Theorem isoeq1 5384

Description: Equality theorem for isomorphisms. (Contributed by NM, 17-May-2004.)

Assertion

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

Proof of Theorem isoeq1

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

Step	Hyp	Ref	Expression
1		f1oeq1 5060	. . 3 ⊢ (𝐻 = 𝐺 → (𝐻:A–1-1-onto→B ↔ 𝐺:A–1-1-onto→B))
2		fveq1 5120	. . . . . 6 ⊢ (𝐻 = 𝐺 → (𝐻‘x) = (𝐺‘x))
3		fveq1 5120	. . . . . 6 ⊢ (𝐻 = 𝐺 → (𝐻‘y) = (𝐺‘y))
4	2, 3	breq12d 3768	. . . . 5 ⊢ (𝐻 = 𝐺 → ((𝐻‘x)𝑆(𝐻‘y) ↔ (𝐺‘x)𝑆(𝐺‘y)))
5	4	bibi2d 221	. . . 4 ⊢ (𝐻 = 𝐺 → ((x𝑅y ↔ (𝐻‘x)𝑆(𝐻‘y)) ↔ (x𝑅y ↔ (𝐺‘x)𝑆(𝐺‘y))))
6	5	2ralbidv 2342	. . 3 ⊢ (𝐻 = 𝐺 → (∀x ∈ A ∀y ∈ A (x𝑅y ↔ (𝐻‘x)𝑆(𝐻‘y)) ↔ ∀x ∈ A ∀y ∈ A (x𝑅y ↔ (𝐺‘x)𝑆(𝐺‘y))))
7	1, 6	anbi12d 442	. 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)))))
8		df-isom 4854	. 2 ⊢ (𝐻 Isom 𝑅, 𝑆 (A, B) ↔ (𝐻:A–1-1-onto→B ∧ ∀x ∈ A ∀y ∈ A (x𝑅y ↔ (𝐻‘x)𝑆(𝐻‘y))))
9		df-isom 4854	. 2 ⊢ (𝐺 Isom 𝑅, 𝑆 (A, B) ↔ (𝐺:A–1-1-onto→B ∧ ∀x ∈ A ∀y ∈ A (x𝑅y ↔ (𝐺‘x)𝑆(𝐺‘y))))
10	7, 8, 9	3bitr4g 212	1 ⊢ (𝐻 = 𝐺 → (𝐻 Isom 𝑅, 𝑆 (A, B) ↔ 𝐺 Isom 𝑅, 𝑆 (A, B)))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   = wceq 1242  ∀wral 2300   class class class wbr 3755  –1-1-onto→wf1o 4844  ‘cfv 4845   Isom wiso 4846

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-isom 4854

This theorem is referenced by:  isores1  5397