Theorem isorel 5490

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 ↔ (H ‘C)S(H ‘D)))

Proof of Theorem isorel

Dummy variables x y are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-iso 4797	. . 3 ⊢ (H Isom R, S (A, B) ↔ (H:A–1-1-onto→B ∧ ∀x ∈ A ∀y ∈ A (xRy ↔ (H ‘x)S(H ‘y))))
2	1	simprbi 450	. 2 ⊢ (H Isom R, S (A, B) → ∀x ∈ A ∀y ∈ A (xRy ↔ (H ‘x)S(H ‘y)))
3		breq1 4643	. . . 4 ⊢ (x = C → (xRy ↔ CRy))
4		fveq2 5329	. . . . 5 ⊢ (x = C → (H ‘x) = (H ‘C))
5	4	breq1d 4650	. . . 4 ⊢ (x = C → ((H ‘x)S(H ‘y) ↔ (H ‘C)S(H ‘y)))
6	3, 5	bibi12d 312	. . 3 ⊢ (x = C → ((xRy ↔ (H ‘x)S(H ‘y)) ↔ (CRy ↔ (H ‘C)S(H ‘y))))
7		breq2 4644	. . . 4 ⊢ (y = D → (CRy ↔ CRD))
8		fveq2 5329	. . . . 5 ⊢ (y = D → (H ‘y) = (H ‘D))
9	8	breq2d 4652	. . . 4 ⊢ (y = D → ((H ‘C)S(H ‘y) ↔ (H ‘C)S(H ‘D)))
10	7, 9	bibi12d 312	. . 3 ⊢ (y = D → ((CRy ↔ (H ‘C)S(H ‘y)) ↔ (CRD ↔ (H ‘C)S(H ‘D))))
11	6, 10	rspc2v 2962	. 2 ⊢ ((C ∈ A ∧ D ∈ A) → (∀x ∈ A ∀y ∈ A (xRy ↔ (H ‘x)S(H ‘y)) → (CRD ↔ (H ‘C)S(H ‘D))))
12	2, 11	mpan9 455	1 ⊢ ((H Isom R, S (A, B) ∧ (C ∈ A ∧ D ∈ A)) → (CRD ↔ (H ‘C)S(H ‘D)))

Syntax hints:   → wi 4   ↔ wb 176   ∧ wa 358   = wceq 1642   ∈ wcel 1710  ∀wral 2615   class class class wbr 4640  –1-1-onto→wf1o 4781   ‘cfv 4782   Isom wiso 4783

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 4079  ax-xp 4080  ax-cnv 4081  ax-1c 4082  ax-sset 4083  ax-si 4084  ax-ins2 4085  ax-ins3 4086  ax-typlower 4087  ax-sn 4088

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 2479  df-ne 2519  df-ral 2620  df-rex 2621  df-v 2862  df-sbc 3048  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-symdif 3217  df-ss 3260  df-nul 3552  df-if 3664  df-pw 3725  df-sn 3742  df-pr 3743  df-uni 3893  df-int 3928  df-opk 4059  df-1c 4137  df-pw1 4138  df-uni1 4139  df-xpk 4186  df-cnvk 4187  df-ins2k 4188  df-ins3k 4189  df-imak 4190  df-cok 4191  df-p6 4192  df-sik 4193  df-ssetk 4194  df-imagek 4195  df-idk 4196  df-iota 4340  df-addc 4379  df-nnc 4380  df-phi 4566  df-op 4567  df-br 4641  df-fv 4796  df-iso 4797

This theorem is referenced by:  isomin  5497  isoini  5498