Theorem isorel 2932

Description: An isomorphism connects binary relations via its function values.

Assertion

Ref	Expression
isorel	⊢ ((H Isom R, S (A, B) ∧ (C ∈ A ∧ D ∈ A)) → (CRD ↔ (H ‘C)S(H ‘D)))

Proof of Theorem isorel

Step	Hyp	Ref	Expression
1		df-iso 2439	. . . 4 ⊢ (H Isom R, S (A, B) ↔ (H:A–1-1-onto→B ∧ ∀x ∈ A ∀y ∈ A (xRy ↔ (H ‘x)S(H ‘y))))
2	1	pm3.27bd 263	. . 3 ⊢ (H Isom R, S (A, B) → ∀x ∈ A ∀y ∈ A (xRy ↔ (H ‘x)S(H ‘y)))
3		breq1 2065	. . . . 5 ⊢ (x = C → (xRy ↔ CRy))
4		fveq2 2832	. . . . . 6 ⊢ (x = C → (H ‘x) = (H ‘C))
5	4	breq1d 2071	. . . . 5 ⊢ (x = C → ((H ‘x)S(H ‘y) ↔ (H ‘C)S(H ‘y)))
6	3, 5	bibi12d 477	. . . 4 ⊢ (x = C → ((xRy ↔ (H ‘x)S(H ‘y)) ↔ (CRy ↔ (H ‘C)S(H ‘y))))
7		breq2 2066	. . . . 5 ⊢ (y = D → (CRy ↔ CRD))
8		fveq2 2832	. . . . . 6 ⊢ (y = D → (H ‘y) = (H ‘D))
9	8	breq2d 2072	. . . . 5 ⊢ (y = D → ((H ‘C)S(H ‘y) ↔ (H ‘C)S(H ‘D)))
10	7, 9	bibi12d 477	. . . 4 ⊢ (y = D → ((CRy ↔ (H ‘C)S(H ‘y)) ↔ (CRD ↔ (H ‘C)S(H ‘D))))
11	6, 10	rcla42v 1404	. . 3 ⊢ (∀x ∈ A ∀y ∈ A (xRy ↔ (H ‘x)S(H ‘y)) → ((C ∈ A ∧ D ∈ A) → (CRD ↔ (H ‘C)S(H ‘D))))
12	2, 11	syl 12	. 2 ⊢ (H Isom R, S (A, B) → ((C ∈ A ∧ D ∈ A) → (CRD ↔ (H ‘C)S(H ‘D))))
13	12	imp 277	1 ⊢ ((H Isom R, S (A, B) ∧ (C ∈ A ∧ D ∈ A)) → (CRD ↔ (H ‘C)S(H ‘D)))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196   = wceq 1091   ∈ wcel 1092  ∀wral 1201   class class class wbr 2054  –1-1-onto→wf1o 2421   ‘cfv 2422   Isom wiso 2423

This theorem is referenced by:  isomin 2937  isoini 2938  isowe 2941

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-xp 2424  df-cnv 2426  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fv 2438  df-iso 2439

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