Theorem haushmphlem 23725

Description: Lemma for haushmph 23730 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 23720	. 2 ⊢ (𝐽 ≃ 𝐾 → 𝐾 ≃ 𝐽)
2		hmph 23714	. . 3 ⊢ (𝐾 ≃ 𝐽 ↔ (𝐾Homeo𝐽) ≠ ∅)
3		n0 4328	. . . 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 23702	. . . . . . . . 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 6817	. . . . . . . 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 23698	. . . . . . . 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 1930	. . . 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 1779   ∈ wcel 2108   ≠ wne 2932  ∅c0 4308  ∪ cuni 4883   class class class wbr 5119  –1-1→wf1 6528  –1-1-onto→wf1o 6530  (class class class)co 7405  Topctop 22831   Cn ccn 23162  Homeochmeo 23691   ≃ chmph 23692

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5266  ax-nul 5276  ax-pow 5335  ax-pr 5402  ax-un 7729

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-sbc 3766  df-csb 3875  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-pw 4577  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-iun 4969  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-ima 5667  df-suc 6358  df-iota 6484  df-fun 6533  df-fn 6534  df-f 6535  df-f1 6536  df-fo 6537  df-f1o 6538  df-fv 6539  df-ov 7408  df-oprab 7409  df-mpo 7410  df-1st 7988  df-2nd 7989  df-1o 8480  df-map 8842  df-top 22832  df-topon 22849  df-cn 23165  df-hmeo 23693  df-hmph 23694

This theorem is referenced by:  t0hmph  23728  t1hmph  23729  haushmph  23730