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Theorem isoid 7275
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 6823 . 2 ( I ↾ 𝐴):𝐴1-1-onto𝐴
2 fvresi 7120 . . . . 5 (𝑥𝐴 → (( I ↾ 𝐴)‘𝑥) = 𝑥)
3 fvresi 7120 . . . . 5 (𝑦𝐴 → (( I ↾ 𝐴)‘𝑦) = 𝑦)
42, 3breqan12d 5122 . . . 4 ((𝑥𝐴𝑦𝐴) → ((( I ↾ 𝐴)‘𝑥)𝑅(( I ↾ 𝐴)‘𝑦) ↔ 𝑥𝑅𝑦))
54bicomd 222 . . 3 ((𝑥𝐴𝑦𝐴) → (𝑥𝑅𝑦 ↔ (( I ↾ 𝐴)‘𝑥)𝑅(( I ↾ 𝐴)‘𝑦)))
65rgen2 3191 . 2 𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (( I ↾ 𝐴)‘𝑥)𝑅(( I ↾ 𝐴)‘𝑦))
7 df-isom 6506 . 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 397  wcel 2107  wral 3061   class class class wbr 5106   I cid 5531  cres 5636  1-1-ontowf1o 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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