻‘𝐶)𝑆(𝐻‘𝑦)) ↔ (𝐶𝑅𝐷 ↔ (𝐻‘𝐶)𝑆(𝐻‘𝐷)))) |
11 | 6, 10 | rspc2v 3593 | . 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 508 | 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 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ∀wral 3065 class class class wbr 5110 –1-1-onto→wf1o 6500 ‘cfv 6501 Isom wiso 6502 |
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 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-ral 3066 df-rab 3411 df-v 3450 df-dif 3918 df-un 3920 df-in 3922 df-ss 3932 df-nul 4288 df-if 4492 df-sn 4592 df-pr 4594 df-op 4598 df-uni 4871 df-br 5111 df-iota 6453 df-fv 6509 df-isom 6510 |
This theorem is referenced by: soisores 7277 isomin 7287 isoini 7288 isopolem 7295 isosolem 7297 weniso 7304 smoiso 8313 supisolem 9416 ordiso2 9458 cantnflt 9615 cantnfp1lem3 9623 cantnflem1b 9629 cantnflem1 9632 wemapwe 9640 cnfcomlem 9642 cnfcom 9643 cnfcom3lem 9646 fpwwe2lem5 10578 fpwwe2lem6 10579 fpwwe2lem8 10581 leisorel 14366 seqcoll 14370 seqcoll2 14371 isercoll 15559 ordthmeolem 23168 iccpnfhmeo 24324 xrhmeo 24325 dvcnvrelem1 25397 dvcvx 25400 isoun 31657 erdszelem8 33832 erdsze2lem2 33838 cantnfresb 41688 fourierdlem20 44442 fourierdlem46 44467 fourierdlem50 44471 fourierdlem63 44484 fourierdlem64 44485 fourierdlem65 44486 fourierdlem76 44497 fourierdlem79 44500 fourierdlem102 44523 fourierdlem103 44524 fourierdlem104 44525 fourierdlem114 44535 |
Copyright terms: Public domain | W3C validator |