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Theorem isoresbr 5813
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 5812 . . . 4 (((𝐹𝐴) Isom 𝑅, 𝑆 (𝐴, (𝐹𝐴)) ∧ (𝑥𝐴𝑦𝐴)) → (𝑥𝑅𝑦 ↔ ((𝐹𝐴)‘𝑥)𝑆((𝐹𝐴)‘𝑦)))
2 fvres 5541 . . . . . 6 (𝑥𝐴 → ((𝐹𝐴)‘𝑥) = (𝐹𝑥))
3 fvres 5541 . . . . . 6 (𝑦𝐴 → ((𝐹𝐴)‘𝑦) = (𝐹𝑦))
42, 3breqan12d 4021 . . . . 5 ((𝑥𝐴𝑦𝐴) → (((𝐹𝐴)‘𝑥)𝑆((𝐹𝐴)‘𝑦) ↔ (𝐹𝑥)𝑆(𝐹𝑦)))
54adantl 277 . . . 4 (((𝐹𝐴) Isom 𝑅, 𝑆 (𝐴, (𝐹𝐴)) ∧ (𝑥𝐴𝑦𝐴)) → (((𝐹𝐴)‘𝑥)𝑆((𝐹𝐴)‘𝑦) ↔ (𝐹𝑥)𝑆(𝐹𝑦)))
61, 5bitrd 188 . . 3 (((𝐹𝐴) Isom 𝑅, 𝑆 (𝐴, (𝐹𝐴)) ∧ (𝑥𝐴𝑦𝐴)) → (𝑥𝑅𝑦 ↔ (𝐹𝑥)𝑆(𝐹𝑦)))
76biimpd 144 . 2 (((𝐹𝐴) Isom 𝑅, 𝑆 (𝐴, (𝐹𝐴)) ∧ (𝑥𝐴𝑦𝐴)) → (𝑥𝑅𝑦 → (𝐹𝑥)𝑆(𝐹𝑦)))
87ralrimivva 2559 1 ((𝐹𝐴) Isom 𝑅, 𝑆 (𝐴, (𝐹𝐴)) → ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 → (𝐹𝑥)𝑆(𝐹𝑦)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  wcel 2148  wral 2455   class class class wbr 4005  cres 4630  cima 4631  cfv 5218   Isom wiso 5219
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-v 2741  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-xp 4634  df-res 4640  df-iota 5180  df-fv 5226  df-isom 5227
This theorem is referenced by: (None)
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