Theorem isoini2 5720

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 5708	. . . . . 6 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴–1-1-onto→𝐵)
2		f1of1 5366	. . . . . 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 274	. . . 4 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑋 ∈ 𝐴) → 𝐻:𝐴–1-1→𝐵)
5		isoini2.1	. . . . 5 ⊢ 𝐶 = (𝐴 ∩ (◡𝑅 “ {𝑋}))
6		inss1 3296	. . . . 5 ⊢ (𝐴 ∩ (◡𝑅 “ {𝑋})) ⊆ 𝐴
7	5, 6	eqsstri 3129	. . . 4 ⊢ 𝐶 ⊆ 𝐴
8		f1ores 5382	. . . 4 ⊢ ((𝐻:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐻 ↾ 𝐶):𝐶–1-1-onto→(𝐻 “ 𝐶))
9	4, 7, 8	sylancl 409	. . 3 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑋 ∈ 𝐴) → (𝐻 ↾ 𝐶):𝐶–1-1-onto→(𝐻 “ 𝐶))
10		isoini 5719	. . . . 5 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑋 ∈ 𝐴) → (𝐻 “ (𝐴 ∩ (◡𝑅 “ {𝑋}))) = (𝐵 ∩ (◡𝑆 “ {(𝐻‘𝑋)})))
11	5	imaeq2i 4879	. . . . 5 ⊢ (𝐻 “ 𝐶) = (𝐻 “ (𝐴 ∩ (◡𝑅 “ {𝑋})))
12		isoini2.2	. . . . 5 ⊢ 𝐷 = (𝐵 ∩ (◡𝑆 “ {(𝐻‘𝑋)}))
13	10, 11, 12	3eqtr4g 2197	. . . 4 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑋 ∈ 𝐴) → (𝐻 “ 𝐶) = 𝐷)
14		f1oeq3 5358	. . . 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 146	. 2 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑋 ∈ 𝐴) → (𝐻 ↾ 𝐶):𝐶–1-1-onto→𝐷)
17		df-isom 5132	. . . . . . 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 273	. . . . . 6 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)))
19	18	adantr 274	. . . . 5 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑋 ∈ 𝐴) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)))
20		ssralv 3161	. . . . . 6 ⊢ (𝐶 ⊆ 𝐴 → (∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)) → ∀𝑦 ∈ 𝐶 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦))))
21	20	ralimdv 2500	. . . . 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 3161	. . . 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 5445	. . . . . . 7 ⊢ (𝑥 ∈ 𝐶 → ((𝐻 ↾ 𝐶)‘𝑥) = (𝐻‘𝑥))
26		fvres 5445	. . . . . . 7 ⊢ (𝑦 ∈ 𝐶 → ((𝐻 ↾ 𝐶)‘𝑦) = (𝐻‘𝑦))
27	25, 26	breqan12d 3945	. . . . . 6 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → (((𝐻 ↾ 𝐶)‘𝑥)𝑆((𝐻 ↾ 𝐶)‘𝑦) ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)))
28	27	bibi2d 231	. . . . 5 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → ((𝑥𝑅𝑦 ↔ ((𝐻 ↾ 𝐶)‘𝑥)𝑆((𝐻 ↾ 𝐶)‘𝑦)) ↔ (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦))))
29	28	ralbidva 2433	. . . 4 ⊢ (𝑥 ∈ 𝐶 → (∀𝑦 ∈ 𝐶 (𝑥𝑅𝑦 ↔ ((𝐻 ↾ 𝐶)‘𝑥)𝑆((𝐻 ↾ 𝐶)‘𝑦)) ↔ ∀𝑦 ∈ 𝐶 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦))))
30	29	ralbiia 2449	. . 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 133	. 2 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑋 ∈ 𝐴) → ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐶 (𝑥𝑅𝑦 ↔ ((𝐻 ↾ 𝐶)‘𝑥)𝑆((𝐻 ↾ 𝐶)‘𝑦)))
32		df-isom 5132	. 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 413	1 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑋 ∈ 𝐴) → (𝐻 ↾ 𝐶) Isom 𝑅, 𝑆 (𝐶, 𝐷))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1331   ∈ wcel 1480  ∀wral 2416   ∩ cin 3070   ⊆ wss 3071  {csn 3527   class class class wbr 3929  ^◡ccnv 4538   ↾ cres 4541   “ cima 4542  –1-1→wf1 5120  –1-1-onto→wf1o 5122  ‘cfv 5123   Isom wiso 5124

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-sbc 2910  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-isom 5132

This theorem is referenced by: (None)