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Theorem ordtrest2 20918
Description: An interval-closed set 𝐴 in a total order has the same subspace topology as the restricted order topology. (An interval-closed set is the same thing as an open or half-open or closed interval in , but in other sets like there are interval-closed sets like (π, +∞) ∩ ℚ that are not intervals.) (Contributed by Mario Carneiro, 9-Sep-2015.)
Hypotheses
Ref Expression
ordtrest2.1 𝑋 = dom 𝑅
ordtrest2.2 (𝜑𝑅 ∈ TosetRel )
ordtrest2.3 (𝜑𝐴𝑋)
ordtrest2.4 ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → {𝑧𝑋 ∣ (𝑥𝑅𝑧𝑧𝑅𝑦)} ⊆ 𝐴)
Assertion
Ref Expression
ordtrest2 (𝜑 → (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) = ((ordTop‘𝑅) ↾t 𝐴))
Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝜑,𝑥,𝑦,𝑧   𝑥,𝑅,𝑦,𝑧   𝑥,𝑋,𝑦,𝑧

Proof of Theorem ordtrest2
Dummy variables 𝑤 𝑣 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ordtrest2.2 . . . 4 (𝜑𝑅 ∈ TosetRel )
2 tsrps 17142 . . . 4 (𝑅 ∈ TosetRel → 𝑅 ∈ PosetRel)
31, 2syl 17 . . 3 (𝜑𝑅 ∈ PosetRel)
4 ordtrest2.1 . . . . 5 𝑋 = dom 𝑅
5 dmexg 7044 . . . . . 6 (𝑅 ∈ TosetRel → dom 𝑅 ∈ V)
61, 5syl 17 . . . . 5 (𝜑 → dom 𝑅 ∈ V)
74, 6syl5eqel 2702 . . . 4 (𝜑𝑋 ∈ V)
8 ordtrest2.3 . . . 4 (𝜑𝐴𝑋)
97, 8ssexd 4765 . . 3 (𝜑𝐴 ∈ V)
10 ordtrest 20916 . . 3 ((𝑅 ∈ PosetRel ∧ 𝐴 ∈ V) → (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ⊆ ((ordTop‘𝑅) ↾t 𝐴))
113, 9, 10syl2anc 692 . 2 (𝜑 → (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ⊆ ((ordTop‘𝑅) ↾t 𝐴))
12 eqid 2621 . . . . . . . 8 ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) = ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧})
13 eqid 2621 . . . . . . . 8 ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}) = ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤})
144, 12, 13ordtval 20903 . . . . . . 7 (𝑅 ∈ TosetRel → (ordTop‘𝑅) = (topGen‘(fi‘({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}))))))
151, 14syl 17 . . . . . 6 (𝜑 → (ordTop‘𝑅) = (topGen‘(fi‘({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}))))))
1615oveq1d 6619 . . . . 5 (𝜑 → ((ordTop‘𝑅) ↾t 𝐴) = ((topGen‘(fi‘({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}))))) ↾t 𝐴))
17 fibas 20692 . . . . . 6 (fi‘({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤})))) ∈ TopBases
18 tgrest 20873 . . . . . 6 (((fi‘({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤})))) ∈ TopBases ∧ 𝐴 ∈ V) → (topGen‘((fi‘({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤})))) ↾t 𝐴)) = ((topGen‘(fi‘({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}))))) ↾t 𝐴))
1917, 9, 18sylancr 694 . . . . 5 (𝜑 → (topGen‘((fi‘({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤})))) ↾t 𝐴)) = ((topGen‘(fi‘({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}))))) ↾t 𝐴))
2016, 19eqtr4d 2658 . . . 4 (𝜑 → ((ordTop‘𝑅) ↾t 𝐴) = (topGen‘((fi‘({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤})))) ↾t 𝐴)))
21 firest 16014 . . . . 5 (fi‘(({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}))) ↾t 𝐴)) = ((fi‘({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤})))) ↾t 𝐴)
2221fveq2i 6151 . . . 4 (topGen‘(fi‘(({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}))) ↾t 𝐴))) = (topGen‘((fi‘({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤})))) ↾t 𝐴))
2320, 22syl6eqr 2673 . . 3 (𝜑 → ((ordTop‘𝑅) ↾t 𝐴) = (topGen‘(fi‘(({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}))) ↾t 𝐴))))
24 inex1g 4761 . . . . . 6 (𝑅 ∈ TosetRel → (𝑅 ∩ (𝐴 × 𝐴)) ∈ V)
251, 24syl 17 . . . . 5 (𝜑 → (𝑅 ∩ (𝐴 × 𝐴)) ∈ V)
26 ordttop 20914 . . . . 5 ((𝑅 ∩ (𝐴 × 𝐴)) ∈ V → (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ∈ Top)
2725, 26syl 17 . . . 4 (𝜑 → (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ∈ Top)
284, 12, 13ordtuni 20904 . . . . . . . . 9 (𝑅 ∈ TosetRel → 𝑋 = ({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}))))
291, 28syl 17 . . . . . . . 8 (𝜑𝑋 = ({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}))))
3029, 7eqeltrrd 2699 . . . . . . 7 (𝜑 ({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}))) ∈ V)
31 uniexb 6921 . . . . . . 7 (({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}))) ∈ V ↔ ({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}))) ∈ V)
3230, 31sylibr 224 . . . . . 6 (𝜑 → ({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}))) ∈ V)
33 restval 16008 . . . . . 6 ((({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}))) ∈ V ∧ 𝐴 ∈ V) → (({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}))) ↾t 𝐴) = ran (𝑣 ∈ ({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}))) ↦ (𝑣𝐴)))
3432, 9, 33syl2anc 692 . . . . 5 (𝜑 → (({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}))) ↾t 𝐴) = ran (𝑣 ∈ ({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}))) ↦ (𝑣𝐴)))
35 sseqin2 3795 . . . . . . . . . . . 12 (𝐴𝑋 ↔ (𝑋𝐴) = 𝐴)
368, 35sylib 208 . . . . . . . . . . 11 (𝜑 → (𝑋𝐴) = 𝐴)
37 eqid 2621 . . . . . . . . . . . . . . 15 dom (𝑅 ∩ (𝐴 × 𝐴)) = dom (𝑅 ∩ (𝐴 × 𝐴))
3837ordttopon 20907 . . . . . . . . . . . . . 14 ((𝑅 ∩ (𝐴 × 𝐴)) ∈ V → (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ∈ (TopOn‘dom (𝑅 ∩ (𝐴 × 𝐴))))
3925, 38syl 17 . . . . . . . . . . . . 13 (𝜑 → (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ∈ (TopOn‘dom (𝑅 ∩ (𝐴 × 𝐴))))
404psssdm 17137 . . . . . . . . . . . . . . 15 ((𝑅 ∈ PosetRel ∧ 𝐴𝑋) → dom (𝑅 ∩ (𝐴 × 𝐴)) = 𝐴)
413, 8, 40syl2anc 692 . . . . . . . . . . . . . 14 (𝜑 → dom (𝑅 ∩ (𝐴 × 𝐴)) = 𝐴)
4241fveq2d 6152 . . . . . . . . . . . . 13 (𝜑 → (TopOn‘dom (𝑅 ∩ (𝐴 × 𝐴))) = (TopOn‘𝐴))
4339, 42eleqtrd 2700 . . . . . . . . . . . 12 (𝜑 → (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ∈ (TopOn‘𝐴))
44 toponmax 20643 . . . . . . . . . . . 12 ((ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ∈ (TopOn‘𝐴) → 𝐴 ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
4543, 44syl 17 . . . . . . . . . . 11 (𝜑𝐴 ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
4636, 45eqeltrd 2698 . . . . . . . . . 10 (𝜑 → (𝑋𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
47 elsni 4165 . . . . . . . . . . . 12 (𝑣 ∈ {𝑋} → 𝑣 = 𝑋)
4847ineq1d 3791 . . . . . . . . . . 11 (𝑣 ∈ {𝑋} → (𝑣𝐴) = (𝑋𝐴))
4948eleq1d 2683 . . . . . . . . . 10 (𝑣 ∈ {𝑋} → ((𝑣𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ↔ (𝑋𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴)))))
5046, 49syl5ibrcom 237 . . . . . . . . 9 (𝜑 → (𝑣 ∈ {𝑋} → (𝑣𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴)))))
5150ralrimiv 2959 . . . . . . . 8 (𝜑 → ∀𝑣 ∈ {𝑋} (𝑣𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
52 ordtrest2.4 . . . . . . . . . 10 ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → {𝑧𝑋 ∣ (𝑥𝑅𝑧𝑧𝑅𝑦)} ⊆ 𝐴)
534, 1, 8, 52ordtrest2lem 20917 . . . . . . . . 9 (𝜑 → ∀𝑣 ∈ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧})(𝑣𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
54 df-rn 5085 . . . . . . . . . . 11 ran 𝑅 = dom 𝑅
55 cnvtsr 17143 . . . . . . . . . . . 12 (𝑅 ∈ TosetRel → 𝑅 ∈ TosetRel )
561, 55syl 17 . . . . . . . . . . 11 (𝜑𝑅 ∈ TosetRel )
574psrn 17130 . . . . . . . . . . . . 13 (𝑅 ∈ PosetRel → 𝑋 = ran 𝑅)
583, 57syl 17 . . . . . . . . . . . 12 (𝜑𝑋 = ran 𝑅)
598, 58sseqtrd 3620 . . . . . . . . . . 11 (𝜑𝐴 ⊆ ran 𝑅)
6058adantr 481 . . . . . . . . . . . . . . 15 ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → 𝑋 = ran 𝑅)
61 rabeq 3179 . . . . . . . . . . . . . . 15 (𝑋 = ran 𝑅 → {𝑧𝑋 ∣ (𝑥𝑅𝑧𝑧𝑅𝑦)} = {𝑧 ∈ ran 𝑅 ∣ (𝑥𝑅𝑧𝑧𝑅𝑦)})
6260, 61syl 17 . . . . . . . . . . . . . 14 ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → {𝑧𝑋 ∣ (𝑥𝑅𝑧𝑧𝑅𝑦)} = {𝑧 ∈ ran 𝑅 ∣ (𝑥𝑅𝑧𝑧𝑅𝑦)})
63 vex 3189 . . . . . . . . . . . . . . . . . 18 𝑦 ∈ V
64 vex 3189 . . . . . . . . . . . . . . . . . 18 𝑧 ∈ V
6563, 64brcnv 5265 . . . . . . . . . . . . . . . . 17 (𝑦𝑅𝑧𝑧𝑅𝑦)
66 vex 3189 . . . . . . . . . . . . . . . . . 18 𝑥 ∈ V
6764, 66brcnv 5265 . . . . . . . . . . . . . . . . 17 (𝑧𝑅𝑥𝑥𝑅𝑧)
6865, 67anbi12ci 733 . . . . . . . . . . . . . . . 16 ((𝑦𝑅𝑧𝑧𝑅𝑥) ↔ (𝑥𝑅𝑧𝑧𝑅𝑦))
6968a1i 11 . . . . . . . . . . . . . . 15 (𝑧 ∈ ran 𝑅 → ((𝑦𝑅𝑧𝑧𝑅𝑥) ↔ (𝑥𝑅𝑧𝑧𝑅𝑦)))
7069rabbiia 3173 . . . . . . . . . . . . . 14 {𝑧 ∈ ran 𝑅 ∣ (𝑦𝑅𝑧𝑧𝑅𝑥)} = {𝑧 ∈ ran 𝑅 ∣ (𝑥𝑅𝑧𝑧𝑅𝑦)}
7162, 70syl6eqr 2673 . . . . . . . . . . . . 13 ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → {𝑧𝑋 ∣ (𝑥𝑅𝑧𝑧𝑅𝑦)} = {𝑧 ∈ ran 𝑅 ∣ (𝑦𝑅𝑧𝑧𝑅𝑥)})
7271, 52eqsstr3d 3619 . . . . . . . . . . . 12 ((𝜑 ∧ (𝑥𝐴𝑦𝐴)) → {𝑧 ∈ ran 𝑅 ∣ (𝑦𝑅𝑧𝑧𝑅𝑥)} ⊆ 𝐴)
7372ancom2s 843 . . . . . . . . . . 11 ((𝜑 ∧ (𝑦𝐴𝑥𝐴)) → {𝑧 ∈ ran 𝑅 ∣ (𝑦𝑅𝑧𝑧𝑅𝑥)} ⊆ 𝐴)
7454, 56, 59, 73ordtrest2lem 20917 . . . . . . . . . 10 (𝜑 → ∀𝑣 ∈ ran (𝑧 ∈ ran 𝑅 ↦ {𝑤 ∈ ran 𝑅 ∣ ¬ 𝑤𝑅𝑧})(𝑣𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
75 vex 3189 . . . . . . . . . . . . . . . . . 18 𝑤 ∈ V
7675, 64brcnv 5265 . . . . . . . . . . . . . . . . 17 (𝑤𝑅𝑧𝑧𝑅𝑤)
7776bicomi 214 . . . . . . . . . . . . . . . 16 (𝑧𝑅𝑤𝑤𝑅𝑧)
7877a1i 11 . . . . . . . . . . . . . . 15 (𝜑 → (𝑧𝑅𝑤𝑤𝑅𝑧))
7978notbid 308 . . . . . . . . . . . . . 14 (𝜑 → (¬ 𝑧𝑅𝑤 ↔ ¬ 𝑤𝑅𝑧))
8058, 79rabeqbidv 3181 . . . . . . . . . . . . 13 (𝜑 → {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤} = {𝑤 ∈ ran 𝑅 ∣ ¬ 𝑤𝑅𝑧})
8158, 80mpteq12dv 4693 . . . . . . . . . . . 12 (𝜑 → (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}) = (𝑧 ∈ ran 𝑅 ↦ {𝑤 ∈ ran 𝑅 ∣ ¬ 𝑤𝑅𝑧}))
8281rneqd 5313 . . . . . . . . . . 11 (𝜑 → ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}) = ran (𝑧 ∈ ran 𝑅 ↦ {𝑤 ∈ ran 𝑅 ∣ ¬ 𝑤𝑅𝑧}))
83 cnvin 5499 . . . . . . . . . . . . . . 15 (𝑅 ∩ (𝐴 × 𝐴)) = (𝑅(𝐴 × 𝐴))
84 cnvxp 5510 . . . . . . . . . . . . . . . 16 (𝐴 × 𝐴) = (𝐴 × 𝐴)
8584ineq2i 3789 . . . . . . . . . . . . . . 15 (𝑅(𝐴 × 𝐴)) = (𝑅 ∩ (𝐴 × 𝐴))
8683, 85eqtri 2643 . . . . . . . . . . . . . 14 (𝑅 ∩ (𝐴 × 𝐴)) = (𝑅 ∩ (𝐴 × 𝐴))
8786fveq2i 6151 . . . . . . . . . . . . 13 (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) = (ordTop‘(𝑅 ∩ (𝐴 × 𝐴)))
88 psss 17135 . . . . . . . . . . . . . . 15 (𝑅 ∈ PosetRel → (𝑅 ∩ (𝐴 × 𝐴)) ∈ PosetRel)
893, 88syl 17 . . . . . . . . . . . . . 14 (𝜑 → (𝑅 ∩ (𝐴 × 𝐴)) ∈ PosetRel)
90 ordtcnv 20915 . . . . . . . . . . . . . 14 ((𝑅 ∩ (𝐴 × 𝐴)) ∈ PosetRel → (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) = (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
9189, 90syl 17 . . . . . . . . . . . . 13 (𝜑 → (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) = (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
9287, 91syl5reqr 2670 . . . . . . . . . . . 12 (𝜑 → (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) = (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
9392eleq2d 2684 . . . . . . . . . . 11 (𝜑 → ((𝑣𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ↔ (𝑣𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴)))))
9482, 93raleqbidv 3141 . . . . . . . . . 10 (𝜑 → (∀𝑣 ∈ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤})(𝑣𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ↔ ∀𝑣 ∈ ran (𝑧 ∈ ran 𝑅 ↦ {𝑤 ∈ ran 𝑅 ∣ ¬ 𝑤𝑅𝑧})(𝑣𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴)))))
9574, 94mpbird 247 . . . . . . . . 9 (𝜑 → ∀𝑣 ∈ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤})(𝑣𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
96 ralunb 3772 . . . . . . . . 9 (∀𝑣 ∈ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}))(𝑣𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ↔ (∀𝑣 ∈ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧})(𝑣𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ∧ ∀𝑣 ∈ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤})(𝑣𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴)))))
9753, 95, 96sylanbrc 697 . . . . . . . 8 (𝜑 → ∀𝑣 ∈ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}))(𝑣𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
98 ralunb 3772 . . . . . . . 8 (∀𝑣 ∈ ({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤})))(𝑣𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ↔ (∀𝑣 ∈ {𝑋} (𝑣𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ∧ ∀𝑣 ∈ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}))(𝑣𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴)))))
9951, 97, 98sylanbrc 697 . . . . . . 7 (𝜑 → ∀𝑣 ∈ ({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤})))(𝑣𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
100 eqid 2621 . . . . . . . 8 (𝑣 ∈ ({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}))) ↦ (𝑣𝐴)) = (𝑣 ∈ ({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}))) ↦ (𝑣𝐴))
101100fmpt 6337 . . . . . . 7 (∀𝑣 ∈ ({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤})))(𝑣𝐴) ∈ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ↔ (𝑣 ∈ ({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}))) ↦ (𝑣𝐴)):({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤})))⟶(ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
10299, 101sylib 208 . . . . . 6 (𝜑 → (𝑣 ∈ ({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}))) ↦ (𝑣𝐴)):({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤})))⟶(ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
103 frn 6010 . . . . . 6 ((𝑣 ∈ ({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}))) ↦ (𝑣𝐴)):({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤})))⟶(ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) → ran (𝑣 ∈ ({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}))) ↦ (𝑣𝐴)) ⊆ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
104102, 103syl 17 . . . . 5 (𝜑 → ran (𝑣 ∈ ({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}))) ↦ (𝑣𝐴)) ⊆ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
10534, 104eqsstrd 3618 . . . 4 (𝜑 → (({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}))) ↾t 𝐴) ⊆ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
106 tgfiss 20706 . . . 4 (((ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) ∈ Top ∧ (({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}))) ↾t 𝐴) ⊆ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴)))) → (topGen‘(fi‘(({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}))) ↾t 𝐴))) ⊆ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
10727, 105, 106syl2anc 692 . . 3 (𝜑 → (topGen‘(fi‘(({𝑋} ∪ (ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑤𝑅𝑧}) ∪ ran (𝑧𝑋 ↦ {𝑤𝑋 ∣ ¬ 𝑧𝑅𝑤}))) ↾t 𝐴))) ⊆ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
10823, 107eqsstrd 3618 . 2 (𝜑 → ((ordTop‘𝑅) ↾t 𝐴) ⊆ (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))))
10911, 108eqssd 3600 1 (𝜑 → (ordTop‘(𝑅 ∩ (𝐴 × 𝐴))) = ((ordTop‘𝑅) ↾t 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  wral 2907  {crab 2911  Vcvv 3186  cun 3553  cin 3554  wss 3555  {csn 4148   cuni 4402   class class class wbr 4613  cmpt 4673   × cxp 5072  ccnv 5073  dom cdm 5074  ran crn 5075  wf 5843  cfv 5847  (class class class)co 6604  ficfi 8260  t crest 16002  topGenctg 16019  ordTopcordt 16080  PosetRelcps 17119   TosetRel ctsr 17120  Topctop 20617  TopOnctopon 20618  TopBasesctb 20620
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-rep 4731  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-reu 2914  df-rab 2916  df-v 3188  df-sbc 3418  df-csb 3515  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-pss 3571  df-nul 3892  df-if 4059  df-pw 4132  df-sn 4149  df-pr 4151  df-tp 4153  df-op 4155  df-uni 4403  df-int 4441  df-iun 4487  df-iin 4488  df-br 4614  df-opab 4674  df-mpt 4675  df-tr 4713  df-eprel 4985  df-id 4989  df-po 4995  df-so 4996  df-fr 5033  df-we 5035  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-pred 5639  df-ord 5685  df-on 5686  df-lim 5687  df-suc 5688  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-fv 5855  df-ov 6607  df-oprab 6608  df-mpt2 6609  df-om 7013  df-1st 7113  df-2nd 7114  df-wrecs 7352  df-recs 7413  df-rdg 7451  df-1o 7505  df-oadd 7509  df-er 7687  df-en 7900  df-dom 7901  df-fin 7903  df-fi 8261  df-rest 16004  df-topgen 16025  df-ordt 16082  df-ps 17121  df-tsr 17122  df-top 20621  df-bases 20622  df-topon 20623
This theorem is referenced by:  ordtrestixx  20936  cnvordtrestixx  29741
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