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Theorem isoresbr 5717
Description: A consequence of isomorphism on two relations for a function's restriction. (Contributed by Jim Kingdon, 11-Jan-2019.)
Assertion
Ref Expression
isoresbr ((𝐹𝐴) Isom 𝑅, 𝑆 (𝐴, (𝐹𝐴)) → ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐹𝑥)𝑆(𝐹𝑦)))
Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐹,𝑦   𝑥,𝑅,𝑦   𝑥,𝑆,𝑦

Proof of Theorem isoresbr
StepHypRef Expression
1 isorel 5716 . . . 4 (((𝐹𝐴) Isom 𝑅, 𝑆 (𝐴, (𝐹𝐴)) ∧ (𝑥𝐴𝑦𝐴)) → (𝑥𝑅𝑦 ↔ ((𝐹𝐴)‘𝑥)𝑆((𝐹𝐴)‘𝑦)))
2 fvres 5452 . . . . . 6 (𝑥𝐴 → ((𝐹𝐴)‘𝑥) = (𝐹𝑥))
3 fvres 5452 . . . . . 6 (𝑦𝐴 → ((𝐹𝐴)‘𝑦) = (𝐹𝑦))
42, 3breqan12d 3952 . . . . 5 ((𝑥𝐴𝑦𝐴) → (((𝐹𝐴)‘𝑥)𝑆((𝐹𝐴)‘𝑦) ↔ (𝐹𝑥)𝑆(𝐹𝑦)))
54adantl 275 . . . 4 (((𝐹𝐴) Isom 𝑅, 𝑆 (𝐴, (𝐹𝐴)) ∧ (𝑥𝐴𝑦𝐴)) → (((𝐹𝐴)‘𝑥)𝑆((𝐹𝐴)‘𝑦) ↔ (𝐹𝑥)𝑆(𝐹𝑦)))
61, 5bitrd 187 . . 3 (((𝐹𝐴) Isom 𝑅, 𝑆 (𝐴, (𝐹𝐴)) ∧ (𝑥𝐴𝑦𝐴)) → (𝑥𝑅𝑦 ↔ (𝐹𝑥)𝑆(𝐹𝑦)))
76biimpd 143 . 2 (((𝐹𝐴) Isom 𝑅, 𝑆 (𝐴, (𝐹𝐴)) ∧ (𝑥𝐴𝑦𝐴)) → (𝑥𝑅𝑦 → (𝐹𝑥)𝑆(𝐹𝑦)))
87ralrimivva 2517 1 ((𝐹𝐴) Isom 𝑅, 𝑆 (𝐴, (𝐹𝐴)) → ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐹𝑥)𝑆(𝐹𝑦)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wcel 1481  wral 2417   class class class wbr 3936  cres 4548  cima 4549  cfv 5130   Isom wiso 5131
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-br 3937  df-opab 3997  df-xp 4552  df-res 4558  df-iota 5095  df-fv 5138  df-isom 5139
This theorem is referenced by: (None)
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