Theorem hmeoimaf1o 15417

Description: The function mapping open sets to their images under a homeomorphism is a bijection of topologies. (Contributed by Mario Carneiro, 10-Sep-2015.)

Hypothesis

Ref	Expression
hmeoimaf1o.1	⊢ 𝐺 = (𝑥 ∈ 𝐽 ↦ (𝐹 “ 𝑥))

Assertion

Ref	Expression
hmeoimaf1o	⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐺:𝐽–1-1-onto→𝐾)

Distinct variable groups:   𝑥,𝐹   𝑥,𝐽   𝑥,𝐾

Allowed substitution hint:   𝐺(𝑥)

Proof of Theorem hmeoimaf1o

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		hmeoimaf1o.1	. 2 ⊢ 𝐺 = (𝑥 ∈ 𝐽 ↦ (𝐹 “ 𝑥))
2		hmeoima 15413	. 2 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑥 ∈ 𝐽) → (𝐹 “ 𝑥) ∈ 𝐾)
3		hmeocn 15408	. . 3 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾))
4		cnima 15323	. . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑦 ∈ 𝐾) → (◡𝐹 “ 𝑦) ∈ 𝐽)
5	3, 4	sylan 283	. 2 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑦 ∈ 𝐾) → (◡𝐹 “ 𝑦) ∈ 𝐽)
6		eqid 2238	. . . . . . 7 ⊢ ∪ 𝐽 = ∪ 𝐽
7		eqid 2238	. . . . . . 7 ⊢ ∪ 𝐾 = ∪ 𝐾
8	6, 7	hmeof1o 15412	. . . . . 6 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹:∪ 𝐽–1-1-onto→∪ 𝐾)
9	8	adantr 276	. . . . 5 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) → 𝐹:∪ 𝐽–1-1-onto→∪ 𝐾)
10		f1of1 5638	. . . . 5 ⊢ (𝐹:∪ 𝐽–1-1-onto→∪ 𝐾 → 𝐹:∪ 𝐽–1-1→∪ 𝐾)
11	9, 10	syl 14	. . . 4 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) → 𝐹:∪ 𝐽–1-1→∪ 𝐾)
12		elssuni 3963	. . . . 5 ⊢ (𝑥 ∈ 𝐽 → 𝑥 ⊆ ∪ 𝐽)
13	12	ad2antrl 494	. . . 4 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) → 𝑥 ⊆ ∪ 𝐽)
14		cnvimass 5150	. . . . 5 ⊢ (◡𝐹 “ 𝑦) ⊆ dom 𝐹
15		f1dm 5603	. . . . . 6 ⊢ (𝐹:∪ 𝐽–1-1→∪ 𝐾 → dom 𝐹 = ∪ 𝐽)
16	11, 15	syl 14	. . . . 5 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) → dom 𝐹 = ∪ 𝐽)
17	14, 16	sseqtrid 3298	. . . 4 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) → (◡𝐹 “ 𝑦) ⊆ ∪ 𝐽)
18		f1imaeq 5981	. . . 4 ⊢ ((𝐹:∪ 𝐽–1-1→∪ 𝐾 ∧ (𝑥 ⊆ ∪ 𝐽 ∧ (◡𝐹 “ 𝑦) ⊆ ∪ 𝐽)) → ((𝐹 “ 𝑥) = (𝐹 “ (◡𝐹 “ 𝑦)) ↔ 𝑥 = (◡𝐹 “ 𝑦)))
19	11, 13, 17, 18	syl12anc 1276	. . 3 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) → ((𝐹 “ 𝑥) = (𝐹 “ (◡𝐹 “ 𝑦)) ↔ 𝑥 = (◡𝐹 “ 𝑦)))
20		f1ofo 5646	. . . . . . 7 ⊢ (𝐹:∪ 𝐽–1-1-onto→∪ 𝐾 → 𝐹:∪ 𝐽–onto→∪ 𝐾)
21	9, 20	syl 14	. . . . . 6 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) → 𝐹:∪ 𝐽–onto→∪ 𝐾)
22		elssuni 3963	. . . . . . 7 ⊢ (𝑦 ∈ 𝐾 → 𝑦 ⊆ ∪ 𝐾)
23	22	ad2antll 495	. . . . . 6 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) → 𝑦 ⊆ ∪ 𝐾)
24		foimacnv 5657	. . . . . 6 ⊢ ((𝐹:∪ 𝐽–onto→∪ 𝐾 ∧ 𝑦 ⊆ ∪ 𝐾) → (𝐹 “ (◡𝐹 “ 𝑦)) = 𝑦)
25	21, 23, 24	syl2anc 415	. . . . 5 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) → (𝐹 “ (◡𝐹 “ 𝑦)) = 𝑦)
26	25	eqeq2d 2250	. . . 4 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) → ((𝐹 “ 𝑥) = (𝐹 “ (◡𝐹 “ 𝑦)) ↔ (𝐹 “ 𝑥) = 𝑦))
27		eqcom 2240	. . . 4 ⊢ ((𝐹 “ 𝑥) = 𝑦 ↔ 𝑦 = (𝐹 “ 𝑥))
28	26, 27	bitrdi 196	. . 3 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) → ((𝐹 “ 𝑥) = (𝐹 “ (◡𝐹 “ 𝑦)) ↔ 𝑦 = (𝐹 “ 𝑥)))
29	19, 28	bitr3d 190	. 2 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) → (𝑥 = (◡𝐹 “ 𝑦) ↔ 𝑦 = (𝐹 “ 𝑥)))
30	1, 2, 5, 29	f1o2d 6295	1 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐺:𝐽–1-1-onto→𝐾)

This proof depends on syntax axioms:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1402   ∈ wcel 2209   ⊆ wss 3220  ∪ cuni 3935   ↦ cmpt 4192  ^◡ccnv 4773  dom cdm 4774   “ cima 4777  –1-1→wf1 5374  –onto→wfo 5375  –1-1-onto→wf1o 5376  (class class class)co 6085   Cn ccn 15288  Homeochmeo 15403

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4249  ax-pow 4311  ax-pr 4346  ax-un 4578  ax-setind 4684

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3690  df-sn 3715  df-pr 3716  df-op 3718  df-uni 3936  df-iun 4014  df-br 4131  df-opab 4193  df-mpt 4194  df-id 4438  df-xp 4780  df-rel 4781  df-cnv 4782  df-co 4783  df-dm 4784  df-rn 4785  df-res 4786  df-ima 4787  df-iota 5337  df-fun 5379  df-fn 5380  df-f 5381  df-f1 5382  df-fo 5383  df-f1o 5384  df-fv 5385  df-ov 6088  df-oprab 6089  df-mpo 6090  df-1st 6374  df-2nd 6375  df-map 6924  df-top 15101  df-topon 15114  df-cn 15291  df-hmeo 15404

This theorem is used by: (None)