Theorem hmeoimaf1o 24089

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 24084	. 2 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑥 ∈ 𝐽) → (𝐹 “ 𝑥) ∈ 𝐾)
3		hmeocn 24079	. . 3 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹 ∈ (𝐽 Cn 𝐾))
4		cnima 23583	. . 3 ⊢ ((𝐹 ∈ (𝐽 Cn 𝐾) ∧ 𝑦 ∈ 𝐾) → (◡𝐹 “ 𝑦) ∈ 𝐽)
5	3, 4	sylan 592	. 2 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ 𝑦 ∈ 𝐾) → (◡𝐹 “ 𝑦) ∈ 𝐽)
6		eqid 2761	. . . . . . 7 ⊢ ∪ 𝐽 = ∪ 𝐽
7		eqid 2761	. . . . . . 7 ⊢ ∪ 𝐾 = ∪ 𝐾
8	6, 7	hmeof1o 24083	. . . . . 6 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐹:∪ 𝐽–1-1-onto→∪ 𝐾)
9	8	adantr 486	. . . . 5 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) → 𝐹:∪ 𝐽–1-1-onto→∪ 𝐾)
10		f1of1 6823	. . . . 5 ⊢ (𝐹:∪ 𝐽–1-1-onto→∪ 𝐾 → 𝐹:∪ 𝐽–1-1→∪ 𝐾)
11	9, 10	syl 18	. . . 4 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) → 𝐹:∪ 𝐽–1-1→∪ 𝐾)
12		elssuni 4899	. . . . 5 ⊢ (𝑥 ∈ 𝐽 → 𝑥 ⊆ ∪ 𝐽)
13	12	ad2antrl 741	. . . 4 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) → 𝑥 ⊆ ∪ 𝐽)
14		cnvimass 6198	. . . . 5 ⊢ (◡𝐹 “ 𝑦) ⊆ dom 𝐹
15		f1dm 6784	. . . . . 6 ⊢ (𝐹:∪ 𝐽–1-1→∪ 𝐾 → dom 𝐹 = ∪ 𝐽)
16	11, 15	syl 18	. . . . 5 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) → dom 𝐹 = ∪ 𝐽)
17	14, 16	sseqtrid 3973	. . . 4 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) → (◡𝐹 “ 𝑦) ⊆ ∪ 𝐽)
18		f1imaeq 7269	. . . 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 850	. . 3 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) → ((𝐹 “ 𝑥) = (𝐹 “ (◡𝐹 “ 𝑦)) ↔ 𝑥 = (◡𝐹 “ 𝑦)))
20		f1ofo 6832	. . . . . . 7 ⊢ (𝐹:∪ 𝐽–1-1-onto→∪ 𝐾 → 𝐹:∪ 𝐽–onto→∪ 𝐾)
21	9, 20	syl 18	. . . . . 6 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) → 𝐹:∪ 𝐽–onto→∪ 𝐾)
22		elssuni 4899	. . . . . . 7 ⊢ (𝑦 ∈ 𝐾 → 𝑦 ⊆ ∪ 𝐾)
23	22	ad2antll 742	. . . . . 6 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) → 𝑦 ⊆ ∪ 𝐾)
24		foimacnv 6842	. . . . . 6 ⊢ ((𝐹:∪ 𝐽–onto→∪ 𝐾 ∧ 𝑦 ⊆ ∪ 𝐾) → (𝐹 “ (◡𝐹 “ 𝑦)) = 𝑦)
25	21, 23, 24	syl2anc 596	. . . . 5 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) → (𝐹 “ (◡𝐹 “ 𝑦)) = 𝑦)
26	25	eqeq2d 2772	. . . 4 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) → ((𝐹 “ 𝑥) = (𝐹 “ (◡𝐹 “ 𝑦)) ↔ (𝐹 “ 𝑥) = 𝑦))
27		eqcom 2768	. . . 4 ⊢ ((𝐹 “ 𝑥) = 𝑦 ↔ 𝑦 = (𝐹 “ 𝑥))
28	26, 27	bitrdi 290	. . 3 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) → ((𝐹 “ 𝑥) = (𝐹 “ (◡𝐹 “ 𝑦)) ↔ 𝑦 = (𝐹 “ 𝑥)))
29	19, 28	bitr3d 284	. 2 ⊢ ((𝐹 ∈ (𝐽Homeo𝐾) ∧ (𝑥 ∈ 𝐽 ∧ 𝑦 ∈ 𝐾)) → (𝑥 = (◡𝐹 “ 𝑦) ↔ 𝑦 = (𝐹 “ 𝑥)))
30	1, 2, 5, 29	f1o2d 7675	1 ⊢ (𝐹 ∈ (𝐽Homeo𝐾) → 𝐺:𝐽–1-1-onto→𝐾)

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145   ⊆ wss 3899  ∪ cuni 4867   ↦ cmpt 5186  ^◡ccnv 5650  dom cdm 5651   “ cima 5654  –1-1→wf1 6535  –onto→wfo 6536  –1-1-onto→wf1o 6537  (class class class)co 7420   Cn ccn 23542  Homeochmeo 24072

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-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-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-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-map 8849  df-top 23212  df-topon 23229  df-cn 23545  df-hmeo 24074

This theorem is used by:  hmphen  24104