Theorem resthauslem 21963

Description: Lemma for resthaus 21968 and similar theorems. If the topological property 𝐴 is preserved under injective preimages, then property 𝐴 passes to subspaces. (Contributed by Mario Carneiro, 25-Aug-2015.)

Hypotheses

Ref	Expression
resthauslem.1	⊢ (𝐽 ∈ 𝐴 → 𝐽 ∈ Top)
resthauslem.2	⊢ ((𝐽 ∈ 𝐴 ∧ ( I ↾ (𝑆 ∩ ∪ 𝐽)):(𝑆 ∩ ∪ 𝐽)–1-1→(𝑆 ∩ ∪ 𝐽) ∧ ( I ↾ (𝑆 ∩ ∪ 𝐽)) ∈ ((𝐽 ↾t 𝑆) Cn 𝐽)) → (𝐽 ↾t 𝑆) ∈ 𝐴)

Assertion

Ref	Expression
resthauslem	⊢ ((𝐽 ∈ 𝐴 ∧ 𝑆 ∈ 𝑉) → (𝐽 ↾t 𝑆) ∈ 𝐴)

Proof of Theorem resthauslem

Step	Hyp	Ref	Expression
1		simpl 485	. 2 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝑆 ∈ 𝑉) → 𝐽 ∈ 𝐴)
2		f1oi 6645	. . 3 ⊢ ( I ↾ (𝑆 ∩ ∪ 𝐽)):(𝑆 ∩ ∪ 𝐽)–1-1-onto→(𝑆 ∩ ∪ 𝐽)
3		f1of1 6607	. . 3 ⊢ (( I ↾ (𝑆 ∩ ∪ 𝐽)):(𝑆 ∩ ∪ 𝐽)–1-1-onto→(𝑆 ∩ ∪ 𝐽) → ( I ↾ (𝑆 ∩ ∪ 𝐽)):(𝑆 ∩ ∪ 𝐽)–1-1→(𝑆 ∩ ∪ 𝐽))
4	2, 3	mp1i 13	. 2 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝑆 ∈ 𝑉) → ( I ↾ (𝑆 ∩ ∪ 𝐽)):(𝑆 ∩ ∪ 𝐽)–1-1→(𝑆 ∩ ∪ 𝐽))
5		inss2 4204	. . . . 5 ⊢ (𝑆 ∩ ∪ 𝐽) ⊆ ∪ 𝐽
6		resabs1 5876	. . . . 5 ⊢ ((𝑆 ∩ ∪ 𝐽) ⊆ ∪ 𝐽 → (( I ↾ ∪ 𝐽) ↾ (𝑆 ∩ ∪ 𝐽)) = ( I ↾ (𝑆 ∩ ∪ 𝐽)))
7	5, 6	ax-mp 5	. . . 4 ⊢ (( I ↾ ∪ 𝐽) ↾ (𝑆 ∩ ∪ 𝐽)) = ( I ↾ (𝑆 ∩ ∪ 𝐽))
8		resthauslem.1	. . . . . . . 8 ⊢ (𝐽 ∈ 𝐴 → 𝐽 ∈ Top)
9	8	adantr 483	. . . . . . 7 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝑆 ∈ 𝑉) → 𝐽 ∈ Top)
10		toptopon2 21518	. . . . . . 7 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘∪ 𝐽))
11	9, 10	sylib 220	. . . . . 6 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝑆 ∈ 𝑉) → 𝐽 ∈ (TopOn‘∪ 𝐽))
12		idcn 21857	. . . . . 6 ⊢ (𝐽 ∈ (TopOn‘∪ 𝐽) → ( I ↾ ∪ 𝐽) ∈ (𝐽 Cn 𝐽))
13	11, 12	syl 17	. . . . 5 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝑆 ∈ 𝑉) → ( I ↾ ∪ 𝐽) ∈ (𝐽 Cn 𝐽))
14		eqid 2819	. . . . . 6 ⊢ ∪ 𝐽 = ∪ 𝐽
15	14	cnrest 21885	. . . . 5 ⊢ ((( I ↾ ∪ 𝐽) ∈ (𝐽 Cn 𝐽) ∧ (𝑆 ∩ ∪ 𝐽) ⊆ ∪ 𝐽) → (( I ↾ ∪ 𝐽) ↾ (𝑆 ∩ ∪ 𝐽)) ∈ ((𝐽 ↾t (𝑆 ∩ ∪ 𝐽)) Cn 𝐽))
16	13, 5, 15	sylancl 588	. . . 4 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝑆 ∈ 𝑉) → (( I ↾ ∪ 𝐽) ↾ (𝑆 ∩ ∪ 𝐽)) ∈ ((𝐽 ↾t (𝑆 ∩ ∪ 𝐽)) Cn 𝐽))
17	7, 16	eqeltrrid 2916	. . 3 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝑆 ∈ 𝑉) → ( I ↾ (𝑆 ∩ ∪ 𝐽)) ∈ ((𝐽 ↾t (𝑆 ∩ ∪ 𝐽)) Cn 𝐽))
18	14	restin 21766	. . . 4 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝑆 ∈ 𝑉) → (𝐽 ↾t 𝑆) = (𝐽 ↾t (𝑆 ∩ ∪ 𝐽)))
19	18	oveq1d 7163	. . 3 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝑆 ∈ 𝑉) → ((𝐽 ↾t 𝑆) Cn 𝐽) = ((𝐽 ↾t (𝑆 ∩ ∪ 𝐽)) Cn 𝐽))
20	17, 19	eleqtrrd 2914	. 2 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝑆 ∈ 𝑉) → ( I ↾ (𝑆 ∩ ∪ 𝐽)) ∈ ((𝐽 ↾t 𝑆) Cn 𝐽))
21		resthauslem.2	. 2 ⊢ ((𝐽 ∈ 𝐴 ∧ ( I ↾ (𝑆 ∩ ∪ 𝐽)):(𝑆 ∩ ∪ 𝐽)–1-1→(𝑆 ∩ ∪ 𝐽) ∧ ( I ↾ (𝑆 ∩ ∪ 𝐽)) ∈ ((𝐽 ↾t 𝑆) Cn 𝐽)) → (𝐽 ↾t 𝑆) ∈ 𝐴)
22	1, 4, 20, 21	syl3anc 1366	1 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝑆 ∈ 𝑉) → (𝐽 ↾t 𝑆) ∈ 𝐴)

Syntax hints:   → wi 4   ∧ wa 398   ∧ w3a 1082   = wceq 1531   ∈ wcel 2108   ∩ cin 3933   ⊆ wss 3934  ∪ cuni 4830   I cid 5452   ↾ cres 5550  –1-1→wf1 6345  –1-1-onto→wf1o 6347  ‘cfv 6348  (class class class)co 7148   ↾_t crest 16686  Topctop 21493  TopOnctopon 21510   Cn ccn 21824

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1905  ax-6 1964  ax-7 2009  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2154  ax-12 2170  ax-ext 2791  ax-rep 5181  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7453

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1083  df-3an 1084  df-tru 1534  df-ex 1775  df-nf 1779  df-sb 2064  df-mo 2616  df-eu 2648  df-clab 2798  df-cleq 2812  df-clel 2891  df-nfc 2961  df-ne 3015  df-ral 3141  df-rex 3142  df-reu 3143  df-rab 3145  df-v 3495  df-sbc 3771  df-csb 3882  df-dif 3937  df-un 3939  df-in 3941  df-ss 3950  df-pss 3952  df-nul 4290  df-if 4466  df-pw 4539  df-sn 4560  df-pr 4562  df-tp 4564  df-op 4566  df-uni 4831  df-int 4868  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-tr 5164  df-id 5453  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-pred 6141  df-ord 6187  df-on 6188  df-lim 6189  df-suc 6190  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7151  df-oprab 7152  df-mpo 7153  df-om 7573  df-1st 7681  df-2nd 7682  df-wrecs 7939  df-recs 8000  df-rdg 8038  df-oadd 8098  df-er 8281  df-map 8400  df-en 8502  df-fin 8505  df-fi 8867  df-rest 16688  df-topgen 16709  df-top 21494  df-topon 21511  df-bases 21546  df-cn 21827

This theorem is referenced by:  restt0  21966  restt1  21967  resthaus  21968