Theorem haushmphlem 23811

Description: Lemma for haushmph 23816 and similar theorems. If the topological property 𝐴 is preserved under injective preimages, then property 𝐴 is preserved under homeomorphisms. (Contributed by Mario Carneiro, 25-Aug-2015.)

Hypotheses

Ref	Expression
haushmphlem.1	⊢ (𝐽 ∈ 𝐴 → 𝐽 ∈ Top)
haushmphlem.2	⊢ ((𝐽 ∈ 𝐴 ∧ 𝑓:∪ 𝐾–1-1→∪ 𝐽 ∧ 𝑓 ∈ (𝐾 Cn 𝐽)) → 𝐾 ∈ 𝐴)

Assertion

Ref	Expression
haushmphlem	⊢ (𝐽 ≃ 𝐾 → (𝐽 ∈ 𝐴 → 𝐾 ∈ 𝐴))

Distinct variable groups:   𝐴,𝑓   𝑓,𝐽   𝑓,𝐾

Proof of Theorem haushmphlem

Step	Hyp	Ref	Expression
1		hmphsym 23806	. 2 ⊢ (𝐽 ≃ 𝐾 → 𝐾 ≃ 𝐽)
2		hmph 23800	. . 3 ⊢ (𝐾 ≃ 𝐽 ↔ (𝐾Homeo𝐽) ≠ ∅)
3		n0 4359	. . . 4 ⊢ ((𝐾Homeo𝐽) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝐾Homeo𝐽))
4		simpl 482	. . . . . . 7 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝑓 ∈ (𝐾Homeo𝐽)) → 𝐽 ∈ 𝐴)
5		eqid 2735	. . . . . . . . . 10 ⊢ ∪ 𝐾 = ∪ 𝐾
6		eqid 2735	. . . . . . . . . 10 ⊢ ∪ 𝐽 = ∪ 𝐽
7	5, 6	hmeof1o 23788	. . . . . . . . 9 ⊢ (𝑓 ∈ (𝐾Homeo𝐽) → 𝑓:∪ 𝐾–1-1-onto→∪ 𝐽)
8	7	adantl 481	. . . . . . . 8 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝑓 ∈ (𝐾Homeo𝐽)) → 𝑓:∪ 𝐾–1-1-onto→∪ 𝐽)
9		f1of1 6848	. . . . . . . 8 ⊢ (𝑓:∪ 𝐾–1-1-onto→∪ 𝐽 → 𝑓:∪ 𝐾–1-1→∪ 𝐽)
10	8, 9	syl 17	. . . . . . 7 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝑓 ∈ (𝐾Homeo𝐽)) → 𝑓:∪ 𝐾–1-1→∪ 𝐽)
11		hmeocn 23784	. . . . . . . 8 ⊢ (𝑓 ∈ (𝐾Homeo𝐽) → 𝑓 ∈ (𝐾 Cn 𝐽))
12	11	adantl 481	. . . . . . 7 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝑓 ∈ (𝐾Homeo𝐽)) → 𝑓 ∈ (𝐾 Cn 𝐽))
13		haushmphlem.2	. . . . . . 7 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝑓:∪ 𝐾–1-1→∪ 𝐽 ∧ 𝑓 ∈ (𝐾 Cn 𝐽)) → 𝐾 ∈ 𝐴)
14	4, 10, 12, 13	syl3anc 1370	. . . . . 6 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝑓 ∈ (𝐾Homeo𝐽)) → 𝐾 ∈ 𝐴)
15	14	expcom 413	. . . . 5 ⊢ (𝑓 ∈ (𝐾Homeo𝐽) → (𝐽 ∈ 𝐴 → 𝐾 ∈ 𝐴))
16	15	exlimiv 1928	. . . 4 ⊢ (∃𝑓 𝑓 ∈ (𝐾Homeo𝐽) → (𝐽 ∈ 𝐴 → 𝐾 ∈ 𝐴))
17	3, 16	sylbi 217	. . 3 ⊢ ((𝐾Homeo𝐽) ≠ ∅ → (𝐽 ∈ 𝐴 → 𝐾 ∈ 𝐴))
18	2, 17	sylbi 217	. 2 ⊢ (𝐾 ≃ 𝐽 → (𝐽 ∈ 𝐴 → 𝐾 ∈ 𝐴))
19	1, 18	syl 17	1 ⊢ (𝐽 ≃ 𝐾 → (𝐽 ∈ 𝐴 → 𝐾 ∈ 𝐴))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086  ∃wex 1776   ∈ wcel 2106   ≠ wne 2938  ∅c0 4339  ∪ cuni 4912   class class class wbr 5148  –1-1→wf1 6560  –1-1-onto→wf1o 6562  (class class class)co 7431  Topctop 22915   Cn ccn 23248  Homeochmeo 23777   ≃ chmph 23778

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-pow 5371  ax-pr 5438  ax-un 7754

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-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  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-suc 6392  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-ov 7434  df-oprab 7435  df-mpo 7436  df-1st 8013  df-2nd 8014  df-1o 8505  df-map 8867  df-top 22916  df-topon 22933  df-cn 23251  df-hmeo 23779  df-hmph 23780

This theorem is referenced by:  t0hmph  23814  t1hmph  23815  haushmph  23816