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Theorem isorel 5391
 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) (𝐶 A 𝐷 A)) → (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 4854 . . 3 (𝐻 Isom 𝑅, 𝑆 (A, B) ↔ (𝐻:A1-1-ontoB x A y A (x𝑅y ↔ (𝐻x)𝑆(𝐻y))))
21simprbi 260 . 2 (𝐻 Isom 𝑅, 𝑆 (A, B) → x A y A (x𝑅y ↔ (𝐻x)𝑆(𝐻y)))
3 breq1 3758 . . . 4 (x = 𝐶 → (x𝑅y𝐶𝑅y))
4 fveq2 5121 . . . . 5 (x = 𝐶 → (𝐻x) = (𝐻𝐶))
54breq1d 3765 . . . 4 (x = 𝐶 → ((𝐻x)𝑆(𝐻y) ↔ (𝐻𝐶)𝑆(𝐻y)))
63, 5bibi12d 224 . . 3 (x = 𝐶 → ((x𝑅y ↔ (𝐻x)𝑆(𝐻y)) ↔ (𝐶𝑅y ↔ (𝐻𝐶)𝑆(𝐻y))))
7 breq2 3759 . . . 4 (y = 𝐷 → (𝐶𝑅y𝐶𝑅𝐷))
8 fveq2 5121 . . . . 5 (y = 𝐷 → (𝐻y) = (𝐻𝐷))
98breq2d 3767 . . . 4 (y = 𝐷 → ((𝐻𝐶)𝑆(𝐻y) ↔ (𝐻𝐶)𝑆(𝐻𝐷)))
107, 9bibi12d 224 . . 3 (y = 𝐷 → ((𝐶𝑅y ↔ (𝐻𝐶)𝑆(𝐻y)) ↔ (𝐶𝑅𝐷 ↔ (𝐻𝐶)𝑆(𝐻𝐷))))
116, 10rspc2v 2656 . 2 ((𝐶 A 𝐷 A) → (x A y A (x𝑅y ↔ (𝐻x)𝑆(𝐻y)) → (𝐶𝑅𝐷 ↔ (𝐻𝐶)𝑆(𝐻𝐷))))
122, 11mpan9 265 1 ((𝐻 Isom 𝑅, 𝑆 (A, B) (𝐶 A 𝐷 A)) → (𝐶𝑅𝐷 ↔ (𝐻𝐶)𝑆(𝐻𝐷)))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242   ∈ wcel 1390  ∀wral 2300   class class class wbr 3755  –1-1-onto→wf1o 4844  ‘cfv 4845   Isom wiso 4846 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-iota 4810  df-fv 4853  df-isom 4854 This theorem is referenced by:  isoresbr  5392  isoini  5400  isopolem  5404  isosolem  5406  smoiso  5858
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