Theorem hmeof1o 22823

Description: A homeomorphism is a 1-1-onto mapping. (Contributed by FL, 5-Mar-2007.) (Revised by Mario Carneiro, 30-May-2014.)

Hypotheses

Ref	Expression
hmeof1o.1	⊢ 𝑋 = ∪ 𝐽
hmeof1o.2	⊢ 𝑌 = ∪ 𝐾

Assertion

Ref	Expression
hmeof1o	⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹:𝑋–1-1-onto→𝑌)

Proof of Theorem hmeof1o

Step	Hyp	Ref	Expression
1		hmeocn 22819	. . 3 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾))
2		cntop1 22299	. . . . 5 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ Top)
3		hmeof1o.1	. . . . . 6 ⊢ 𝑋 = ∪ 𝐽
4	3	toptopon 21974	. . . . 5 ⊢ (𝐽 ∈ Top ↔ 𝐽 ∈ (TopOn‘𝑋))
5	2, 4	sylib 217	. . . 4 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐽 ∈ (TopOn‘𝑋))
6		cntop2 22300	. . . . 5 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ Top)
7		hmeof1o.2	. . . . . 6 ⊢ 𝑌 = ∪ 𝐾
8	7	toptopon 21974	. . . . 5 ⊢ (𝐾 ∈ Top ↔ 𝐾 ∈ (TopOn‘𝑌))
9	6, 8	sylib 217	. . . 4 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → 𝐾 ∈ (TopOn‘𝑌))
10	5, 9	jca 511	. . 3 ⊢ (𝐹 ∈ (𝐽 Cn 𝐾) → (𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)))
11	1, 10	syl 17	. 2 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → (𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)))
12		hmeof1o2 22822	. . 3 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌) ∧ 𝐹 ∈ (𝐽Homeo𝐾)) → 𝐹:𝑋–1-1-onto→𝑌)
13	12	3expia 1119	. 2 ⊢ ((𝐽 ∈ (TopOn‘𝑋) ∧ 𝐾 ∈ (TopOn‘𝑌)) → (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹:𝑋–1-1-onto→𝑌))
14	11, 13	mpcom 38	1 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹:𝑋–1-1-onto→𝑌)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1539   ∈ wcel 2108  ∪ cuni 4836  –1-1-onto→wf1o 6417  ‘cfv 6418  (class class class)co 7255  Topctop 21950  TopOnctopon 21967   Cn ccn 22283  Homeochmeo 22812

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pow 5283  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-sbc 3712  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4837  df-br 5071  df-opab 5133  df-mpt 5154  df-id 5480  df-xp 5586  df-rel 5587  df-cnv 5588  df-co 5589  df-dm 5590  df-rn 5591  df-res 5592  df-ima 5593  df-iota 6376  df-fun 6420  df-fn 6421  df-f 6422  df-f1 6423  df-fo 6424  df-f1o 6425  df-fv 6426  df-ov 7258  df-oprab 7259  df-mpo 7260  df-map 8575  df-top 21951  df-topon 21968  df-cn 22286  df-hmeo 22814

This theorem is referenced by:  hmeoopn  22825  hmeocld  22826  hmeontr  22828  hmeoimaf1o  22829  hmeoqtop  22834  haushmphlem  22846  cmphmph  22847  connhmph  22848  reghmph  22852  nrmhmph  22853  hmphdis  22855  hmphen2  22858  cmphaushmeo  22859  txhmeo  22862  tpr2rico  31764  mndpluscn  31778