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Theorem isorel 5489
 Description: An isomorphism connects binary relations via its function values. (Contributed by set.mm contributors, 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-iso 4796 . . 3 (H Isom R, S (A, B) ↔ (H:A1-1-ontoB x A y A (xRy ↔ (Hx)S(Hy))))
21simprbi 450 . 2 (H Isom R, S (A, B) → x A y A (xRy ↔ (Hx)S(Hy)))
3 breq1 4642 . . . 4 (x = C → (xRyCRy))
4 fveq2 5328 . . . . 5 (x = C → (Hx) = (HC))
54breq1d 4649 . . . 4 (x = C → ((Hx)S(Hy) ↔ (HC)S(Hy)))
63, 5bibi12d 312 . . 3 (x = C → ((xRy ↔ (Hx)S(Hy)) ↔ (CRy ↔ (HC)S(Hy))))
7 breq2 4643 . . . 4 (y = D → (CRyCRD))
8 fveq2 5328 . . . . 5 (y = D → (Hy) = (HD))
98breq2d 4651 . . . 4 (y = D → ((HC)S(Hy) ↔ (HC)S(HD)))
107, 9bibi12d 312 . . 3 (y = D → ((CRy ↔ (HC)S(Hy)) ↔ (CRD ↔ (HC)S(HD))))
116, 10rspc2v 2961 . 2 ((C A D A) → (x A y A (xRy ↔ (Hx)S(Hy)) → (CRD ↔ (HC)S(HD))))
122, 11mpan9 455 1 ((H Isom R, S (A, B) (C A D A)) → (CRD ↔ (HC)S(HD)))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∀wral 2614   class class class wbr 4639  –1-1-onto→wf1o 4780   ‘cfv 4781   Isom wiso 4782 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334  ax-nin 4078  ax-xp 4079  ax-cnv 4080  ax-1c 4081  ax-sset 4082  ax-si 4083  ax-ins2 4084  ax-ins3 4085  ax-typlower 4086  ax-sn 4087 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ne 2518  df-ral 2619  df-rex 2620  df-v 2861  df-sbc 3047  df-nin 3211  df-compl 3212  df-in 3213  df-un 3214  df-dif 3215  df-symdif 3216  df-ss 3259  df-nul 3551  df-if 3663  df-pw 3724  df-sn 3741  df-pr 3742  df-uni 3892  df-int 3927  df-opk 4058  df-1c 4136  df-pw1 4137  df-uni1 4138  df-xpk 4185  df-cnvk 4186  df-ins2k 4187  df-ins3k 4188  df-imak 4189  df-cok 4190  df-p6 4191  df-sik 4192  df-ssetk 4193  df-imagek 4194  df-idk 4195  df-iota 4339  df-addc 4378  df-nnc 4379  df-phi 4565  df-op 4566  df-br 4640  df-fv 4795  df-iso 4796 This theorem is referenced by:  isomin  5496  isoini  5497
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