Theorem hmeoimaf1o 14901

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 14897	. 2 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑥 ∈ 𝐽) → (𝐹 “ 𝑥) ∈ 𝐾)
3		hmeocn 14892	. . 3 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾))
4		cnima 14807	. . 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 2207	. . . . . . 7 ⊢ ∪ 𝐽 = ∪ 𝐽
7		eqid 2207	. . . . . . 7 ⊢ ∪ 𝐾 = ∪ 𝐾
8	6, 7	hmeof1o 14896	. . . . . 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 5543	. . . . 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 3892	. . . . 5 ⊢ (𝑥 ∈ 𝐽 → 𝑥 ⊆ ∪ 𝐽)
13	12	ad2antrl 490	. . . 4 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) → 𝑥 ⊆ ∪ 𝐽)
14		cnvimass 5064	. . . . 5 ⊢ (◡𝐹 “ 𝑦) ⊆ dom 𝐹
15		f1dm 5508	. . . . . 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 3251	. . . 4 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) → (◡𝐹 “ 𝑦) ⊆ ∪ 𝐽)
18		f1imaeq 5867	. . . 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 1248	. . 3 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) → ((𝐹 “ 𝑥) = (𝐹 “ (◡𝐹 “ 𝑦)) ↔ 𝑥 = (◡𝐹 “ 𝑦)))
20		f1ofo 5551	. . . . . . 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 3892	. . . . . . 7 ⊢ (𝑦 ∈ 𝐾 → 𝑦 ⊆ ∪ 𝐾)
23	22	ad2antll 491	. . . . . 6 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) → 𝑦 ⊆ ∪ 𝐾)
24		foimacnv 5562	. . . . . 6 ⊢ ((𝐹:∪ 𝐽–onto→∪ 𝐾 ∧ 𝑦 ⊆ ∪ 𝐾) → (𝐹 “ (◡𝐹 “ 𝑦)) = 𝑦)
25	21, 23, 24	syl2anc 411	. . . . 5 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) → (𝐹 “ (◡𝐹 “ 𝑦)) = 𝑦)
26	25	eqeq2d 2219	. . . 4 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) → ((𝐹 “ 𝑥) = (𝐹 “ (◡𝐹 “ 𝑦)) ↔ (𝐹 “ 𝑥) = 𝑦))
27		eqcom 2209	. . . 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 6174	1 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐺:𝐽–1-1-onto→𝐾)

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1373   ∈ wcel 2178   ⊆ wss 3174  ∪ cuni 3864   ↦ cmpt 4121  ^◡ccnv 4692  dom cdm 4693   “ cima 4696  –1-1→wf1 5287  –onto→wfo 5288  –1-1-onto→wf1o 5289  (class class class)co 5967   Cn ccn 14772  Homeochmeo 14887

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-ral 2491  df-rex 2492  df-rab 2495  df-v 2778  df-sbc 3006  df-csb 3102  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-iun 3943  df-br 4060  df-opab 4122  df-mpt 4123  df-id 4358  df-xp 4699  df-rel 4700  df-cnv 4701  df-co 4702  df-dm 4703  df-rn 4704  df-res 4705  df-ima 4706  df-iota 5251  df-fun 5292  df-fn 5293  df-f 5294  df-f1 5295  df-fo 5296  df-f1o 5297  df-fv 5298  df-ov 5970  df-oprab 5971  df-mpo 5972  df-1st 6249  df-2nd 6250  df-map 6760  df-top 14585  df-topon 14598  df-cn 14775  df-hmeo 14888

This theorem is referenced by: (None)