Theorem isoeq4 2928

Description: Equality theorem for isomorphisms.

Assertion

Ref	Expression
isoeq4	⊢ (A = C → (H Isom R, S (A, B) ↔ H Isom R, S (C, B)))

Proof of Theorem isoeq4

Step	Hyp	Ref	Expression
1		f1oeq2 2796	. . 3 ⊢ (A = C → (H:A–1-1-onto→B ↔ H:C–1-1-onto→B))
2		raleq 1324	. . . 4 ⊢ (A = C → (∀y ∈ A (xRy ↔ (H ‘x)S(H ‘y)) ↔ ∀y ∈ C (xRy ↔ (H ‘x)S(H ‘y))))
3	2	raleqd 1327	. . 3 ⊢ (A = C → (∀x ∈ A ∀y ∈ A (xRy ↔ (H ‘x)S(H ‘y)) ↔ ∀x ∈ C ∀y ∈ C (xRy ↔ (H ‘x)S(H ‘y))))
4	1, 3	anbi12d 476	. 2 ⊢ (A = C → ((H:A–1-1-onto→B ∧ ∀x ∈ A ∀y ∈ A (xRy ↔ (H ‘x)S(H ‘y))) ↔ (H:C–1-1-onto→B ∧ ∀x ∈ C ∀y ∈ C (xRy ↔ (H ‘x)S(H ‘y)))))
5		df-iso 2439	. 2 ⊢ (H Isom R, S (A, B) ↔ (H:A–1-1-onto→B ∧ ∀x ∈ A ∀y ∈ A (xRy ↔ (H ‘x)S(H ‘y))))
6		df-iso 2439	. 2 ⊢ (H Isom R, S (C, B) ↔ (H:C–1-1-onto→B ∧ ∀x ∈ C ∀y ∈ C (xRy ↔ (H ‘x)S(H ‘y))))
7	4, 5, 6	3bitr4g 428	1 ⊢ (A = C → (H Isom R, S (A, B) ↔ H Isom R, S (C, B)))

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

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-9 799  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-cleq 1097  df-clel 1099  df-ral 1205  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-iso 2439

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