Theorem sshauslem 21977

Description: Lemma for sshaus 21980 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 1133	. 2 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) → 𝐽 ∈ 𝐴)
2		f1oi 6627	. . 3 ⊢ ( I ↾ 𝑋):𝑋–1-1-onto→𝑋
3		f1of1 6589	. . 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 1135	. . 3 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) → 𝐽 ⊆ 𝐾)
6		simp2 1134	. . . 4 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) → 𝐾 ∈ (TopOn‘𝑋))
7		sshauslem.2	. . . . . 6 ⊢ (𝐽 ∈ 𝐴 → 𝐽 ∈ Top)
8	7	3ad2ant1 1130	. . . . 5 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) → 𝐽 ∈ Top)
9		t1sep.1	. . . . . 6 ⊢ 𝑋 = ∪ 𝐽
10	9	toptopon 21522	. . . . 5 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
11	8, 10	sylib 221	. . . 4 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) → 𝐽 ∈ (TopOn‘𝑋))
12		ssidcn 21860	. . . 4 ⊢ ((𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ∈ (TopOn‘𝑋)) → (( I ↾ 𝑋) ∈ (𝐾 Cn 𝐽) ↔ 𝐽 ⊆ 𝐾))
13	6, 11, 12	syl2anc 587	. . 3 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) → (( I ↾ 𝑋) ∈ (𝐾 Cn 𝐽) ↔ 𝐽 ⊆ 𝐾))
14	5, 13	mpbird 260	. 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 1368	1 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝐾 ∈ (TopOn‘𝑋) ∧ 𝐽 ⊆ 𝐾) → 𝐾 ∈ 𝐴)

Syntax hints:   → wi 4   ↔ wb 209   ∧ w3a 1084   = wceq 1538   ∈ wcel 2111   ⊆ wss 3881  ∪ cuni 4800   I cid 5424   ↾ cres 5521  –1-1→wf1 6321  –1-1-onto→wf1o 6323  ‘cfv 6324  (class class class)co 7135  Topctop 21498  TopOnctopon 21515   Cn ccn 21829

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pow 5231  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-res 5531  df-ima 5532  df-iota 6283  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331  df-fv 6332  df-ov 7138  df-oprab 7139  df-mpo 7140  df-map 8391  df-top 21499  df-topon 21516  df-cn 21832

This theorem is referenced by:  sst0  21978  sst1  21979  sshaus  21980