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Theorem tgpconncompeqg 23272
Description: The connected component containing 𝐴 is the left coset of the identity component containing 𝐴. (Contributed by Mario Carneiro, 17-Sep-2015.)
Hypotheses
Ref Expression
tgpconncomp.x 𝑋 = (Base‘𝐺)
tgpconncomp.z 0 = (0g𝐺)
tgpconncomp.j 𝐽 = (TopOpen‘𝐺)
tgpconncomp.s 𝑆 = {𝑥 ∈ 𝒫 𝑋 ∣ ( 0𝑥 ∧ (𝐽t 𝑥) ∈ Conn)}
tgpconncompeqg.r = (𝐺 ~QG 𝑆)
Assertion
Ref Expression
tgpconncompeqg ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → [𝐴] = {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)})
Distinct variable groups:   𝑥, 0   𝑥,𝐴   𝑥,𝐽   𝑥,𝐺   𝑥,𝑋
Allowed substitution hints:   (𝑥)   𝑆(𝑥)

Proof of Theorem tgpconncompeqg
Dummy variables 𝑦 𝑧 𝑔 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dfec2 8510 . . . . 5 (𝐴𝑋 → [𝐴] = {𝑧𝐴 𝑧})
21adantl 482 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → [𝐴] = {𝑧𝐴 𝑧})
3 tgpconncomp.s . . . . . . . . 9 𝑆 = {𝑥 ∈ 𝒫 𝑋 ∣ ( 0𝑥 ∧ (𝐽t 𝑥) ∈ Conn)}
4 ssrab2 4014 . . . . . . . . . 10 {𝑥 ∈ 𝒫 𝑋 ∣ ( 0𝑥 ∧ (𝐽t 𝑥) ∈ Conn)} ⊆ 𝒫 𝑋
5 sspwuni 5030 . . . . . . . . . 10 ({𝑥 ∈ 𝒫 𝑋 ∣ ( 0𝑥 ∧ (𝐽t 𝑥) ∈ Conn)} ⊆ 𝒫 𝑋 {𝑥 ∈ 𝒫 𝑋 ∣ ( 0𝑥 ∧ (𝐽t 𝑥) ∈ Conn)} ⊆ 𝑋)
64, 5mpbi 229 . . . . . . . . 9 {𝑥 ∈ 𝒫 𝑋 ∣ ( 0𝑥 ∧ (𝐽t 𝑥) ∈ Conn)} ⊆ 𝑋
73, 6eqsstri 3956 . . . . . . . 8 𝑆𝑋
87a1i 11 . . . . . . 7 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → 𝑆𝑋)
9 tgpconncomp.x . . . . . . . 8 𝑋 = (Base‘𝐺)
10 eqid 2739 . . . . . . . 8 (invg𝐺) = (invg𝐺)
11 eqid 2739 . . . . . . . 8 (+g𝐺) = (+g𝐺)
12 tgpconncompeqg.r . . . . . . . 8 = (𝐺 ~QG 𝑆)
139, 10, 11, 12eqgval 18814 . . . . . . 7 ((𝐺 ∈ TopGrp ∧ 𝑆𝑋) → (𝐴 𝑧 ↔ (𝐴𝑋𝑧𝑋 ∧ (((invg𝐺)‘𝐴)(+g𝐺)𝑧) ∈ 𝑆)))
148, 13syldan 591 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → (𝐴 𝑧 ↔ (𝐴𝑋𝑧𝑋 ∧ (((invg𝐺)‘𝐴)(+g𝐺)𝑧) ∈ 𝑆)))
15 simp2 1136 . . . . . 6 ((𝐴𝑋𝑧𝑋 ∧ (((invg𝐺)‘𝐴)(+g𝐺)𝑧) ∈ 𝑆) → 𝑧𝑋)
1614, 15syl6bi 252 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → (𝐴 𝑧𝑧𝑋))
1716abssdv 4003 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → {𝑧𝐴 𝑧} ⊆ 𝑋)
182, 17eqsstrd 3960 . . 3 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → [𝐴] 𝑋)
19 simpr 485 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → 𝐴𝑋)
20 tgpgrp 23238 . . . . . . 7 (𝐺 ∈ TopGrp → 𝐺 ∈ Grp)
21 tgpconncomp.z . . . . . . . 8 0 = (0g𝐺)
229, 11, 21, 10grplinv 18637 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (((invg𝐺)‘𝐴)(+g𝐺)𝐴) = 0 )
2320, 22sylan 580 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → (((invg𝐺)‘𝐴)(+g𝐺)𝐴) = 0 )
24 tgpconncomp.j . . . . . . . . 9 𝐽 = (TopOpen‘𝐺)
2524, 9tgptopon 23242 . . . . . . . 8 (𝐺 ∈ TopGrp → 𝐽 ∈ (TopOn‘𝑋))
2625adantr 481 . . . . . . 7 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → 𝐽 ∈ (TopOn‘𝑋))
2720adantr 481 . . . . . . . 8 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → 𝐺 ∈ Grp)
289, 21grpidcl 18616 . . . . . . . 8 (𝐺 ∈ Grp → 0𝑋)
2927, 28syl 17 . . . . . . 7 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → 0𝑋)
303conncompid 22591 . . . . . . 7 ((𝐽 ∈ (TopOn‘𝑋) ∧ 0𝑋) → 0𝑆)
3126, 29, 30syl2anc 584 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → 0𝑆)
3223, 31eqeltrd 2840 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → (((invg𝐺)‘𝐴)(+g𝐺)𝐴) ∈ 𝑆)
339, 10, 11, 12eqgval 18814 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝑆𝑋) → (𝐴 𝐴 ↔ (𝐴𝑋𝐴𝑋 ∧ (((invg𝐺)‘𝐴)(+g𝐺)𝐴) ∈ 𝑆)))
348, 33syldan 591 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → (𝐴 𝐴 ↔ (𝐴𝑋𝐴𝑋 ∧ (((invg𝐺)‘𝐴)(+g𝐺)𝐴) ∈ 𝑆)))
3519, 19, 32, 34mpbir3and 1341 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → 𝐴 𝐴)
36 elecg 8550 . . . . 5 ((𝐴𝑋𝐴𝑋) → (𝐴 ∈ [𝐴] 𝐴 𝐴))
3719, 19, 36syl2anc 584 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → (𝐴 ∈ [𝐴] 𝐴 𝐴))
3835, 37mpbird 256 . . 3 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → 𝐴 ∈ [𝐴] )
399, 12, 11eqglact 18816 . . . . . . 7 ((𝐺 ∈ Grp ∧ 𝑆𝑋𝐴𝑋) → [𝐴] = ((𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) “ 𝑆))
407, 39mp3an2 1448 . . . . . 6 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → [𝐴] = ((𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) “ 𝑆))
4120, 40sylan 580 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → [𝐴] = ((𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) “ 𝑆))
4241oveq2d 7300 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → (𝐽t [𝐴] ) = (𝐽t ((𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) “ 𝑆)))
43 eqid 2739 . . . . 5 𝐽 = 𝐽
44 eqid 2739 . . . . . . 7 (𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) = (𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧))
4544, 9, 11, 24tgplacthmeo 23263 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → (𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) ∈ (𝐽Homeo𝐽))
46 hmeocn 22920 . . . . . 6 ((𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) ∈ (𝐽Homeo𝐽) → (𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) ∈ (𝐽 Cn 𝐽))
4745, 46syl 17 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → (𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) ∈ (𝐽 Cn 𝐽))
48 toponuni 22072 . . . . . . 7 (𝐽 ∈ (TopOn‘𝑋) → 𝑋 = 𝐽)
4926, 48syl 17 . . . . . 6 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → 𝑋 = 𝐽)
507, 49sseqtrid 3974 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → 𝑆 𝐽)
513conncompconn 22592 . . . . . 6 ((𝐽 ∈ (TopOn‘𝑋) ∧ 0𝑋) → (𝐽t 𝑆) ∈ Conn)
5226, 29, 51syl2anc 584 . . . . 5 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → (𝐽t 𝑆) ∈ Conn)
5343, 47, 50, 52connima 22585 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → (𝐽t ((𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) “ 𝑆)) ∈ Conn)
5442, 53eqeltrd 2840 . . 3 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → (𝐽t [𝐴] ) ∈ Conn)
55 eqid 2739 . . . 4 {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)} = {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)}
5655conncompss 22593 . . 3 (([𝐴] 𝑋𝐴 ∈ [𝐴] ∧ (𝐽t [𝐴] ) ∈ Conn) → [𝐴] {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)})
5718, 38, 54, 56syl3anc 1370 . 2 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → [𝐴] {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)})
58 elpwi 4543 . . . . . 6 (𝑦 ∈ 𝒫 𝑋𝑦𝑋)
5944mptpreima 6146 . . . . . . . . . . 11 ((𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) “ 𝑦) = {𝑧𝑋 ∣ (𝐴(+g𝐺)𝑧) ∈ 𝑦}
6059ssrab3 4016 . . . . . . . . . 10 ((𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) “ 𝑦) ⊆ 𝑋
6129adantr 481 . . . . . . . . . . 11 (((𝐺 ∈ TopGrp ∧ 𝐴𝑋) ∧ (𝑦𝑋 ∧ (𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn))) → 0𝑋)
629, 11, 21grprid 18619 . . . . . . . . . . . . . 14 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (𝐴(+g𝐺) 0 ) = 𝐴)
6320, 62sylan 580 . . . . . . . . . . . . 13 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → (𝐴(+g𝐺) 0 ) = 𝐴)
6463adantr 481 . . . . . . . . . . . 12 (((𝐺 ∈ TopGrp ∧ 𝐴𝑋) ∧ (𝑦𝑋 ∧ (𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn))) → (𝐴(+g𝐺) 0 ) = 𝐴)
65 simprrl 778 . . . . . . . . . . . 12 (((𝐺 ∈ TopGrp ∧ 𝐴𝑋) ∧ (𝑦𝑋 ∧ (𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn))) → 𝐴𝑦)
6664, 65eqeltrd 2840 . . . . . . . . . . 11 (((𝐺 ∈ TopGrp ∧ 𝐴𝑋) ∧ (𝑦𝑋 ∧ (𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn))) → (𝐴(+g𝐺) 0 ) ∈ 𝑦)
67 oveq2 7292 . . . . . . . . . . . . 13 (𝑧 = 0 → (𝐴(+g𝐺)𝑧) = (𝐴(+g𝐺) 0 ))
6867eleq1d 2824 . . . . . . . . . . . 12 (𝑧 = 0 → ((𝐴(+g𝐺)𝑧) ∈ 𝑦 ↔ (𝐴(+g𝐺) 0 ) ∈ 𝑦))
6968, 59elrab2 3628 . . . . . . . . . . 11 ( 0 ∈ ((𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) “ 𝑦) ↔ ( 0𝑋 ∧ (𝐴(+g𝐺) 0 ) ∈ 𝑦))
7061, 66, 69sylanbrc 583 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝐴𝑋) ∧ (𝑦𝑋 ∧ (𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn))) → 0 ∈ ((𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) “ 𝑦))
71 hmeocnvcn 22921 . . . . . . . . . . . . 13 ((𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) ∈ (𝐽Homeo𝐽) → (𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) ∈ (𝐽 Cn 𝐽))
7245, 71syl 17 . . . . . . . . . . . 12 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → (𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) ∈ (𝐽 Cn 𝐽))
7372adantr 481 . . . . . . . . . . 11 (((𝐺 ∈ TopGrp ∧ 𝐴𝑋) ∧ (𝑦𝑋 ∧ (𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn))) → (𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) ∈ (𝐽 Cn 𝐽))
74 simprl 768 . . . . . . . . . . . 12 (((𝐺 ∈ TopGrp ∧ 𝐴𝑋) ∧ (𝑦𝑋 ∧ (𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn))) → 𝑦𝑋)
7549adantr 481 . . . . . . . . . . . 12 (((𝐺 ∈ TopGrp ∧ 𝐴𝑋) ∧ (𝑦𝑋 ∧ (𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn))) → 𝑋 = 𝐽)
7674, 75sseqtrd 3962 . . . . . . . . . . 11 (((𝐺 ∈ TopGrp ∧ 𝐴𝑋) ∧ (𝑦𝑋 ∧ (𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn))) → 𝑦 𝐽)
77 simprrr 779 . . . . . . . . . . 11 (((𝐺 ∈ TopGrp ∧ 𝐴𝑋) ∧ (𝑦𝑋 ∧ (𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn))) → (𝐽t 𝑦) ∈ Conn)
7843, 73, 76, 77connima 22585 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝐴𝑋) ∧ (𝑦𝑋 ∧ (𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn))) → (𝐽t ((𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) “ 𝑦)) ∈ Conn)
793conncompss 22593 . . . . . . . . . 10 ((((𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) “ 𝑦) ⊆ 𝑋0 ∈ ((𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) “ 𝑦) ∧ (𝐽t ((𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) “ 𝑦)) ∈ Conn) → ((𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) “ 𝑦) ⊆ 𝑆)
8060, 70, 78, 79mp3an2i 1465 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝐴𝑋) ∧ (𝑦𝑋 ∧ (𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn))) → ((𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) “ 𝑦) ⊆ 𝑆)
81 eqid 2739 . . . . . . . . . . . . . . . 16 (𝑔𝑋 ↦ (𝑧𝑋 ↦ (𝑔(+g𝐺)𝑧))) = (𝑔𝑋 ↦ (𝑧𝑋 ↦ (𝑔(+g𝐺)𝑧)))
8281, 9, 11, 10grplactcnv 18687 . . . . . . . . . . . . . . 15 ((𝐺 ∈ Grp ∧ 𝐴𝑋) → (((𝑔𝑋 ↦ (𝑧𝑋 ↦ (𝑔(+g𝐺)𝑧)))‘𝐴):𝑋1-1-onto𝑋((𝑔𝑋 ↦ (𝑧𝑋 ↦ (𝑔(+g𝐺)𝑧)))‘𝐴) = ((𝑔𝑋 ↦ (𝑧𝑋 ↦ (𝑔(+g𝐺)𝑧)))‘((invg𝐺)‘𝐴))))
8320, 82sylan 580 . . . . . . . . . . . . . 14 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → (((𝑔𝑋 ↦ (𝑧𝑋 ↦ (𝑔(+g𝐺)𝑧)))‘𝐴):𝑋1-1-onto𝑋((𝑔𝑋 ↦ (𝑧𝑋 ↦ (𝑔(+g𝐺)𝑧)))‘𝐴) = ((𝑔𝑋 ↦ (𝑧𝑋 ↦ (𝑔(+g𝐺)𝑧)))‘((invg𝐺)‘𝐴))))
8483simpld 495 . . . . . . . . . . . . 13 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → ((𝑔𝑋 ↦ (𝑧𝑋 ↦ (𝑔(+g𝐺)𝑧)))‘𝐴):𝑋1-1-onto𝑋)
8581, 9grplactfval 18685 . . . . . . . . . . . . . . 15 (𝐴𝑋 → ((𝑔𝑋 ↦ (𝑧𝑋 ↦ (𝑔(+g𝐺)𝑧)))‘𝐴) = (𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)))
8685adantl 482 . . . . . . . . . . . . . 14 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → ((𝑔𝑋 ↦ (𝑧𝑋 ↦ (𝑔(+g𝐺)𝑧)))‘𝐴) = (𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)))
8786f1oeq1d 6720 . . . . . . . . . . . . 13 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → (((𝑔𝑋 ↦ (𝑧𝑋 ↦ (𝑔(+g𝐺)𝑧)))‘𝐴):𝑋1-1-onto𝑋 ↔ (𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)):𝑋1-1-onto𝑋))
8884, 87mpbid 231 . . . . . . . . . . . 12 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → (𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)):𝑋1-1-onto𝑋)
8988adantr 481 . . . . . . . . . . 11 (((𝐺 ∈ TopGrp ∧ 𝐴𝑋) ∧ (𝑦𝑋 ∧ (𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn))) → (𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)):𝑋1-1-onto𝑋)
90 f1ocnv 6737 . . . . . . . . . . 11 ((𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)):𝑋1-1-onto𝑋(𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)):𝑋1-1-onto𝑋)
91 f1ofun 6727 . . . . . . . . . . 11 ((𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)):𝑋1-1-onto𝑋 → Fun (𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)))
9289, 90, 913syl 18 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝐴𝑋) ∧ (𝑦𝑋 ∧ (𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn))) → Fun (𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)))
93 f1odm 6729 . . . . . . . . . . . 12 ((𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)):𝑋1-1-onto𝑋 → dom (𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) = 𝑋)
9489, 90, 933syl 18 . . . . . . . . . . 11 (((𝐺 ∈ TopGrp ∧ 𝐴𝑋) ∧ (𝑦𝑋 ∧ (𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn))) → dom (𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) = 𝑋)
9574, 94sseqtrrd 3963 . . . . . . . . . 10 (((𝐺 ∈ TopGrp ∧ 𝐴𝑋) ∧ (𝑦𝑋 ∧ (𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn))) → 𝑦 ⊆ dom (𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)))
96 funimass3 6940 . . . . . . . . . 10 ((Fun (𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) ∧ 𝑦 ⊆ dom (𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧))) → (((𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) “ 𝑦) ⊆ 𝑆𝑦 ⊆ ((𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) “ 𝑆)))
9792, 95, 96syl2anc 584 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝐴𝑋) ∧ (𝑦𝑋 ∧ (𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn))) → (((𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) “ 𝑦) ⊆ 𝑆𝑦 ⊆ ((𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) “ 𝑆)))
9880, 97mpbid 231 . . . . . . . 8 (((𝐺 ∈ TopGrp ∧ 𝐴𝑋) ∧ (𝑦𝑋 ∧ (𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn))) → 𝑦 ⊆ ((𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) “ 𝑆))
9941adantr 481 . . . . . . . . 9 (((𝐺 ∈ TopGrp ∧ 𝐴𝑋) ∧ (𝑦𝑋 ∧ (𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn))) → [𝐴] = ((𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) “ 𝑆))
100 imacnvcnv 6114 . . . . . . . . 9 ((𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) “ 𝑆) = ((𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) “ 𝑆)
10199, 100eqtr4di 2797 . . . . . . . 8 (((𝐺 ∈ TopGrp ∧ 𝐴𝑋) ∧ (𝑦𝑋 ∧ (𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn))) → [𝐴] = ((𝑧𝑋 ↦ (𝐴(+g𝐺)𝑧)) “ 𝑆))
10298, 101sseqtrrd 3963 . . . . . . 7 (((𝐺 ∈ TopGrp ∧ 𝐴𝑋) ∧ (𝑦𝑋 ∧ (𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn))) → 𝑦 ⊆ [𝐴] )
103102expr 457 . . . . . 6 (((𝐺 ∈ TopGrp ∧ 𝐴𝑋) ∧ 𝑦𝑋) → ((𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn) → 𝑦 ⊆ [𝐴] ))
10458, 103sylan2 593 . . . . 5 (((𝐺 ∈ TopGrp ∧ 𝐴𝑋) ∧ 𝑦 ∈ 𝒫 𝑋) → ((𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn) → 𝑦 ⊆ [𝐴] ))
105104ralrimiva 3104 . . . 4 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → ∀𝑦 ∈ 𝒫 𝑋((𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn) → 𝑦 ⊆ [𝐴] ))
106 eleq2w 2823 . . . . . 6 (𝑥 = 𝑦 → (𝐴𝑥𝐴𝑦))
107 oveq2 7292 . . . . . . 7 (𝑥 = 𝑦 → (𝐽t 𝑥) = (𝐽t 𝑦))
108107eleq1d 2824 . . . . . 6 (𝑥 = 𝑦 → ((𝐽t 𝑥) ∈ Conn ↔ (𝐽t 𝑦) ∈ Conn))
109106, 108anbi12d 631 . . . . 5 (𝑥 = 𝑦 → ((𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn) ↔ (𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn)))
110109ralrab 3631 . . . 4 (∀𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)}𝑦 ⊆ [𝐴] ↔ ∀𝑦 ∈ 𝒫 𝑋((𝐴𝑦 ∧ (𝐽t 𝑦) ∈ Conn) → 𝑦 ⊆ [𝐴] ))
111105, 110sylibr 233 . . 3 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → ∀𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)}𝑦 ⊆ [𝐴] )
112 unissb 4874 . . 3 ( {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)} ⊆ [𝐴] ↔ ∀𝑦 ∈ {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)}𝑦 ⊆ [𝐴] )
113111, 112sylibr 233 . 2 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)} ⊆ [𝐴] )
11457, 113eqssd 3939 1 ((𝐺 ∈ TopGrp ∧ 𝐴𝑋) → [𝐴] = {𝑥 ∈ 𝒫 𝑋 ∣ (𝐴𝑥 ∧ (𝐽t 𝑥) ∈ Conn)})
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  w3a 1086   = wceq 1539  wcel 2107  {cab 2716  wral 3065  {crab 3069  wss 3888  𝒫 cpw 4534   cuni 4840   class class class wbr 5075  cmpt 5158  ccnv 5589  dom cdm 5590  cima 5593  Fun wfun 6431  1-1-ontowf1o 6436  cfv 6437  (class class class)co 7284  [cec 8505  Basecbs 16921  +gcplusg 16971  t crest 17140  TopOpenctopn 17141  0gc0g 17159  Grpcgrp 18586  invgcminusg 18587   ~QG cqg 18760  TopOnctopon 22068   Cn ccn 22384  Conncconn 22571  Homeochmeo 22913  TopGrpctgp 23231
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2710  ax-rep 5210  ax-sep 5224  ax-nul 5231  ax-pow 5289  ax-pr 5353  ax-un 7597
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2541  df-eu 2570  df-clab 2717  df-cleq 2731  df-clel 2817  df-nfc 2890  df-ne 2945  df-ral 3070  df-rex 3071  df-rmo 3072  df-reu 3073  df-rab 3074  df-v 3435  df-sbc 3718  df-csb 3834  df-dif 3891  df-un 3893  df-in 3895  df-ss 3905  df-pss 3907  df-nul 4258  df-if 4461  df-pw 4536  df-sn 4563  df-pr 4565  df-op 4569  df-uni 4841  df-int 4881  df-iun 4927  df-br 5076  df-opab 5138  df-mpt 5159  df-tr 5193  df-id 5490  df-eprel 5496  df-po 5504  df-so 5505  df-fr 5545  df-we 5547  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-ord 6273  df-on 6274  df-lim 6275  df-suc 6276  df-iota 6395  df-fun 6439  df-fn 6440  df-f 6441  df-f1 6442  df-fo 6443  df-f1o 6444  df-fv 6445  df-riota 7241  df-ov 7287  df-oprab 7288  df-mpo 7289  df-om 7722  df-1st 7840  df-2nd 7841  df-ec 8509  df-map 8626  df-en 8743  df-fin 8746  df-fi 9179  df-rest 17142  df-0g 17161  df-topgen 17163  df-plusf 18334  df-mgm 18335  df-sgrp 18384  df-mnd 18395  df-grp 18589  df-minusg 18590  df-eqg 18763  df-top 22052  df-topon 22069  df-topsp 22091  df-bases 22105  df-cld 22179  df-cn 22387  df-cnp 22388  df-conn 22572  df-tx 22722  df-hmeo 22915  df-tmd 23232  df-tgp 23233
This theorem is referenced by:  tgpconncomp  23273
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