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Theorem isoid 6564
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 6161 . 2 ( I ↾ 𝐴):𝐴1-1-onto𝐴
2 fvresi 6424 . . . . 5 (𝑥𝐴 → (( I ↾ 𝐴)‘𝑥) = 𝑥)
3 fvresi 6424 . . . . 5 (𝑦𝐴 → (( I ↾ 𝐴)‘𝑦) = 𝑦)
42, 3breqan12d 4660 . . . 4 ((𝑥𝐴𝑦𝐴) → ((( I ↾ 𝐴)‘𝑥)𝑅(( I ↾ 𝐴)‘𝑦) ↔ 𝑥𝑅𝑦))
54bicomd 213 . . 3 ((𝑥𝐴𝑦𝐴) → (𝑥𝑅𝑦 ↔ (( I ↾ 𝐴)‘𝑥)𝑅(( I ↾ 𝐴)‘𝑦)))
65rgen2a 2974 . 2 𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (( I ↾ 𝐴)‘𝑥)𝑅(( I ↾ 𝐴)‘𝑦))
7 df-isom 5885 . 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 954 1 ( I ↾ 𝐴) Isom 𝑅, 𝑅 (𝐴, 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wb 196  wa 384  wcel 1988  wral 2909   class class class wbr 4644   I cid 5013  cres 5106  1-1-ontowf1o 5875  cfv 5876   Isom wiso 5877
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pr 4897
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ral 2914  df-rex 2915  df-rab 2918  df-v 3197  df-sbc 3430  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-nul 3908  df-if 4078  df-sn 4169  df-pr 4171  df-op 4175  df-uni 4428  df-br 4645  df-opab 4704  df-id 5014  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-isom 5885
This theorem is referenced by:  ordiso  8406
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