Theorem isoini2 5970

Description: Isomorphisms are isomorphisms on their initial segments. (Contributed by Mario Carneiro, 29-Mar-2014.)

Hypotheses

Ref	Expression
isoini2.1	⊢ 𝐶 = (𝐴 ∩ (◡𝑅 “ {𝑋}))
isoini2.2	⊢ 𝐷 = (𝐵 ∩ (◡𝑆 “ {(𝐻‘𝑋)}))

Assertion

Ref	Expression
isoini2	⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑋 ∈ 𝐴) → (𝐻 ↾ 𝐶) Isom 𝑅, 𝑆 (𝐶, 𝐷))

Proof of Theorem isoini2

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

Step	Hyp	Ref	Expression
1		isof1o 5958	. . . . . 6 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴–1-1-onto→𝐵)
2		f1of1 5591	. . . . . 6 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → 𝐻:𝐴–1-1→𝐵)
3	1, 2	syl 14	. . . . 5 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴–1-1→𝐵)
4	3	adantr 276	. . . 4 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑋 ∈ 𝐴) → 𝐻:𝐴–1-1→𝐵)
5		isoini2.1	. . . . 5 ⊢ 𝐶 = (𝐴 ∩ (◡𝑅 “ {𝑋}))
6		inss1 3429	. . . . 5 ⊢ (𝐴 ∩ (◡𝑅 “ {𝑋})) ⊆ 𝐴
7	5, 6	eqsstri 3260	. . . 4 ⊢ 𝐶 ⊆ 𝐴
8		f1ores 5607	. . . 4 ⊢ ((𝐻:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐻 ↾ 𝐶):𝐶–1-1-onto→(𝐻 “ 𝐶))
9	4, 7, 8	sylancl 413	. . 3 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑋 ∈ 𝐴) → (𝐻 ↾ 𝐶):𝐶–1-1-onto→(𝐻 “ 𝐶))
10		isoini 5969	. . . . 5 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑋 ∈ 𝐴) → (𝐻 “ (𝐴 ∩ (◡𝑅 “ {𝑋}))) = (𝐵 ∩ (◡𝑆 “ {(𝐻‘𝑋)})))
11	5	imaeq2i 5080	. . . . 5 ⊢ (𝐻 “ 𝐶) = (𝐻 “ (𝐴 ∩ (◡𝑅 “ {𝑋})))
12		isoini2.2	. . . . 5 ⊢ 𝐷 = (𝐵 ∩ (◡𝑆 “ {(𝐻‘𝑋)}))
13	10, 11, 12	3eqtr4g 2289	. . . 4 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑋 ∈ 𝐴) → (𝐻 “ 𝐶) = 𝐷)
14		f1oeq3 5582	. . . 4 ⊢ ((𝐻 “ 𝐶) = 𝐷 → ((𝐻 ↾ 𝐶):𝐶–1-1-onto→(𝐻 “ 𝐶) ↔ (𝐻 ↾ 𝐶):𝐶–1-1-onto→𝐷))
15	13, 14	syl 14	. . 3 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑋 ∈ 𝐴) → ((𝐻 ↾ 𝐶):𝐶–1-1-onto→(𝐻 “ 𝐶) ↔ (𝐻 ↾ 𝐶):𝐶–1-1-onto→𝐷))
16	9, 15	mpbid 147	. 2 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑋 ∈ 𝐴) → (𝐻 ↾ 𝐶):𝐶–1-1-onto→𝐷)
17		df-isom 5342	. . . . . . 7 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐻:𝐴–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦))))
18	17	simprbi 275	. . . . . 6 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)))
19	18	adantr 276	. . . . 5 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑋 ∈ 𝐴) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)))
20		ssralv 3292	. . . . . 6 ⊢ (𝐶 ⊆ 𝐴 → (∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)) → ∀𝑦 ∈ 𝐶 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦))))
21	20	ralimdv 2601	. . . . 5 ⊢ (𝐶 ⊆ 𝐴 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐶 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦))))
22	7, 19, 21	mpsyl 65	. . . 4 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑋 ∈ 𝐴) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐶 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)))
23		ssralv 3292	. . . 4 ⊢ (𝐶 ⊆ 𝐴 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐶 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)) → ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐶 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦))))
24	7, 22, 23	mpsyl 65	. . 3 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑋 ∈ 𝐴) → ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐶 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)))
25		fvres 5672	. . . . . . 7 ⊢ (𝑥 ∈ 𝐶 → ((𝐻 ↾ 𝐶)‘𝑥) = (𝐻‘𝑥))
26		fvres 5672	. . . . . . 7 ⊢ (𝑦 ∈ 𝐶 → ((𝐻 ↾ 𝐶)‘𝑦) = (𝐻‘𝑦))
27	25, 26	breqan12d 4109	. . . . . 6 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → (((𝐻 ↾ 𝐶)‘𝑥)𝑆((𝐻 ↾ 𝐶)‘𝑦) ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)))
28	27	bibi2d 232	. . . . 5 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → ((𝑥𝑅𝑦 ↔ ((𝐻 ↾ 𝐶)‘𝑥)𝑆((𝐻 ↾ 𝐶)‘𝑦)) ↔ (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦))))
29	28	ralbidva 2529	. . . 4 ⊢ (𝑥 ∈ 𝐶 → (∀𝑦 ∈ 𝐶 (𝑥𝑅𝑦 ↔ ((𝐻 ↾ 𝐶)‘𝑥)𝑆((𝐻 ↾ 𝐶)‘𝑦)) ↔ ∀𝑦 ∈ 𝐶 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦))))
30	29	ralbiia 2547	. . 3 ⊢ (∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐶 (𝑥𝑅𝑦 ↔ ((𝐻 ↾ 𝐶)‘𝑥)𝑆((𝐻 ↾ 𝐶)‘𝑦)) ↔ ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐶 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)))
31	24, 30	sylibr 134	. 2 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑋 ∈ 𝐴) → ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐶 (𝑥𝑅𝑦 ↔ ((𝐻 ↾ 𝐶)‘𝑥)𝑆((𝐻 ↾ 𝐶)‘𝑦)))
32		df-isom 5342	. 2 ⊢ ((𝐻 ↾ 𝐶) Isom 𝑅, 𝑆 (𝐶, 𝐷) ↔ ((𝐻 ↾ 𝐶):𝐶–1-1-onto→𝐷 ∧ ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐶 (𝑥𝑅𝑦 ↔ ((𝐻 ↾ 𝐶)‘𝑥)𝑆((𝐻 ↾ 𝐶)‘𝑦))))
33	16, 31, 32	sylanbrc 417	1 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑋 ∈ 𝐴) → (𝐻 ↾ 𝐶) Isom 𝑅, 𝑆 (𝐶, 𝐷))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398   ∈ wcel 2202  ∀wral 2511   ∩ cin 3200   ⊆ wss 3201  {csn 3673   class class class wbr 4093  ^◡ccnv 4730   ↾ cres 4733   “ cima 4734  –1-1→wf1 5330  –1-1-onto→wf1o 5332  ‘cfv 5333   Isom wiso 5334

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-sbc 3033  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-isom 5342

This theorem is referenced by: (None)