Theorem haushmphlem 24106

Description: Lemma for haushmph 24111 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 24101	. 2 ⊢ (𝐽 ≃ 𝐾 → 𝐾 ≃ 𝐽)
2		hmph 24095	. . 3 ⊢ (𝐾 ≃ 𝐽 ↔ (𝐾Homeo𝐽) ≠ ∅)
3		n0 4300	. . . 4 ⊢ ((𝐾Homeo𝐽) ≠ ∅ ↔ ∃𝑓 𝑓 ∈ (𝐾Homeo𝐽))
4		simpl 488	. . . . . . 7 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝑓 ∈ (𝐾Homeo𝐽)) → 𝐽 ∈ 𝐴)
5		eqid 2761	. . . . . . . . . 10 ⊢ ∪ 𝐾 = ∪ 𝐾
6		eqid 2761	. . . . . . . . . 10 ⊢ ∪ 𝐽 = ∪ 𝐽
7	5, 6	hmeof1o 24083	. . . . . . . . 9 ⊢ (𝑓 ∈ (𝐾Homeo𝐽) → 𝑓:∪ 𝐾–1-1-onto→∪ 𝐽)
8	7	adantl 487	. . . . . . . 8 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝑓 ∈ (𝐾Homeo𝐽)) → 𝑓:∪ 𝐾–1-1-onto→∪ 𝐽)
9		f1of1 6823	. . . . . . . 8 ⊢ (𝑓:∪ 𝐾–1-1-onto→∪ 𝐽 → 𝑓:∪ 𝐾–1-1→∪ 𝐽)
10	8, 9	syl 18	. . . . . . 7 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝑓 ∈ (𝐾Homeo𝐽)) → 𝑓:∪ 𝐾–1-1→∪ 𝐽)
11		hmeocn 24079	. . . . . . . 8 ⊢ (𝑓 ∈ (𝐾Homeo𝐽) → 𝑓 ∈ (𝐾 Cn 𝐽))
12	11	adantl 487	. . . . . . 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 1398	. . . . . 6 ⊢ ((𝐽 ∈ 𝐴 ∧ 𝑓 ∈ (𝐾Homeo𝐽)) → 𝐾 ∈ 𝐴)
15	14	expcom 419	. . . . 5 ⊢ (𝑓 ∈ (𝐾Homeo𝐽) → (𝐽 ∈ 𝐴 → 𝐾 ∈ 𝐴))
16	15	exlimiv 1963	. . . 4 ⊢ (∃𝑓 𝑓 ∈ (𝐾Homeo𝐽) → (𝐽 ∈ 𝐴 → 𝐾 ∈ 𝐴))
17	3, 16	sylbi 220	. . 3 ⊢ ((𝐾Homeo𝐽) ≠ ∅ → (𝐽 ∈ 𝐴 → 𝐾 ∈ 𝐴))
18	2, 17	sylbi 220	. 2 ⊢ (𝐾 ≃ 𝐽 → (𝐽 ∈ 𝐴 → 𝐾 ∈ 𝐴))
19	1, 18	syl 18	1 ⊢ (𝐽 ≃ 𝐾 → (𝐽 ∈ 𝐴 → 𝐾 ∈ 𝐴))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103  ∃wex 1812   ∈ wcel 2145   ≠ wne 2956  ∅c0 4279  ∪ cuni 4867   class class class wbr 5103  –1-1→wf1 6535  –1-1-onto→wf1o 6537  (class class class)co 7420  Topctop 23211   Cn ccn 23542  Homeochmeo 24072   ≃ chmph 24073

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2733  ax-sep 5249  ax-nul 5260  ax-pow 5327  ax-pr 5391  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-id 5546  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-ov 7423  df-oprab 7424  df-mpo 7425  df-1st 8001  df-2nd 8002  df-1o 8476  df-map 8849  df-top 23212  df-topon 23229  df-cn 23545  df-hmeo 24074  df-hmph 24075

This theorem is used by:  t0hmph  24109  t1hmph  24110  haushmph  24111