Theorem haushmphlem 23729

Description: Lemma for haushmph 23734 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 23724	. 2 ⊢ (𝐽 ≃ 𝐾 → 𝐾 ≃ 𝐽)
2		hmph 23718	. . 3 ⊢ (𝐾 ≃ 𝐽 ↔ (𝐾Homeo𝐽) ≠ ∅)
3		n0 4303	. . . 4 ⊢ ((𝐾Homeo𝐽) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝐾Homeo𝐽))
4		simpl 482	. . . . . . 7 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝑓 ∈ (𝐾Homeo𝐽)) → 𝐽 ∈ 𝐴)
5		eqid 2734	. . . . . . . . . 10 ⊢ ∪ 𝐾 = ∪ 𝐾
6		eqid 2734	. . . . . . . . . 10 ⊢ ∪ 𝐽 = ∪ 𝐽
7	5, 6	hmeof1o 23706	. . . . . . . . 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 6771	. . . . . . . 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 23702	. . . . . . . 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 1373	. . . . . 6 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝑓 ∈ (𝐾Homeo𝐽)) → 𝐾 ∈ 𝐴)
15	14	expcom 413	. . . . 5 ⊢ (𝑓 ∈ (𝐾Homeo𝐽) → (𝐽 ∈ 𝐴 → 𝐾 ∈ 𝐴))
16	15	exlimiv 1931	. . . 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 1780   ∈ wcel 2113   ≠ wne 2930  ∅c0 4283  ∪ cuni 4861   class class class wbr 5096  –1-1→wf1 6487  –1-1-onto→wf1o 6489  (class class class)co 7356  Topctop 22835   Cn ccn 23166  Homeochmeo 23695   ≃ chmph 23696

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2706  ax-sep 5239  ax-nul 5249  ax-pow 5308  ax-pr 5375  ax-un 7678

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2809  df-nfc 2883  df-ne 2931  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-sbc 3739  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-pw 4554  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-iun 4946  df-br 5097  df-opab 5159  df-mpt 5178  df-id 5517  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-suc 6321  df-iota 6446  df-fun 6492  df-fn 6493  df-f 6494  df-f1 6495  df-fo 6496  df-f1o 6497  df-fv 6498  df-ov 7359  df-oprab 7360  df-mpo 7361  df-1st 7931  df-2nd 7932  df-1o 8395  df-map 8763  df-top 22836  df-topon 22853  df-cn 23169  df-hmeo 23697  df-hmph 23698

This theorem is referenced by:  t0hmph  23732  t1hmph  23733  haushmph  23734