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Mirrors > Home > MPE Home > Th. List > isoid | Structured version Visualization version GIF version |
Description: Identity law for isomorphism. Proposition 6.30(1) of [TakeutiZaring] p. 33. (Contributed by NM, 27-Apr-2004.) |
Ref | Expression |
---|---|
isoid | ⊢ ( I ↾ 𝐴) Isom 𝑅, 𝑅 (𝐴, 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1oi 6823 | . 2 ⊢ ( I ↾ 𝐴):𝐴–1-1-onto→𝐴 | |
2 | fvresi 7120 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 → (( I ↾ 𝐴)‘𝑥) = 𝑥) | |
3 | fvresi 7120 | . . . . 5 ⊢ (𝑦 ∈ 𝐴 → (( I ↾ 𝐴)‘𝑦) = 𝑦) | |
4 | 2, 3 | breqan12d 5122 | . . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((( I ↾ 𝐴)‘𝑥)𝑅(( I ↾ 𝐴)‘𝑦) ↔ 𝑥𝑅𝑦)) |
5 | 4 | bicomd 222 | . . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥𝑅𝑦 ↔ (( I ↾ 𝐴)‘𝑥)𝑅(( I ↾ 𝐴)‘𝑦))) |
6 | 5 | rgen2 3191 | . 2 ⊢ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (( I ↾ 𝐴)‘𝑥)𝑅(( I ↾ 𝐴)‘𝑦)) |
7 | df-isom 6506 | . 2 ⊢ (( I ↾ 𝐴) Isom 𝑅, 𝑅 (𝐴, 𝐴) ↔ (( I ↾ 𝐴):𝐴–1-1-onto→𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (( I ↾ 𝐴)‘𝑥)𝑅(( I ↾ 𝐴)‘𝑦)))) | |
8 | 1, 6, 7 | mpbir2an 710 | 1 ⊢ ( I ↾ 𝐴) Isom 𝑅, 𝑅 (𝐴, 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∧ wa 397 ∈ wcel 2107 ∀wral 3061 class class class wbr 5106 I cid 5531 ↾ cres 5636 –1-1-onto→wf1o 6496 ‘cfv 6497 Isom wiso 6498 |
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-10 2138 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
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-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-f 6501 df-f1 6502 df-fo 6503 df-f1o 6504 df-fv 6505 df-isom 6506 |
This theorem is referenced by: ordiso 9457 |
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