MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  isorel Structured version   Visualization version   GIF version

Theorem isorel 7340
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 6562 . . 3 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))))
21simprbi 495 . 2 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)))
3 breq1 5155 . . . 4 (𝑥 = 𝐶 → (𝑥𝑅𝑦𝐶𝑅𝑦))
4 fveq2 6902 . . . . 5 (𝑥 = 𝐶 → (𝐻𝑥) = (𝐻𝐶))
54breq1d 5162 . . . 4 (𝑥 = 𝐶 → ((𝐻𝑥)𝑆(𝐻𝑦) ↔ (𝐻𝐶)𝑆(𝐻𝑦)))
63, 5bibi12d 344 . . 3 (𝑥 = 𝐶 → ((𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)) ↔ (𝐶𝑅𝑦 ↔ (𝐻𝐶)𝑆(𝐻𝑦))))
7 breq2 5156 . . . 4 (𝑦 = 𝐷 → (𝐶𝑅𝑦𝐶𝑅𝐷))
8 fveq2 6902 . . . . 5 (𝑦 = 𝐷 → (𝐻𝑦) = (𝐻𝐷))
98breq2d 5164 . . . 4 (𝑦 = 𝐷 → ((𝐻𝐶)𝑆(𝐻𝑦) ↔ (𝐻𝐶)𝑆(𝐻𝐷)))
107, 9bibi12d 344 . . 3 (𝑦 = 𝐷 → ((𝐶𝑅𝑦 ↔ (𝐻𝐶)𝑆(𝐻𝑦)) ↔ (𝐶𝑅𝐷 ↔ (𝐻𝐶)𝑆(𝐻𝐷))))
116, 10rspc2v 3622 . 2 ((𝐶𝐴𝐷𝐴) → (∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)) → (𝐶𝑅𝐷 ↔ (𝐻𝐶)𝑆(𝐻𝐷))))
122, 11mpan9 505 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 394   = wceq 1533  wcel 2098  wral 3058   class class class wbr 5152  1-1-ontowf1o 6552  cfv 6553   Isom wiso 6554
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2699
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3059  df-rab 3431  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4327  df-if 4533  df-sn 4633  df-pr 4635  df-op 4639  df-uni 4913  df-br 5153  df-iota 6505  df-fv 6561  df-isom 6562
This theorem is referenced by:  soisores  7341  isomin  7351  isoini  7352  isopolem  7359  isosolem  7361  weniso  7368  smoiso  8389  supisolem  9504  ordiso2  9546  cantnflt  9703  cantnfp1lem3  9711  cantnflem1b  9717  cantnflem1  9720  wemapwe  9728  cnfcomlem  9730  cnfcom  9731  cnfcom3lem  9734  fpwwe2lem5  10666  fpwwe2lem6  10667  fpwwe2lem8  10669  leisorel  14461  seqcoll  14465  seqcoll2  14466  isercoll  15654  ordthmeolem  23725  iccpnfhmeo  24890  xrhmeo  24891  dvcnvrelem1  25970  dvcvx  25973  isoun  32502  erdszelem8  34841  erdsze2lem2  34847  cantnfresb  42784  fourierdlem20  45544  fourierdlem46  45569  fourierdlem50  45573  fourierdlem63  45586  fourierdlem64  45587  fourierdlem65  45588  fourierdlem76  45599  fourierdlem79  45602  fourierdlem102  45625  fourierdlem103  45626  fourierdlem104  45627  fourierdlem114  45637
  Copyright terms: Public domain W3C validator