Theorem isores3 7206

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 6715	. . . . . . 7 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → 𝐻:𝐴–1-1→𝐵)
2		f1ores 6730	. . . . . . . 8 ⊢ ((𝐻:𝐴–1-1→𝐵 ∧ 𝐾 ⊆ 𝐴) → (𝐻 ↾ 𝐾):𝐾–1-1-onto→(𝐻 “ 𝐾))
3	2	expcom 414	. . . . . . 7 ⊢ (𝐾 ⊆ 𝐴 → (𝐻:𝐴–1-1→𝐵 → (𝐻 ↾ 𝐾):𝐾–1-1-onto→(𝐻 “ 𝐾)))
4	1, 3	syl5 34	. . . . . 6 ⊢ (𝐾 ⊆ 𝐴 → (𝐻:𝐴–1-1-onto→𝐵 → (𝐻 ↾ 𝐾):𝐾–1-1-onto→(𝐻 “ 𝐾)))
5		ssralv 3987	. . . . . . 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 3987	. . . . . . . . . 10 ⊢ (𝐾 ⊆ 𝐴 → (∀𝑏 ∈ 𝐴 (𝑎𝑅𝑏 ↔ (𝐻‘𝑎)𝑆(𝐻‘𝑏)) → ∀𝑏 ∈ 𝐾 (𝑎𝑅𝑏 ↔ (𝐻‘𝑎)𝑆(𝐻‘𝑏))))
7	6	adantr 481	. . . . . . . . 9 ⊢ ((𝐾 ⊆ 𝐴 ∧ 𝑎 ∈ 𝐾) → (∀𝑏 ∈ 𝐴 (𝑎𝑅𝑏 ↔ (𝐻‘𝑎)𝑆(𝐻‘𝑏)) → ∀𝑏 ∈ 𝐾 (𝑎𝑅𝑏 ↔ (𝐻‘𝑎)𝑆(𝐻‘𝑏))))
8		fvres 6793	. . . . . . . . . . . . . 14 ⊢ (𝑎 ∈ 𝐾 → ((𝐻 ↾ 𝐾)‘𝑎) = (𝐻‘𝑎))
9		fvres 6793	. . . . . . . . . . . . . 14 ⊢ (𝑏 ∈ 𝐾 → ((𝐻 ↾ 𝐾)‘𝑏) = (𝐻‘𝑏))
10	8, 9	breqan12d 5090	. . . . . . . . . . . . 13 ⊢ ((𝑎 ∈ 𝐾 ∧ 𝑏 ∈ 𝐾) → (((𝐻 ↾ 𝐾)‘𝑎)𝑆((𝐻 ↾ 𝐾)‘𝑏) ↔ (𝐻‘𝑎)𝑆(𝐻‘𝑏)))
11	10	adantll 711	. . . . . . . . . . . 12 ⊢ (((𝐾 ⊆ 𝐴 ∧ 𝑎 ∈ 𝐾) ∧ 𝑏 ∈ 𝐾) → (((𝐻 ↾ 𝐾)‘𝑎)𝑆((𝐻 ↾ 𝐾)‘𝑏) ↔ (𝐻‘𝑎)𝑆(𝐻‘𝑏)))
12	11	bibi2d 343	. . . . . . . . . . 11 ⊢ (((𝐾 ⊆ 𝐴 ∧ 𝑎 ∈ 𝐾) ∧ 𝑏 ∈ 𝐾) → ((𝑎𝑅𝑏 ↔ ((𝐻 ↾ 𝐾)‘𝑎)𝑆((𝐻 ↾ 𝐾)‘𝑏)) ↔ (𝑎𝑅𝑏 ↔ (𝐻‘𝑎)𝑆(𝐻‘𝑏))))
13	12	biimprd 247	. . . . . . . . . 10 ⊢ (((𝐾 ⊆ 𝐴 ∧ 𝑎 ∈ 𝐾) ∧ 𝑏 ∈ 𝐾) → ((𝑎𝑅𝑏 ↔ (𝐻‘𝑎)𝑆(𝐻‘𝑏)) → (𝑎𝑅𝑏 ↔ ((𝐻 ↾ 𝐾)‘𝑎)𝑆((𝐻 ↾ 𝐾)‘𝑏))))
14	13	ralimdva 3108	. . . . . . . . 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 47	. . . . . . . 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 3108	. . . . . . 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 47	. . . . . 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 609	. . . . 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 6442	. . . . 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 6442	. . . . 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 296	. . . 4 ⊢ (𝐾 ⊆ 𝐴 → (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝐻 ↾ 𝐾) Isom 𝑅, 𝑆 (𝐾, (𝐻 “ 𝐾))))
22	21	impcom 408	. . 3 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐾 ⊆ 𝐴) → (𝐻 ↾ 𝐾) Isom 𝑅, 𝑆 (𝐾, (𝐻 “ 𝐾)))
23		isoeq5 7192	. . 3 ⊢ (𝑋 = (𝐻 “ 𝐾) → ((𝐻 ↾ 𝐾) Isom 𝑅, 𝑆 (𝐾, 𝑋) ↔ (𝐻 ↾ 𝐾) Isom 𝑅, 𝑆 (𝐾, (𝐻 “ 𝐾))))
24	22, 23	syl5ibrcom 246	. 2 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐾 ⊆ 𝐴) → (𝑋 = (𝐻 “ 𝐾) → (𝐻 ↾ 𝐾) Isom 𝑅, 𝑆 (𝐾, 𝑋)))
25	24	3impia 1116	1 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝐾 ⊆ 𝐴 ∧ 𝑋 = (𝐻 “ 𝐾)) → (𝐻 ↾ 𝐾) Isom 𝑅, 𝑆 (𝐾, 𝑋))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  ∀wral 3064   ⊆ wss 3887   class class class wbr 5074   ↾ cres 5591   “ cima 5592  –1-1→wf1 6430  –1-1-onto→wf1o 6432  ‘cfv 6433   Isom wiso 6434

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-br 5075  df-opab 5137  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-isom 6442

This theorem is referenced by:  cantnfp1lem3  9438  fpwwe2lem8  10394  efcvx  25608