Theorem isoeq5 2929

Description: Equality theorem for isomorphisms.

Assertion

Ref	Expression
isoeq5	⊢ (B = C → (H Isom R, S (A, B) ↔ H Isom R, S (A, C)))

Proof of Theorem isoeq5

Step	Hyp	Ref	Expression
1		f1oeq3 2797	. . 3 ⊢ (B = C → (H:A–1-1-onto→B ↔ H:A–1-1-onto→C))
2	1	anbi1d 469	. 2 ⊢ (B = C → ((H:A–1-1-onto→B ∧ ∀x ∈ A ∀y ∈ A (xRy ↔ (H ‘x)S(H ‘y))) ↔ (H:A–1-1-onto→C ∧ ∀x ∈ A ∀y ∈ A (xRy ↔ (H ‘x)S(H ‘y)))))
3		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))))
4		df-iso 2439	. 2 ⊢ (H Isom R, S (A, C) ↔ (H:A–1-1-onto→C ∧ ∀x ∈ A ∀y ∈ A (xRy ↔ (H ‘x)S(H ‘y))))
5	2, 3, 4	3bitr4g 428	1 ⊢ (B = C → (H Isom R, S (A, B) ↔ H Isom R, S (A, C)))

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-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-in 1491  df-ss 1492  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-iso 2439

metamath.org