Theorem isoini2 7359

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 7343	. . . . . 6 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴–1-1-onto→𝐵)
2		f1of1 6848	. . . . . 6 ⊢ (𝐻:𝐴–1-1-onto→𝐵 → 𝐻:𝐴–1-1→𝐵)
3	1, 2	syl 17	. . . . 5 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → 𝐻:𝐴–1-1→𝐵)
4	3	adantr 480	. . . 4 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑋 ∈ 𝐴) → 𝐻:𝐴–1-1→𝐵)
5		isoini2.1	. . . . 5 ⊢ 𝐶 = (𝐴 ∩ (◡𝑅 “ {𝑋}))
6		inss1 4245	. . . . 5 ⊢ (𝐴 ∩ (◡𝑅 “ {𝑋})) ⊆ 𝐴
7	5, 6	eqsstri 4030	. . . 4 ⊢ 𝐶 ⊆ 𝐴
8		f1ores 6863	. . . 4 ⊢ ((𝐻:𝐴–1-1→𝐵 ∧ 𝐶 ⊆ 𝐴) → (𝐻 ↾ 𝐶):𝐶–1-1-onto→(𝐻 “ 𝐶))
9	4, 7, 8	sylancl 586	. . 3 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑋 ∈ 𝐴) → (𝐻 ↾ 𝐶):𝐶–1-1-onto→(𝐻 “ 𝐶))
10		isoini 7358	. . . . 5 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑋 ∈ 𝐴) → (𝐻 “ (𝐴 ∩ (◡𝑅 “ {𝑋}))) = (𝐵 ∩ (◡𝑆 “ {(𝐻‘𝑋)})))
11	5	imaeq2i 6078	. . . . 5 ⊢ (𝐻 “ 𝐶) = (𝐻 “ (𝐴 ∩ (◡𝑅 “ {𝑋})))
12		isoini2.2	. . . . 5 ⊢ 𝐷 = (𝐵 ∩ (◡𝑆 “ {(𝐻‘𝑋)}))
13	10, 11, 12	3eqtr4g 2800	. . . 4 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑋 ∈ 𝐴) → (𝐻 “ 𝐶) = 𝐷)
14	13	f1oeq3d 6846	. . 3 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑋 ∈ 𝐴) → ((𝐻 ↾ 𝐶):𝐶–1-1-onto→(𝐻 “ 𝐶) ↔ (𝐻 ↾ 𝐶):𝐶–1-1-onto→𝐷))
15	9, 14	mpbid 232	. 2 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑋 ∈ 𝐴) → (𝐻 ↾ 𝐶):𝐶–1-1-onto→𝐷)
16		df-isom 6572	. . . . . . 7 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐻:𝐴–1-1-onto→𝐵 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦))))
17	16	simprbi 496	. . . . . 6 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)))
18	17	adantr 480	. . . . 5 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑋 ∈ 𝐴) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)))
19		ssralv 4064	. . . . . 6 ⊢ (𝐶 ⊆ 𝐴 → (∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)) → ∀𝑦 ∈ 𝐶 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦))))
20	19	ralimdv 3167	. . . . 5 ⊢ (𝐶 ⊆ 𝐴 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐶 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦))))
21	7, 18, 20	mpsyl 68	. . . 4 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑋 ∈ 𝐴) → ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐶 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)))
22		ssralv 4064	. . . 4 ⊢ (𝐶 ⊆ 𝐴 → (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐶 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)) → ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐶 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦))))
23	7, 21, 22	mpsyl 68	. . 3 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑋 ∈ 𝐴) → ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐶 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)))
24		fvres 6926	. . . . . . 7 ⊢ (𝑥 ∈ 𝐶 → ((𝐻 ↾ 𝐶)‘𝑥) = (𝐻‘𝑥))
25		fvres 6926	. . . . . . 7 ⊢ (𝑦 ∈ 𝐶 → ((𝐻 ↾ 𝐶)‘𝑦) = (𝐻‘𝑦))
26	24, 25	breqan12d 5164	. . . . . 6 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → (((𝐻 ↾ 𝐶)‘𝑥)𝑆((𝐻 ↾ 𝐶)‘𝑦) ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)))
27	26	bibi2d 342	. . . . 5 ⊢ ((𝑥 ∈ 𝐶 ∧ 𝑦 ∈ 𝐶) → ((𝑥𝑅𝑦 ↔ ((𝐻 ↾ 𝐶)‘𝑥)𝑆((𝐻 ↾ 𝐶)‘𝑦)) ↔ (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦))))
28	27	ralbidva 3174	. . . 4 ⊢ (𝑥 ∈ 𝐶 → (∀𝑦 ∈ 𝐶 (𝑥𝑅𝑦 ↔ ((𝐻 ↾ 𝐶)‘𝑥)𝑆((𝐻 ↾ 𝐶)‘𝑦)) ↔ ∀𝑦 ∈ 𝐶 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦))))
29	28	ralbiia 3089	. . 3 ⊢ (∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐶 (𝑥𝑅𝑦 ↔ ((𝐻 ↾ 𝐶)‘𝑥)𝑆((𝐻 ↾ 𝐶)‘𝑦)) ↔ ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐶 (𝑥𝑅𝑦 ↔ (𝐻‘𝑥)𝑆(𝐻‘𝑦)))
30	23, 29	sylibr 234	. 2 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑋 ∈ 𝐴) → ∀𝑥 ∈ 𝐶 ∀𝑦 ∈ 𝐶 (𝑥𝑅𝑦 ↔ ((𝐻 ↾ 𝐶)‘𝑥)𝑆((𝐻 ↾ 𝐶)‘𝑦)))
31		df-isom 6572	. 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))))
32	15, 30, 31	sylanbrc 583	1 ⊢ ((𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) ∧ 𝑋 ∈ 𝐴) → (𝐻 ↾ 𝐶) Isom 𝑅, 𝑆 (𝐶, 𝐷))

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   = wceq 1537   ∈ wcel 2106  ∀wral 3059   ∩ cin 3962   ⊆ wss 3963  {csn 4631   class class class wbr 5148  ^◡ccnv 5688   ↾ cres 5691   “ cima 5692  –1-1→wf1 6560  –1-1-onto→wf1o 6562  ‘cfv 6563   Isom wiso 6564

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-isom 6572

This theorem is referenced by:  fz1isolem  14497