Theorem isores3 5938

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 5570	. . . . . . 7 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → 𝐻:𝐴–1-1→𝐵)
2		f1ores 5586	. . . . . . . 8 ⊢ ((𝐻:𝐴–1-1→𝐵 ∧ 𝐾 ⊆ 𝐴) → (𝐻 ↾ 𝐾):𝐾–1-1-onto→(𝐻 “ 𝐾))
3	2	expcom 116	. . . . . . 7 ⊢ (𝐾 ⊆ 𝐴 → (𝐻:𝐴–1-1→𝐵 → (𝐻 ↾ 𝐾):𝐾–1-1-onto→(𝐻 “ 𝐾)))
4	1, 3	syl5 32	. . . . . 6 ⊢ (𝐾 ⊆ 𝐴 → (𝐻:𝐴–1-1-onto→𝐵 → (𝐻 ↾ 𝐾):𝐾–1-1-onto→(𝐻 “ 𝐾)))
5		ssralv 3288	. . . . . . 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 3288	. . . . . . . . . 10 ⊢ (𝐾 ⊆ 𝐴 → (∀𝑏 ∈ 𝐴 (𝑎𝑅𝑏 ↔ (𝐻‘𝑎)𝑆(𝐻‘𝑏)) → ∀𝑏 ∈ 𝐾 (𝑎𝑅𝑏 ↔ (𝐻‘𝑎)𝑆(𝐻‘𝑏))))
7	6	adantr 276	. . . . . . . . 9 ⊢ ((𝐾 ⊆ 𝐴 ∧ 𝑎 ∈ 𝐾) → (∀𝑏 ∈ 𝐴 (𝑎𝑅𝑏 ↔ (𝐻‘𝑎)𝑆(𝐻‘𝑏)) → ∀𝑏 ∈ 𝐾 (𝑎𝑅𝑏 ↔ (𝐻‘𝑎)𝑆(𝐻‘𝑏))))
8		fvres 5650	. . . . . . . . . . . . . 14 ⊢ (𝑎 ∈ 𝐾 → ((𝐻 ↾ 𝐾)‘𝑎) = (𝐻‘𝑎))
9		fvres 5650	. . . . . . . . . . . . . 14 ⊢ (𝑏 ∈ 𝐾 → ((𝐻 ↾ 𝐾)‘𝑏) = (𝐻‘𝑏))
10	8, 9	breqan12d 4098	. . . . . . . . . . . . 13 ⊢ ((𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝐾) → (((𝐻 ↾ 𝐾)‘𝑎)𝑆((𝐻 ↾ 𝐾)‘𝑏) ↔ (𝐻‘𝑎)𝑆(𝐻‘𝑏)))
11	10	adantll 476	. . . . . . . . . . . 12 ⊢ (((𝐾 ⊆ 𝐴 ∧ 𝑎 ∈ 𝐾) ∧ 𝑏 ∈ 𝐾) → (((𝐻 ↾ 𝐾)‘𝑎)𝑆((𝐻 ↾ 𝐾)‘𝑏) ↔ (𝐻‘𝑎)𝑆(𝐻‘𝑏)))
12	11	bibi2d 232	. . . . . . . . . . 11 ⊢ (((𝐾 ⊆ 𝐴 ∧ 𝑎 ∈ 𝐾) ∧ 𝑏 ∈ 𝐾) → ((𝑎𝑅𝑏 ↔ ((𝐻 ↾ 𝐾)‘𝑎)𝑆((𝐻 ↾ 𝐾)‘𝑏)) ↔ (𝑎𝑅𝑏 ↔ (𝐻‘𝑎)𝑆(𝐻‘𝑏))))
13	12	biimprd 158	. . . . . . . . . 10 ⊢ (((𝐾 ⊆ 𝐴 ∧ 𝑎 ∈ 𝐾) ∧ 𝑏 ∈ 𝐾) → ((𝑎𝑅𝑏 ↔ (𝐻‘𝑎)𝑆(𝐻‘𝑏)) → (𝑎𝑅𝑏 ↔ ((𝐻 ↾ 𝐾)‘𝑎)𝑆((𝐻 ↾ 𝐾)‘𝑏))))
14	13	ralimdva 2597	. . . . . . . . 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 45	. . . . . . . 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 2597	. . . . . . 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 45	. . . . . 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 335	. . . . 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 5326	. . . . 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 5326	. . . . 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 205	. . . 4 ⊢ (𝐾 ⊆ 𝐴 → (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝐻 ↾ 𝐾) Isom 𝑅, 𝑆 (𝐾, (𝐻 “ 𝐾))))
22	21	impcom 125	. . 3 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐾 ⊆ 𝐴) → (𝐻 ↾ 𝐾) Isom 𝑅, 𝑆 (𝐾, (𝐻 “ 𝐾)))
23		isoeq5 5928	. . 3 ⊢ (𝑋 = (𝐻 “ 𝐾) → ((𝐻 ↾ 𝐾) Isom 𝑅, 𝑆 (𝐾, 𝑋) ↔ (𝐻 ↾ 𝐾) Isom 𝑅, 𝑆 (𝐾, (𝐻 “ 𝐾))))
24	22, 23	syl5ibrcom 157	. 2 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐾 ⊆ 𝐴) → (𝑋 = (𝐻 “ 𝐾) → (𝐻 ↾ 𝐾) Isom 𝑅, 𝑆 (𝐾, 𝑋)))
25	24	3impia 1224	1 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐾 ⊆ 𝐴 ∧ 𝑋 = (𝐻 “ 𝐾)) → (𝐻 ↾ 𝐾) Isom 𝑅, 𝑆 (𝐾, 𝑋))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   ∧ w3a 1002   = wceq 1395   ∈ wcel 2200  ∀wral 2508   ⊆ wss 3197   class class class wbr 4082   ↾ cres 4720   “ cima 4721  –1-1→wf1 5314  –1-1-onto→wf1o 5316  ‘cfv 5317   Isom wiso 5318

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4201  ax-pow 4257  ax-pr 4292

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-v 2801  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-br 4083  df-opab 4145  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-isom 5326

This theorem is referenced by: (None)