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Theorem liminfequzmpt2 44118
Description: Two functions that are eventually equal to one another have the same superior limit. (Contributed by Glauco Siliprandi, 2-Jan-2022.)
Hypotheses
Ref Expression
liminfequzmpt2.j β„²π‘—πœ‘
liminfequzmpt2.o Ⅎ𝑗𝐴
liminfequzmpt2.p Ⅎ𝑗𝐡
liminfequzmpt2.a 𝐴 = (β„€β‰₯β€˜π‘€)
liminfequzmpt2.b 𝐡 = (β„€β‰₯β€˜π‘)
liminfequzmpt2.k (πœ‘ β†’ 𝐾 ∈ 𝐴)
liminfequzmpt2.e (πœ‘ β†’ 𝐾 ∈ 𝐡)
liminfequzmpt2.c ((πœ‘ ∧ 𝑗 ∈ (β„€β‰₯β€˜πΎ)) β†’ 𝐢 ∈ 𝑉)
Assertion
Ref Expression
liminfequzmpt2 (πœ‘ β†’ (lim infβ€˜(𝑗 ∈ 𝐴 ↦ 𝐢)) = (lim infβ€˜(𝑗 ∈ 𝐡 ↦ 𝐢)))
Distinct variable group:   𝑗,𝐾
Allowed substitution hints:   πœ‘(𝑗)   𝐴(𝑗)   𝐡(𝑗)   𝐢(𝑗)   𝑀(𝑗)   𝑁(𝑗)   𝑉(𝑗)

Proof of Theorem liminfequzmpt2
StepHypRef Expression
1 liminfequzmpt2.j . . . . . . . . 9 β„²π‘—πœ‘
2 liminfequzmpt2.a . . . . . . . . . . . . . . 15 𝐴 = (β„€β‰₯β€˜π‘€)
3 liminfequzmpt2.k . . . . . . . . . . . . . . 15 (πœ‘ β†’ 𝐾 ∈ 𝐴)
42, 3uzssd2 43738 . . . . . . . . . . . . . 14 (πœ‘ β†’ (β„€β‰₯β€˜πΎ) βŠ† 𝐴)
54adantr 482 . . . . . . . . . . . . 13 ((πœ‘ ∧ 𝑗 ∈ (β„€β‰₯β€˜πΎ)) β†’ (β„€β‰₯β€˜πΎ) βŠ† 𝐴)
6 simpr 486 . . . . . . . . . . . . 13 ((πœ‘ ∧ 𝑗 ∈ (β„€β‰₯β€˜πΎ)) β†’ 𝑗 ∈ (β„€β‰₯β€˜πΎ))
75, 6sseldd 3946 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑗 ∈ (β„€β‰₯β€˜πΎ)) β†’ 𝑗 ∈ 𝐴)
8 liminfequzmpt2.c . . . . . . . . . . . . 13 ((πœ‘ ∧ 𝑗 ∈ (β„€β‰₯β€˜πΎ)) β†’ 𝐢 ∈ 𝑉)
98elexd 3464 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑗 ∈ (β„€β‰₯β€˜πΎ)) β†’ 𝐢 ∈ V)
107, 9jca 513 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑗 ∈ (β„€β‰₯β€˜πΎ)) β†’ (𝑗 ∈ 𝐴 ∧ 𝐢 ∈ V))
11 rabid 3426 . . . . . . . . . . 11 (𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐢 ∈ V} ↔ (𝑗 ∈ 𝐴 ∧ 𝐢 ∈ V))
1210, 11sylibr 233 . . . . . . . . . 10 ((πœ‘ ∧ 𝑗 ∈ (β„€β‰₯β€˜πΎ)) β†’ 𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐢 ∈ V})
1312ex 414 . . . . . . . . 9 (πœ‘ β†’ (𝑗 ∈ (β„€β‰₯β€˜πΎ) β†’ 𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐢 ∈ V}))
141, 13ralrimi 3239 . . . . . . . 8 (πœ‘ β†’ βˆ€π‘— ∈ (β„€β‰₯β€˜πΎ)𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐢 ∈ V})
15 nfcv 2904 . . . . . . . . 9 Ⅎ𝑗(β„€β‰₯β€˜πΎ)
16 nfrab1 3425 . . . . . . . . 9 Ⅎ𝑗{𝑗 ∈ 𝐴 ∣ 𝐢 ∈ V}
1715, 16dfss3f 3936 . . . . . . . 8 ((β„€β‰₯β€˜πΎ) βŠ† {𝑗 ∈ 𝐴 ∣ 𝐢 ∈ V} ↔ βˆ€π‘— ∈ (β„€β‰₯β€˜πΎ)𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐢 ∈ V})
1814, 17sylibr 233 . . . . . . 7 (πœ‘ β†’ (β„€β‰₯β€˜πΎ) βŠ† {𝑗 ∈ 𝐴 ∣ 𝐢 ∈ V})
1916, 15resmptf 5994 . . . . . . 7 ((β„€β‰₯β€˜πΎ) βŠ† {𝑗 ∈ 𝐴 ∣ 𝐢 ∈ V} β†’ ((𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐢 ∈ V} ↦ 𝐢) β†Ύ (β„€β‰₯β€˜πΎ)) = (𝑗 ∈ (β„€β‰₯β€˜πΎ) ↦ 𝐢))
2018, 19syl 17 . . . . . 6 (πœ‘ β†’ ((𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐢 ∈ V} ↦ 𝐢) β†Ύ (β„€β‰₯β€˜πΎ)) = (𝑗 ∈ (β„€β‰₯β€˜πΎ) ↦ 𝐢))
2120eqcomd 2739 . . . . 5 (πœ‘ β†’ (𝑗 ∈ (β„€β‰₯β€˜πΎ) ↦ 𝐢) = ((𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐢 ∈ V} ↦ 𝐢) β†Ύ (β„€β‰₯β€˜πΎ)))
2221fveq2d 6847 . . . 4 (πœ‘ β†’ (lim infβ€˜(𝑗 ∈ (β„€β‰₯β€˜πΎ) ↦ 𝐢)) = (lim infβ€˜((𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐢 ∈ V} ↦ 𝐢) β†Ύ (β„€β‰₯β€˜πΎ))))
232, 3eluzelz2d 43734 . . . . 5 (πœ‘ β†’ 𝐾 ∈ β„€)
24 eqid 2733 . . . . 5 (β„€β‰₯β€˜πΎ) = (β„€β‰₯β€˜πΎ)
25 liminfequzmpt2.o . . . . . . . 8 Ⅎ𝑗𝐴
262fvexi 6857 . . . . . . . 8 𝐴 ∈ V
2725, 26rabexf 43432 . . . . . . 7 {𝑗 ∈ 𝐴 ∣ 𝐢 ∈ V} ∈ V
2816, 27mptexf 43550 . . . . . 6 (𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐢 ∈ V} ↦ 𝐢) ∈ V
2928a1i 11 . . . . 5 (πœ‘ β†’ (𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐢 ∈ V} ↦ 𝐢) ∈ V)
30 eqid 2733 . . . . . . . 8 (𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐢 ∈ V} ↦ 𝐢) = (𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐢 ∈ V} ↦ 𝐢)
3116, 30dmmptssf 43544 . . . . . . 7 dom (𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐢 ∈ V} ↦ 𝐢) βŠ† {𝑗 ∈ 𝐴 ∣ 𝐢 ∈ V}
3225ssrab2f 43415 . . . . . . . 8 {𝑗 ∈ 𝐴 ∣ 𝐢 ∈ V} βŠ† 𝐴
33 uzssz 12789 . . . . . . . . 9 (β„€β‰₯β€˜π‘€) βŠ† β„€
342, 33eqsstri 3979 . . . . . . . 8 𝐴 βŠ† β„€
3532, 34sstri 3954 . . . . . . 7 {𝑗 ∈ 𝐴 ∣ 𝐢 ∈ V} βŠ† β„€
3631, 35sstri 3954 . . . . . 6 dom (𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐢 ∈ V} ↦ 𝐢) βŠ† β„€
3736a1i 11 . . . . 5 (πœ‘ β†’ dom (𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐢 ∈ V} ↦ 𝐢) βŠ† β„€)
3823, 24, 29, 37liminfresuz2 44114 . . . 4 (πœ‘ β†’ (lim infβ€˜((𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐢 ∈ V} ↦ 𝐢) β†Ύ (β„€β‰₯β€˜πΎ))) = (lim infβ€˜(𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐢 ∈ V} ↦ 𝐢)))
3922, 38eqtr2d 2774 . . 3 (πœ‘ β†’ (lim infβ€˜(𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐢 ∈ V} ↦ 𝐢)) = (lim infβ€˜(𝑗 ∈ (β„€β‰₯β€˜πΎ) ↦ 𝐢)))
40 liminfequzmpt2.b . . . . . . . . . . . . . . 15 𝐡 = (β„€β‰₯β€˜π‘)
41 liminfequzmpt2.e . . . . . . . . . . . . . . 15 (πœ‘ β†’ 𝐾 ∈ 𝐡)
4240, 41uzssd2 43738 . . . . . . . . . . . . . 14 (πœ‘ β†’ (β„€β‰₯β€˜πΎ) βŠ† 𝐡)
4342adantr 482 . . . . . . . . . . . . 13 ((πœ‘ ∧ 𝑗 ∈ (β„€β‰₯β€˜πΎ)) β†’ (β„€β‰₯β€˜πΎ) βŠ† 𝐡)
4443, 6sseldd 3946 . . . . . . . . . . . 12 ((πœ‘ ∧ 𝑗 ∈ (β„€β‰₯β€˜πΎ)) β†’ 𝑗 ∈ 𝐡)
4544, 9jca 513 . . . . . . . . . . 11 ((πœ‘ ∧ 𝑗 ∈ (β„€β‰₯β€˜πΎ)) β†’ (𝑗 ∈ 𝐡 ∧ 𝐢 ∈ V))
46 rabid 3426 . . . . . . . . . . 11 (𝑗 ∈ {𝑗 ∈ 𝐡 ∣ 𝐢 ∈ V} ↔ (𝑗 ∈ 𝐡 ∧ 𝐢 ∈ V))
4745, 46sylibr 233 . . . . . . . . . 10 ((πœ‘ ∧ 𝑗 ∈ (β„€β‰₯β€˜πΎ)) β†’ 𝑗 ∈ {𝑗 ∈ 𝐡 ∣ 𝐢 ∈ V})
4847ex 414 . . . . . . . . 9 (πœ‘ β†’ (𝑗 ∈ (β„€β‰₯β€˜πΎ) β†’ 𝑗 ∈ {𝑗 ∈ 𝐡 ∣ 𝐢 ∈ V}))
491, 48ralrimi 3239 . . . . . . . 8 (πœ‘ β†’ βˆ€π‘— ∈ (β„€β‰₯β€˜πΎ)𝑗 ∈ {𝑗 ∈ 𝐡 ∣ 𝐢 ∈ V})
50 nfrab1 3425 . . . . . . . . 9 Ⅎ𝑗{𝑗 ∈ 𝐡 ∣ 𝐢 ∈ V}
5115, 50dfss3f 3936 . . . . . . . 8 ((β„€β‰₯β€˜πΎ) βŠ† {𝑗 ∈ 𝐡 ∣ 𝐢 ∈ V} ↔ βˆ€π‘— ∈ (β„€β‰₯β€˜πΎ)𝑗 ∈ {𝑗 ∈ 𝐡 ∣ 𝐢 ∈ V})
5249, 51sylibr 233 . . . . . . 7 (πœ‘ β†’ (β„€β‰₯β€˜πΎ) βŠ† {𝑗 ∈ 𝐡 ∣ 𝐢 ∈ V})
5350, 15resmptf 5994 . . . . . . 7 ((β„€β‰₯β€˜πΎ) βŠ† {𝑗 ∈ 𝐡 ∣ 𝐢 ∈ V} β†’ ((𝑗 ∈ {𝑗 ∈ 𝐡 ∣ 𝐢 ∈ V} ↦ 𝐢) β†Ύ (β„€β‰₯β€˜πΎ)) = (𝑗 ∈ (β„€β‰₯β€˜πΎ) ↦ 𝐢))
5452, 53syl 17 . . . . . 6 (πœ‘ β†’ ((𝑗 ∈ {𝑗 ∈ 𝐡 ∣ 𝐢 ∈ V} ↦ 𝐢) β†Ύ (β„€β‰₯β€˜πΎ)) = (𝑗 ∈ (β„€β‰₯β€˜πΎ) ↦ 𝐢))
5554eqcomd 2739 . . . . 5 (πœ‘ β†’ (𝑗 ∈ (β„€β‰₯β€˜πΎ) ↦ 𝐢) = ((𝑗 ∈ {𝑗 ∈ 𝐡 ∣ 𝐢 ∈ V} ↦ 𝐢) β†Ύ (β„€β‰₯β€˜πΎ)))
5655fveq2d 6847 . . . 4 (πœ‘ β†’ (lim infβ€˜(𝑗 ∈ (β„€β‰₯β€˜πΎ) ↦ 𝐢)) = (lim infβ€˜((𝑗 ∈ {𝑗 ∈ 𝐡 ∣ 𝐢 ∈ V} ↦ 𝐢) β†Ύ (β„€β‰₯β€˜πΎ))))
57 liminfequzmpt2.p . . . . . . . 8 Ⅎ𝑗𝐡
5840fvexi 6857 . . . . . . . 8 𝐡 ∈ V
5957, 58rabexf 43432 . . . . . . 7 {𝑗 ∈ 𝐡 ∣ 𝐢 ∈ V} ∈ V
6050, 59mptexf 43550 . . . . . 6 (𝑗 ∈ {𝑗 ∈ 𝐡 ∣ 𝐢 ∈ V} ↦ 𝐢) ∈ V
6160a1i 11 . . . . 5 (πœ‘ β†’ (𝑗 ∈ {𝑗 ∈ 𝐡 ∣ 𝐢 ∈ V} ↦ 𝐢) ∈ V)
62 eqid 2733 . . . . . . . 8 (𝑗 ∈ {𝑗 ∈ 𝐡 ∣ 𝐢 ∈ V} ↦ 𝐢) = (𝑗 ∈ {𝑗 ∈ 𝐡 ∣ 𝐢 ∈ V} ↦ 𝐢)
6350, 62dmmptssf 43544 . . . . . . 7 dom (𝑗 ∈ {𝑗 ∈ 𝐡 ∣ 𝐢 ∈ V} ↦ 𝐢) βŠ† {𝑗 ∈ 𝐡 ∣ 𝐢 ∈ V}
6457ssrab2f 43415 . . . . . . . 8 {𝑗 ∈ 𝐡 ∣ 𝐢 ∈ V} βŠ† 𝐡
65 uzssz 12789 . . . . . . . . 9 (β„€β‰₯β€˜π‘) βŠ† β„€
6640, 65eqsstri 3979 . . . . . . . 8 𝐡 βŠ† β„€
6764, 66sstri 3954 . . . . . . 7 {𝑗 ∈ 𝐡 ∣ 𝐢 ∈ V} βŠ† β„€
6863, 67sstri 3954 . . . . . 6 dom (𝑗 ∈ {𝑗 ∈ 𝐡 ∣ 𝐢 ∈ V} ↦ 𝐢) βŠ† β„€
6968a1i 11 . . . . 5 (πœ‘ β†’ dom (𝑗 ∈ {𝑗 ∈ 𝐡 ∣ 𝐢 ∈ V} ↦ 𝐢) βŠ† β„€)
7023, 24, 61, 69liminfresuz2 44114 . . . 4 (πœ‘ β†’ (lim infβ€˜((𝑗 ∈ {𝑗 ∈ 𝐡 ∣ 𝐢 ∈ V} ↦ 𝐢) β†Ύ (β„€β‰₯β€˜πΎ))) = (lim infβ€˜(𝑗 ∈ {𝑗 ∈ 𝐡 ∣ 𝐢 ∈ V} ↦ 𝐢)))
7156, 70eqtr2d 2774 . . 3 (πœ‘ β†’ (lim infβ€˜(𝑗 ∈ {𝑗 ∈ 𝐡 ∣ 𝐢 ∈ V} ↦ 𝐢)) = (lim infβ€˜(𝑗 ∈ (β„€β‰₯β€˜πΎ) ↦ 𝐢)))
7239, 71eqtr4d 2776 . 2 (πœ‘ β†’ (lim infβ€˜(𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐢 ∈ V} ↦ 𝐢)) = (lim infβ€˜(𝑗 ∈ {𝑗 ∈ 𝐡 ∣ 𝐢 ∈ V} ↦ 𝐢)))
73 eqid 2733 . . . . 5 {𝑗 ∈ 𝐴 ∣ 𝐢 ∈ V} = {𝑗 ∈ 𝐴 ∣ 𝐢 ∈ V}
7425, 73mptssid 43554 . . . 4 (𝑗 ∈ 𝐴 ↦ 𝐢) = (𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐢 ∈ V} ↦ 𝐢)
7574fveq2i 6846 . . 3 (lim infβ€˜(𝑗 ∈ 𝐴 ↦ 𝐢)) = (lim infβ€˜(𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐢 ∈ V} ↦ 𝐢))
7675a1i 11 . 2 (πœ‘ β†’ (lim infβ€˜(𝑗 ∈ 𝐴 ↦ 𝐢)) = (lim infβ€˜(𝑗 ∈ {𝑗 ∈ 𝐴 ∣ 𝐢 ∈ V} ↦ 𝐢)))
77 eqid 2733 . . . . 5 {𝑗 ∈ 𝐡 ∣ 𝐢 ∈ V} = {𝑗 ∈ 𝐡 ∣ 𝐢 ∈ V}
7857, 77mptssid 43554 . . . 4 (𝑗 ∈ 𝐡 ↦ 𝐢) = (𝑗 ∈ {𝑗 ∈ 𝐡 ∣ 𝐢 ∈ V} ↦ 𝐢)
7978fveq2i 6846 . . 3 (lim infβ€˜(𝑗 ∈ 𝐡 ↦ 𝐢)) = (lim infβ€˜(𝑗 ∈ {𝑗 ∈ 𝐡 ∣ 𝐢 ∈ V} ↦ 𝐢))
8079a1i 11 . 2 (πœ‘ β†’ (lim infβ€˜(𝑗 ∈ 𝐡 ↦ 𝐢)) = (lim infβ€˜(𝑗 ∈ {𝑗 ∈ 𝐡 ∣ 𝐢 ∈ V} ↦ 𝐢)))
8172, 76, 803eqtr4d 2783 1 (πœ‘ β†’ (lim infβ€˜(𝑗 ∈ 𝐴 ↦ 𝐢)) = (lim infβ€˜(𝑗 ∈ 𝐡 ↦ 𝐢)))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ∧ wa 397   = wceq 1542  β„²wnf 1786   ∈ wcel 2107  β„²wnfc 2884  βˆ€wral 3061  {crab 3406  Vcvv 3444   βŠ† wss 3911   ↦ cmpt 5189  dom cdm 5634   β†Ύ cres 5636  β€˜cfv 6497  β„€cz 12504  β„€β‰₯cuz 12768  lim infclsi 44078
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 2704  ax-rep 5243  ax-sep 5257  ax-nul 5264  ax-pow 5321  ax-pr 5385  ax-un 7673  ax-cnex 11112  ax-resscn 11113  ax-1cn 11114  ax-icn 11115  ax-addcl 11116  ax-addrcl 11117  ax-mulcl 11118  ax-mulrcl 11119  ax-mulcom 11120  ax-addass 11121  ax-mulass 11122  ax-distr 11123  ax-i2m1 11124  ax-1ne0 11125  ax-1rid 11126  ax-rnegex 11127  ax-rrecex 11128  ax-cnre 11129  ax-pre-lttri 11130  ax-pre-lttrn 11131  ax-pre-ltadd 11132  ax-pre-mulgt0 11133  ax-pre-sup 11134
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-nel 3047  df-ral 3062  df-rex 3071  df-rmo 3352  df-reu 3353  df-rab 3407  df-v 3446  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-pss 3930  df-nul 4284  df-if 4488  df-pw 4563  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-tr 5224  df-id 5532  df-eprel 5538  df-po 5546  df-so 5547  df-fr 5589  df-we 5591  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-pred 6254  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-riota 7314  df-ov 7361  df-oprab 7362  df-mpo 7363  df-om 7804  df-1st 7922  df-2nd 7923  df-frecs 8213  df-wrecs 8244  df-recs 8318  df-rdg 8357  df-er 8651  df-en 8887  df-dom 8888  df-sdom 8889  df-sup 9383  df-inf 9384  df-pnf 11196  df-mnf 11197  df-xr 11198  df-ltxr 11199  df-le 11200  df-sub 11392  df-neg 11393  df-div 11818  df-nn 12159  df-n0 12419  df-z 12505  df-uz 12769  df-q 12879  df-ico 13276  df-liminf 44079
This theorem is referenced by:  smfliminfmpt  45159
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