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Theorem isoid 6743
Description: Identity law for isomorphism. Proposition 6.30(1) of [TakeutiZaring] p. 33. (Contributed by NM, 27-Apr-2004.)
Assertion
Ref Expression
isoid ( I ↾ 𝐴) Isom 𝑅, 𝑅 (𝐴, 𝐴)

Proof of Theorem isoid
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 f1oi 6336 . 2 ( I ↾ 𝐴):𝐴1-1-onto𝐴
2 fvresi 6604 . . . . 5 (𝑥𝐴 → (( I ↾ 𝐴)‘𝑥) = 𝑥)
3 fvresi 6604 . . . . 5 (𝑦𝐴 → (( I ↾ 𝐴)‘𝑦) = 𝑦)
42, 3breqan12d 4820 . . . 4 ((𝑥𝐴𝑦𝐴) → ((( I ↾ 𝐴)‘𝑥)𝑅(( I ↾ 𝐴)‘𝑦) ↔ 𝑥𝑅𝑦))
54bicomd 213 . . 3 ((𝑥𝐴𝑦𝐴) → (𝑥𝑅𝑦 ↔ (( I ↾ 𝐴)‘𝑥)𝑅(( I ↾ 𝐴)‘𝑦)))
65rgen2a 3115 . 2 𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (( I ↾ 𝐴)‘𝑥)𝑅(( I ↾ 𝐴)‘𝑦))
7 df-isom 6058 . 2 (( I ↾ 𝐴) Isom 𝑅, 𝑅 (𝐴, 𝐴) ↔ (( I ↾ 𝐴):𝐴1-1-onto𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (( I ↾ 𝐴)‘𝑥)𝑅(( I ↾ 𝐴)‘𝑦))))
81, 6, 7mpbir2an 993 1 ( I ↾ 𝐴) Isom 𝑅, 𝑅 (𝐴, 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 383  wcel 2139  wral 3050   class class class wbr 4804   I cid 5173  cres 5268  1-1-ontowf1o 6048  cfv 6049   Isom wiso 6050
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-isom 6058
This theorem is referenced by:  ordiso  8588
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