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Theorem isorel 7323
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 6553 . . 3 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))))
21simprbi 498 . 2 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)))
3 breq1 5152 . . . 4 (𝑥 = 𝐶 → (𝑥𝑅𝑦𝐶𝑅𝑦))
4 fveq2 6892 . . . . 5 (𝑥 = 𝐶 → (𝐻𝑥) = (𝐻𝐶))
54breq1d 5159 . . . 4 (𝑥 = 𝐶 → ((𝐻𝑥)𝑆(𝐻𝑦) ↔ (𝐻𝐶)𝑆(𝐻𝑦)))
63, 5bibi12d 346 . . 3 (𝑥 = 𝐶 → ((𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)) ↔ (𝐶𝑅𝑦 ↔ (𝐻𝐶)𝑆(𝐻𝑦))))
7 breq2 5153 . . . 4 (𝑦 = 𝐷 → (𝐶𝑅𝑦𝐶𝑅𝐷))
8 fveq2 6892 . . . . 5 (𝑦 = 𝐷 → (𝐻𝑦) = (𝐻𝐷))
98breq2d 5161 . . . 4 (𝑦 = 𝐷 → ((𝐻𝐶)𝑆(𝐻𝑦) ↔ (𝐻𝐶)𝑆(𝐻𝐷)))
107, 9bibi12d 346 . . 3 (𝑦 = 𝐷 → ((𝐶𝑅𝑦 ↔ (𝐻𝐶)𝑆(𝐻𝑦)) ↔ (𝐶𝑅𝐷 ↔ (𝐻𝐶)𝑆(𝐻𝐷))))
116, 10rspc2v 3623 . 2 ((𝐶𝐴𝐷𝐴) → (∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)) → (𝐶𝑅𝐷 ↔ (𝐻𝐶)𝑆(𝐻𝐷))))
122, 11mpan9 508 1 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶𝐴𝐷𝐴)) → (𝐶𝑅𝐷 ↔ (𝐻𝐶)𝑆(𝐻𝐷)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 397   = wceq 1542  wcel 2107  wral 3062   class class class wbr 5149  1-1-ontowf1o 6543  cfv 6544   Isom wiso 6545
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-ext 2704
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-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-br 5150  df-iota 6496  df-fv 6552  df-isom 6553
This theorem is referenced by:  soisores  7324  isomin  7334  isoini  7335  isopolem  7342  isosolem  7344  weniso  7351  smoiso  8362  supisolem  9468  ordiso2  9510  cantnflt  9667  cantnfp1lem3  9675  cantnflem1b  9681  cantnflem1  9684  wemapwe  9692  cnfcomlem  9694  cnfcom  9695  cnfcom3lem  9698  fpwwe2lem5  10630  fpwwe2lem6  10631  fpwwe2lem8  10633  leisorel  14421  seqcoll  14425  seqcoll2  14426  isercoll  15614  ordthmeolem  23305  iccpnfhmeo  24461  xrhmeo  24462  dvcnvrelem1  25534  dvcvx  25537  isoun  31923  erdszelem8  34189  erdsze2lem2  34195  cantnfresb  42074  fourierdlem20  44843  fourierdlem46  44868  fourierdlem50  44872  fourierdlem63  44885  fourierdlem64  44886  fourierdlem65  44887  fourierdlem76  44898  fourierdlem79  44901  fourierdlem102  44924  fourierdlem103  44925  fourierdlem104  44926  fourierdlem114  44936
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