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Theorem isoid 7337
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 6877 . 2 ( I ↾ 𝐴):𝐴1-1-onto𝐴
2 fvresi 7182 . . . . 5 (𝑥𝐴 → (( I ↾ 𝐴)‘𝑥) = 𝑥)
3 fvresi 7182 . . . . 5 (𝑦𝐴 → (( I ↾ 𝐴)‘𝑦) = 𝑦)
42, 3breqan12d 5164 . . . 4 ((𝑥𝐴𝑦𝐴) → ((( I ↾ 𝐴)‘𝑥)𝑅(( I ↾ 𝐴)‘𝑦) ↔ 𝑥𝑅𝑦))
54bicomd 222 . . 3 ((𝑥𝐴𝑦𝐴) → (𝑥𝑅𝑦 ↔ (( I ↾ 𝐴)‘𝑥)𝑅(( I ↾ 𝐴)‘𝑦)))
65rgen2 3194 . 2 𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (( I ↾ 𝐴)‘𝑥)𝑅(( I ↾ 𝐴)‘𝑦))
7 df-isom 6557 . 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 710 1 ( I ↾ 𝐴) Isom 𝑅, 𝑅 (𝐴, 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 395  wcel 2099  wral 3058   class class class wbr 5148   I cid 5575  cres 5680  1-1-ontowf1o 6547  cfv 6548   Isom wiso 6549
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-12 2167  ax-ext 2699  ax-sep 5299  ax-nul 5306  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2530  df-eu 2559  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rex 3068  df-rab 3430  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4909  df-br 5149  df-opab 5211  df-id 5576  df-xp 5684  df-rel 5685  df-cnv 5686  df-co 5687  df-dm 5688  df-rn 5689  df-res 5690  df-ima 5691  df-iota 6500  df-fun 6550  df-fn 6551  df-f 6552  df-f1 6553  df-fo 6554  df-f1o 6555  df-fv 6556  df-isom 6557
This theorem is referenced by:  ordiso  9539
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