Theorem isores3 5398

Description: Induced isomorphism on a subset. (Contributed by Stefan O'Rear, 5-Nov-2014.)

Assertion

Ref	Expression
isores3	⊢ ((𝐻 Isom 𝑅, 𝑆 (A, B) ∧ 𝐾 ⊆ A ∧ 𝑋 = (𝐻 “ 𝐾)) → (𝐻 ↾ 𝐾) Isom 𝑅, 𝑆 (𝐾, 𝑋))

Proof of Theorem isores3

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		f1of1 5068	. . . . . . 7 ⊢ (𝐻:A–1-1-onto→B → 𝐻:A–1-1→B)
2		f1ores 5084	. . . . . . . 8 ⊢ ((𝐻:A–1-1→B ∧ 𝐾 ⊆ A) → (𝐻 ↾ 𝐾):𝐾–1-1-onto→(𝐻 “ 𝐾))
3	2	expcom 109	. . . . . . 7 ⊢ (𝐾 ⊆ A → (𝐻:A–1-1→B → (𝐻 ↾ 𝐾):𝐾–1-1-onto→(𝐻 “ 𝐾)))
4	1, 3	syl5 28	. . . . . 6 ⊢ (𝐾 ⊆ A → (𝐻:A–1-1-onto→B → (𝐻 ↾ 𝐾):𝐾–1-1-onto→(𝐻 “ 𝐾)))
5		ssralv 2998	. . . . . . 7 ⊢ (𝐾 ⊆ A → (∀𝑎 ∈ A ∀𝑏 ∈ A (𝑎𝑅𝑏 ↔ (𝐻‘𝑎)𝑆(𝐻‘𝑏)) → ∀𝑎 ∈ 𝐾 ∀𝑏 ∈ A (𝑎𝑅𝑏 ↔ (𝐻‘𝑎)𝑆(𝐻‘𝑏))))
6		ssralv 2998	. . . . . . . . . 10 ⊢ (𝐾 ⊆ A → (∀𝑏 ∈ A (𝑎𝑅𝑏 ↔ (𝐻‘𝑎)𝑆(𝐻‘𝑏)) → ∀𝑏 ∈ 𝐾 (𝑎𝑅𝑏 ↔ (𝐻‘𝑎)𝑆(𝐻‘𝑏))))
7	6	adantr 261	. . . . . . . . 9 ⊢ ((𝐾 ⊆ A ∧ 𝑎 ∈ 𝐾) → (∀𝑏 ∈ A (𝑎𝑅𝑏 ↔ (𝐻‘𝑎)𝑆(𝐻‘𝑏)) → ∀𝑏 ∈ 𝐾 (𝑎𝑅𝑏 ↔ (𝐻‘𝑎)𝑆(𝐻‘𝑏))))
8		fvres 5141	. . . . . . . . . . . . . 14 ⊢ (𝑎 ∈ 𝐾 → ((𝐻 ↾ 𝐾)‘𝑎) = (𝐻‘𝑎))
9		fvres 5141	. . . . . . . . . . . . . 14 ⊢ (𝑏 ∈ 𝐾 → ((𝐻 ↾ 𝐾)‘𝑏) = (𝐻‘𝑏))
10	8, 9	breqan12d 3770	. . . . . . . . . . . . 13 ⊢ ((𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝐾) → (((𝐻 ↾ 𝐾)‘𝑎)𝑆((𝐻 ↾ 𝐾)‘𝑏) ↔ (𝐻‘𝑎)𝑆(𝐻‘𝑏)))
11	10	adantll 445	. . . . . . . . . . . 12 ⊢ (((𝐾 ⊆ A ∧ 𝑎 ∈ 𝐾) ∧ 𝑏 ∈ 𝐾) → (((𝐻 ↾ 𝐾)‘𝑎)𝑆((𝐻 ↾ 𝐾)‘𝑏) ↔ (𝐻‘𝑎)𝑆(𝐻‘𝑏)))
12	11	bibi2d 221	. . . . . . . . . . 11 ⊢ (((𝐾 ⊆ A ∧ 𝑎 ∈ 𝐾) ∧ 𝑏 ∈ 𝐾) → ((𝑎𝑅𝑏 ↔ ((𝐻 ↾ 𝐾)‘𝑎)𝑆((𝐻 ↾ 𝐾)‘𝑏)) ↔ (𝑎𝑅𝑏 ↔ (𝐻‘𝑎)𝑆(𝐻‘𝑏))))
13	12	biimprd 147	. . . . . . . . . 10 ⊢ (((𝐾 ⊆ A ∧ 𝑎 ∈ 𝐾) ∧ 𝑏 ∈ 𝐾) → ((𝑎𝑅𝑏 ↔ (𝐻‘𝑎)𝑆(𝐻‘𝑏)) → (𝑎𝑅𝑏 ↔ ((𝐻 ↾ 𝐾)‘𝑎)𝑆((𝐻 ↾ 𝐾)‘𝑏))))
14	13	ralimdva 2381	. . . . . . . . 9 ⊢ ((𝐾 ⊆ A ∧ 𝑎 ∈ 𝐾) → (∀𝑏 ∈ 𝐾 (𝑎𝑅𝑏 ↔ (𝐻‘𝑎)𝑆(𝐻‘𝑏)) → ∀𝑏 ∈ 𝐾 (𝑎𝑅𝑏 ↔ ((𝐻 ↾ 𝐾)‘𝑎)𝑆((𝐻 ↾ 𝐾)‘𝑏))))
15	7, 14	syld 40	. . . . . . . 8 ⊢ ((𝐾 ⊆ A ∧ 𝑎 ∈ 𝐾) → (∀𝑏 ∈ A (𝑎𝑅𝑏 ↔ (𝐻‘𝑎)𝑆(𝐻‘𝑏)) → ∀𝑏 ∈ 𝐾 (𝑎𝑅𝑏 ↔ ((𝐻 ↾ 𝐾)‘𝑎)𝑆((𝐻 ↾ 𝐾)‘𝑏))))
16	15	ralimdva 2381	. . . . . . 7 ⊢ (𝐾 ⊆ A → (∀𝑎 ∈ 𝐾 ∀𝑏 ∈ A (𝑎𝑅𝑏 ↔ (𝐻‘𝑎)𝑆(𝐻‘𝑏)) → ∀𝑎 ∈ 𝐾 ∀𝑏 ∈ 𝐾 (𝑎𝑅𝑏 ↔ ((𝐻 ↾ 𝐾)‘𝑎)𝑆((𝐻 ↾ 𝐾)‘𝑏))))
17	5, 16	syld 40	. . . . . 6 ⊢ (𝐾 ⊆ A → (∀𝑎 ∈ A ∀𝑏 ∈ A (𝑎𝑅𝑏 ↔ (𝐻‘𝑎)𝑆(𝐻‘𝑏)) → ∀𝑎 ∈ 𝐾 ∀𝑏 ∈ 𝐾 (𝑎𝑅𝑏 ↔ ((𝐻 ↾ 𝐾)‘𝑎)𝑆((𝐻 ↾ 𝐾)‘𝑏))))
18	4, 17	anim12d 318	. . . . 5 ⊢ (𝐾 ⊆ A → ((𝐻:A–1-1-onto→B ∧ ∀𝑎 ∈ A ∀𝑏 ∈ A (𝑎𝑅𝑏 ↔ (𝐻‘𝑎)𝑆(𝐻‘𝑏))) → ((𝐻 ↾ 𝐾):𝐾–1-1-onto→(𝐻 “ 𝐾) ∧ ∀𝑎 ∈ 𝐾 ∀𝑏 ∈ 𝐾 (𝑎𝑅𝑏 ↔ ((𝐻 ↾ 𝐾)‘𝑎)𝑆((𝐻 ↾ 𝐾)‘𝑏)))))
19		df-isom 4854	. . . . 5 ⊢ (𝐻 Isom 𝑅, 𝑆 (A, B) ↔ (𝐻:A–1-1-onto→B ∧ ∀𝑎 ∈ A ∀𝑏 ∈ A (𝑎𝑅𝑏 ↔ (𝐻‘𝑎)𝑆(𝐻‘𝑏))))
20		df-isom 4854	. . . . 5 ⊢ ((𝐻 ↾ 𝐾) Isom 𝑅, 𝑆 (𝐾, (𝐻 “ 𝐾)) ↔ ((𝐻 ↾ 𝐾):𝐾–1-1-onto→(𝐻 “ 𝐾) ∧ ∀𝑎 ∈ 𝐾 ∀𝑏 ∈ 𝐾 (𝑎𝑅𝑏 ↔ ((𝐻 ↾ 𝐾)‘𝑎)𝑆((𝐻 ↾ 𝐾)‘𝑏))))
21	18, 19, 20	3imtr4g 194	. . . 4 ⊢ (𝐾 ⊆ A → (𝐻 Isom 𝑅, 𝑆 (A, B) → (𝐻 ↾ 𝐾) Isom 𝑅, 𝑆 (𝐾, (𝐻 “ 𝐾))))
22	21	impcom 116	. . 3 ⊢ ((𝐻 Isom 𝑅, 𝑆 (A, B) ∧ 𝐾 ⊆ A) → (𝐻 ↾ 𝐾) Isom 𝑅, 𝑆 (𝐾, (𝐻 “ 𝐾)))
23		isoeq5 5388	. . 3 ⊢ (𝑋 = (𝐻 “ 𝐾) → ((𝐻 ↾ 𝐾) Isom 𝑅, 𝑆 (𝐾, 𝑋) ↔ (𝐻 ↾ 𝐾) Isom 𝑅, 𝑆 (𝐾, (𝐻 “ 𝐾))))
24	22, 23	syl5ibrcom 146	. 2 ⊢ ((𝐻 Isom 𝑅, 𝑆 (A, B) ∧ 𝐾 ⊆ A) → (𝑋 = (𝐻 “ 𝐾) → (𝐻 ↾ 𝐾) Isom 𝑅, 𝑆 (𝐾, 𝑋)))
25	24	3impia 1100	1 ⊢ ((𝐻 Isom 𝑅, 𝑆 (A, B) ∧ 𝐾 ⊆ A ∧ 𝑋 = (𝐻 “ 𝐾)) → (𝐻 ↾ 𝐾) Isom 𝑅, 𝑆 (𝐾, 𝑋))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∧ w3a 884   = wceq 1242   ∈ wcel 1390  ∀wral 2300   ⊆ wss 2911   class class class wbr 3755   ↾ cres 4290   “ cima 4291  –1-1→wf1 4842  –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-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

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-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  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: (None)