Theorem sshauslem 23401

Description: Lemma for sshaus 23404 and similar theorems. If the topological property 𝐴 is preserved under injective preimages, then a topology finer than one with property 𝐴 also has property 𝐴. (Contributed by Mario Carneiro, 25-Aug-2015.)

Hypotheses

Ref	Expression
t1sep.1	⊢ 𝑋 = ∪ 𝐽
sshauslem.2	⊢ (𝐽 ∈ 𝐴 → 𝐽 ∈ Top)
sshauslem.3	⊢ ((𝐽 ∈ 𝐴 ∧ ( I ↾ 𝑋):𝑋–1-1→𝑋 ∧ ( I ↾ 𝑋) ∈ (𝐾 Cn 𝐽)) → 𝐾 ∈ 𝐴)

Assertion

Ref	Expression
sshauslem	⊢ ((𝐽 ∈ 𝐴 ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) → 𝐾 ∈ 𝐴)

Proof of Theorem sshauslem

Step	Hyp	Ref	Expression
1		simp1 1136	. 2 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) → 𝐽 ∈ 𝐴)
2		f1oi 6900	. . 3 ⊢ ( I ↾ 𝑋):𝑋–1-1-onto→𝑋
3		f1of1 6861	. . 3 ⊢ (( I ↾ 𝑋):𝑋–1-1-onto→𝑋 → ( I ↾ 𝑋):𝑋–1-1→𝑋)
4	2, 3	mp1i 13	. 2 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) → ( I ↾ 𝑋):𝑋–1-1→𝑋)
5		simp3 1138	. . 3 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) → 𝐽 ⊆ 𝐾)
6		simp2 1137	. . . 4 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) → 𝐾 ∈ (TopOn‘𝑋))
7		sshauslem.2	. . . . . 6 ⊢ (𝐽 ∈ 𝐴 → 𝐽 ∈ Top)
8	7	3ad2ant1 1133	. . . . 5 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) → 𝐽 ∈ Top)
9		t1sep.1	. . . . . 6 ⊢ 𝑋 = ∪ 𝐽
10	9	toptopon 22944	. . . . 5 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
11	8, 10	sylib 218	. . . 4 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) → 𝐽 ∈ (TopOn‘𝑋))
12		ssidcn 23284	. . . 4 ⊢ ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ (TopOn‘𝑋)) → (( I ↾ 𝑋) ∈ (𝐾 Cn 𝐽) ↔ 𝐽 ⊆ 𝐾))
13	6, 11, 12	syl2anc 583	. . 3 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) → (( I ↾ 𝑋) ∈ (𝐾 Cn 𝐽) ↔ 𝐽 ⊆ 𝐾))
14	5, 13	mpbird 257	. 2 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) → ( I ↾ 𝑋) ∈ (𝐾 Cn 𝐽))
15		sshauslem.3	. 2 ⊢ ((𝐽 ∈ 𝐴 ∧ ( I ↾ 𝑋):𝑋–1-1→𝑋 ∧ ( I ↾ 𝑋) ∈ (𝐾 Cn 𝐽)) → 𝐾 ∈ 𝐴)
16	1, 4, 14, 15	syl3anc 1371	1 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) → 𝐾 ∈ 𝐴)

Syntax hints:   → wi 4   ↔ wb 206   ∧ w3a 1087   = wceq 1537   ∈ wcel 2108   ⊆ wss 3976  ∪ cuni 4931   I cid 5592   ↾ cres 5702  –1-1→wf1 6570  –1-1-onto→wf1o 6572  ‘cfv 6573  (class class class)co 7448  Topctop 22920  TopOnctopon 22937   Cn ccn 23253

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-map 8886  df-top 22921  df-topon 22938  df-cn 23256

This theorem is referenced by:  sst0  23402  sst1  23403  sshaus  23404