ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  isorel GIF version

Theorem isorel 5569
Description: An isomorphism connects binary relations via its function values. (Contributed by NM, 27-Apr-2004.)
Assertion
Ref Expression
isorel ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶𝐴𝐷𝐴)) → (𝐶𝑅𝐷 ↔ (𝐻𝐶)𝑆(𝐻𝐷)))

Proof of Theorem isorel
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-isom 5011 . . 3 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))))
21simprbi 269 . 2 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)))
3 breq1 3840 . . . 4 (𝑥 = 𝐶 → (𝑥𝑅𝑦𝐶𝑅𝑦))
4 fveq2 5289 . . . . 5 (𝑥 = 𝐶 → (𝐻𝑥) = (𝐻𝐶))
54breq1d 3847 . . . 4 (𝑥 = 𝐶 → ((𝐻𝑥)𝑆(𝐻𝑦) ↔ (𝐻𝐶)𝑆(𝐻𝑦)))
63, 5bibi12d 233 . . 3 (𝑥 = 𝐶 → ((𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)) ↔ (𝐶𝑅𝑦 ↔ (𝐻𝐶)𝑆(𝐻𝑦))))
7 breq2 3841 . . . 4 (𝑦 = 𝐷 → (𝐶𝑅𝑦𝐶𝑅𝐷))
8 fveq2 5289 . . . . 5 (𝑦 = 𝐷 → (𝐻𝑦) = (𝐻𝐷))
98breq2d 3849 . . . 4 (𝑦 = 𝐷 → ((𝐻𝐶)𝑆(𝐻𝑦) ↔ (𝐻𝐶)𝑆(𝐻𝐷)))
107, 9bibi12d 233 . . 3 (𝑦 = 𝐷 → ((𝐶𝑅𝑦 ↔ (𝐻𝐶)𝑆(𝐻𝑦)) ↔ (𝐶𝑅𝐷 ↔ (𝐻𝐶)𝑆(𝐻𝐷))))
116, 10rspc2v 2733 . 2 ((𝐶𝐴𝐷𝐴) → (∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)) → (𝐶𝑅𝐷 ↔ (𝐻𝐶)𝑆(𝐻𝐷))))
122, 11mpan9 275 1 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶𝐴𝐷𝐴)) → (𝐶𝑅𝐷 ↔ (𝐻𝐶)𝑆(𝐻𝐷)))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103   = wceq 1289  wcel 1438  wral 2359   class class class wbr 3837  1-1-ontowf1o 5001  cfv 5002   Isom wiso 5003
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-un 3001  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-br 3838  df-iota 4967  df-fv 5010  df-isom 5011
This theorem is referenced by:  isoresbr  5570  isoini  5579  isopolem  5583  isosolem  5585  smoiso  6049  isotilem  6680  supisolem  6682  ordiso2  6707  leisorel  10207  zfz1isolemiso  10209  iseqcoll  10212
  Copyright terms: Public domain W3C validator