Theorem isoeq1 2925

Description: Equality theorem for isomorphisms.

Assertion

Ref	Expression
isoeq1	⊢ (H = G → (H Isom R, S (A, B) ↔ G Isom R, S (A, B)))

Proof of Theorem isoeq1

Step	Hyp	Ref	Expression
1		f1oeq1 2795	. . 3 ⊢ (H = G → (H:A–1-1-onto→B ↔ G:A–1-1-onto→B))
2		fveq1 2831	. . . . . . 7 ⊢ (H = G → (H ‘x) = (G ‘x))
3		fveq1 2831	. . . . . . 7 ⊢ (H = G → (H ‘y) = (G ‘y))
4	2, 3	breq12d 2073	. . . . . 6 ⊢ (H = G → ((H ‘x)S(H ‘y) ↔ (G ‘x)S(G ‘y)))
5	4	bibi2d 470	. . . . 5 ⊢ (H = G → ((xRy ↔ (H ‘x)S(H ‘y)) ↔ (xRy ↔ (G ‘x)S(G ‘y))))
6	5	biraldv 1219	. . . 4 ⊢ (H = G → (∀y ∈ A (xRy ↔ (H ‘x)S(H ‘y)) ↔ ∀y ∈ A (xRy ↔ (G ‘x)S(G ‘y))))
7	6	biraldv 1219	. . 3 ⊢ (H = G → (∀x ∈ A ∀y ∈ A (xRy ↔ (H ‘x)S(H ‘y)) ↔ ∀x ∈ A ∀y ∈ A (xRy ↔ (G ‘x)S(G ‘y))))
8	1, 7	anbi12d 476	. 2 ⊢ (H = G → ((H:A–1-1-onto→B ∧ ∀x ∈ A ∀y ∈ A (xRy ↔ (H ‘x)S(H ‘y))) ↔ (G:A–1-1-onto→B ∧ ∀x ∈ A ∀y ∈ A (xRy ↔ (G ‘x)S(G ‘y)))))
9		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))))
10		df-iso 2439	. 2 ⊢ (G Isom R, S (A, B) ↔ (G:A–1-1-onto→B ∧ ∀x ∈ A ∀y ∈ A (xRy ↔ (G ‘x)S(G ‘y))))
11	8, 9, 10	3bitr4g 428	1 ⊢ (H = G → (H Isom R, S (A, B) ↔ G Isom R, S (A, 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-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-eu 1009  df-mo 1010  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-id 2125  df-rel 2425  df-cnv 2426  df-co 2427  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fun 2432  df-fn 2433  df-f 2434  df-f1 2435  df-fo 2436  df-f1o 2437  df-fv 2438  df-iso 2439

metamath.org