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Theorem isorel 6533
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 5858 . . 3 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐻:𝐴1-1-onto𝐵 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦))))
21simprbi 480 . 2 (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)))
3 breq1 4618 . . . 4 (𝑥 = 𝐶 → (𝑥𝑅𝑦𝐶𝑅𝑦))
4 fveq2 6150 . . . . 5 (𝑥 = 𝐶 → (𝐻𝑥) = (𝐻𝐶))
54breq1d 4625 . . . 4 (𝑥 = 𝐶 → ((𝐻𝑥)𝑆(𝐻𝑦) ↔ (𝐻𝐶)𝑆(𝐻𝑦)))
63, 5bibi12d 335 . . 3 (𝑥 = 𝐶 → ((𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)) ↔ (𝐶𝑅𝑦 ↔ (𝐻𝐶)𝑆(𝐻𝑦))))
7 breq2 4619 . . . 4 (𝑦 = 𝐷 → (𝐶𝑅𝑦𝐶𝑅𝐷))
8 fveq2 6150 . . . . 5 (𝑦 = 𝐷 → (𝐻𝑦) = (𝐻𝐷))
98breq2d 4627 . . . 4 (𝑦 = 𝐷 → ((𝐻𝐶)𝑆(𝐻𝑦) ↔ (𝐻𝐶)𝑆(𝐻𝐷)))
107, 9bibi12d 335 . . 3 (𝑦 = 𝐷 → ((𝐶𝑅𝑦 ↔ (𝐻𝐶)𝑆(𝐻𝑦)) ↔ (𝐶𝑅𝐷 ↔ (𝐻𝐶)𝑆(𝐻𝐷))))
116, 10rspc2v 3307 . 2 ((𝐶𝐴𝐷𝐴) → (∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦 ↔ (𝐻𝑥)𝑆(𝐻𝑦)) → (𝐶𝑅𝐷 ↔ (𝐻𝐶)𝑆(𝐻𝐷))))
122, 11mpan9 486 1 ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ (𝐶𝐴𝐷𝐴)) → (𝐶𝑅𝐷 ↔ (𝐻𝐶)𝑆(𝐻𝐷)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2907   class class class wbr 4615  1-1-ontowf1o 5848  cfv 5849   Isom wiso 5850
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-sn 4151  df-pr 4153  df-op 4157  df-uni 4405  df-br 4616  df-iota 5812  df-fv 5857  df-isom 5858
This theorem is referenced by:  soisores  6534  isomin  6544  isoini  6545  isopolem  6552  isosolem  6554  weniso  6561  smoiso  7407  supisolem  8326  ordiso2  8367  cantnflt  8516  cantnfp1lem3  8524  cantnflem1b  8530  cantnflem1  8533  wemapwe  8541  cnfcomlem  8543  cnfcom  8544  cnfcom3lem  8547  fpwwe2lem6  9404  fpwwe2lem7  9405  fpwwe2lem9  9407  leisorel  13185  seqcoll  13189  seqcoll2  13190  isercoll  14335  ordthmeolem  21517  iccpnfhmeo  22657  xrhmeo  22658  dvcnvrelem1  23691  dvcvx  23694  isoun  29334  erdszelem8  30909  erdsze2lem2  30915  fourierdlem20  39667  fourierdlem46  39692  fourierdlem50  39696  fourierdlem63  39709  fourierdlem64  39710  fourierdlem65  39711  fourierdlem76  39722  fourierdlem79  39725  fourierdlem102  39748  fourierdlem103  39749  fourierdlem104  39750  fourierdlem114  39760
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