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Theorem sssmf 45065
Description: The restriction of a sigma-measurable function, is sigma-measurable. (Contributed by Glauco Siliprandi, 26-Jun-2021.)
Hypotheses
Ref Expression
sssmf.s (πœ‘ β†’ 𝑆 ∈ SAlg)
sssmf.f (πœ‘ β†’ 𝐹 ∈ (SMblFnβ€˜π‘†))
Assertion
Ref Expression
sssmf (πœ‘ β†’ (𝐹 β†Ύ 𝐡) ∈ (SMblFnβ€˜π‘†))

Proof of Theorem sssmf
Dummy variables π‘Ž 𝑀 π‘₯ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nfv 1918 . 2 β„²π‘Žπœ‘
2 sssmf.s . 2 (πœ‘ β†’ 𝑆 ∈ SAlg)
3 inss2 4190 . . 3 (𝐡 ∩ dom 𝐹) βŠ† dom 𝐹
4 sssmf.f . . . 4 (πœ‘ β†’ 𝐹 ∈ (SMblFnβ€˜π‘†))
5 eqid 2733 . . . 4 dom 𝐹 = dom 𝐹
62, 4, 5smfdmss 45060 . . 3 (πœ‘ β†’ dom 𝐹 βŠ† βˆͺ 𝑆)
73, 6sstrid 3956 . 2 (πœ‘ β†’ (𝐡 ∩ dom 𝐹) βŠ† βˆͺ 𝑆)
82, 4, 5smff 45059 . . . . 5 (πœ‘ β†’ 𝐹:dom πΉβŸΆβ„)
93a1i 11 . . . . 5 (πœ‘ β†’ (𝐡 ∩ dom 𝐹) βŠ† dom 𝐹)
10 fssres 6709 . . . . 5 ((𝐹:dom πΉβŸΆβ„ ∧ (𝐡 ∩ dom 𝐹) βŠ† dom 𝐹) β†’ (𝐹 β†Ύ (𝐡 ∩ dom 𝐹)):(𝐡 ∩ dom 𝐹)βŸΆβ„)
118, 9, 10syl2anc 585 . . . 4 (πœ‘ β†’ (𝐹 β†Ύ (𝐡 ∩ dom 𝐹)):(𝐡 ∩ dom 𝐹)βŸΆβ„)
128freld 6675 . . . . . . 7 (πœ‘ β†’ Rel 𝐹)
13 resindm 5987 . . . . . . 7 (Rel 𝐹 β†’ (𝐹 β†Ύ (𝐡 ∩ dom 𝐹)) = (𝐹 β†Ύ 𝐡))
1412, 13syl 17 . . . . . 6 (πœ‘ β†’ (𝐹 β†Ύ (𝐡 ∩ dom 𝐹)) = (𝐹 β†Ύ 𝐡))
1514eqcomd 2739 . . . . 5 (πœ‘ β†’ (𝐹 β†Ύ 𝐡) = (𝐹 β†Ύ (𝐡 ∩ dom 𝐹)))
16 dmres 5960 . . . . . 6 dom (𝐹 β†Ύ 𝐡) = (𝐡 ∩ dom 𝐹)
1716a1i 11 . . . . 5 (πœ‘ β†’ dom (𝐹 β†Ύ 𝐡) = (𝐡 ∩ dom 𝐹))
1815, 17feq12d 6657 . . . 4 (πœ‘ β†’ ((𝐹 β†Ύ 𝐡):dom (𝐹 β†Ύ 𝐡)βŸΆβ„ ↔ (𝐹 β†Ύ (𝐡 ∩ dom 𝐹)):(𝐡 ∩ dom 𝐹)βŸΆβ„))
1911, 18mpbird 257 . . 3 (πœ‘ β†’ (𝐹 β†Ύ 𝐡):dom (𝐹 β†Ύ 𝐡)βŸΆβ„)
2017feq2d 6655 . . 3 (πœ‘ β†’ ((𝐹 β†Ύ 𝐡):dom (𝐹 β†Ύ 𝐡)βŸΆβ„ ↔ (𝐹 β†Ύ 𝐡):(𝐡 ∩ dom 𝐹)βŸΆβ„))
2119, 20mpbid 231 . 2 (πœ‘ β†’ (𝐹 β†Ύ 𝐡):(𝐡 ∩ dom 𝐹)βŸΆβ„)
222adantr 482 . . . . 5 ((πœ‘ ∧ π‘Ž ∈ ℝ) β†’ 𝑆 ∈ SAlg)
234adantr 482 . . . . 5 ((πœ‘ ∧ π‘Ž ∈ ℝ) β†’ 𝐹 ∈ (SMblFnβ€˜π‘†))
24 simpr 486 . . . . 5 ((πœ‘ ∧ π‘Ž ∈ ℝ) β†’ π‘Ž ∈ ℝ)
2522, 23, 5, 24smfpreimalt 45058 . . . 4 ((πœ‘ ∧ π‘Ž ∈ ℝ) β†’ {π‘₯ ∈ dom 𝐹 ∣ (πΉβ€˜π‘₯) < π‘Ž} ∈ (𝑆 β†Ύt dom 𝐹))
264dmexd 7843 . . . . . 6 (πœ‘ β†’ dom 𝐹 ∈ V)
27 elrest 17314 . . . . . 6 ((𝑆 ∈ SAlg ∧ dom 𝐹 ∈ V) β†’ ({π‘₯ ∈ dom 𝐹 ∣ (πΉβ€˜π‘₯) < π‘Ž} ∈ (𝑆 β†Ύt dom 𝐹) ↔ βˆƒπ‘€ ∈ 𝑆 {π‘₯ ∈ dom 𝐹 ∣ (πΉβ€˜π‘₯) < π‘Ž} = (𝑀 ∩ dom 𝐹)))
282, 26, 27syl2anc 585 . . . . 5 (πœ‘ β†’ ({π‘₯ ∈ dom 𝐹 ∣ (πΉβ€˜π‘₯) < π‘Ž} ∈ (𝑆 β†Ύt dom 𝐹) ↔ βˆƒπ‘€ ∈ 𝑆 {π‘₯ ∈ dom 𝐹 ∣ (πΉβ€˜π‘₯) < π‘Ž} = (𝑀 ∩ dom 𝐹)))
2928adantr 482 . . . 4 ((πœ‘ ∧ π‘Ž ∈ ℝ) β†’ ({π‘₯ ∈ dom 𝐹 ∣ (πΉβ€˜π‘₯) < π‘Ž} ∈ (𝑆 β†Ύt dom 𝐹) ↔ βˆƒπ‘€ ∈ 𝑆 {π‘₯ ∈ dom 𝐹 ∣ (πΉβ€˜π‘₯) < π‘Ž} = (𝑀 ∩ dom 𝐹)))
3025, 29mpbid 231 . . 3 ((πœ‘ ∧ π‘Ž ∈ ℝ) β†’ βˆƒπ‘€ ∈ 𝑆 {π‘₯ ∈ dom 𝐹 ∣ (πΉβ€˜π‘₯) < π‘Ž} = (𝑀 ∩ dom 𝐹))
31 elinel1 4156 . . . . . . . . . . . . 13 (π‘₯ ∈ (𝐡 ∩ dom 𝐹) β†’ π‘₯ ∈ 𝐡)
3231fvresd 6863 . . . . . . . . . . . 12 (π‘₯ ∈ (𝐡 ∩ dom 𝐹) β†’ ((𝐹 β†Ύ 𝐡)β€˜π‘₯) = (πΉβ€˜π‘₯))
3332breq1d 5116 . . . . . . . . . . 11 (π‘₯ ∈ (𝐡 ∩ dom 𝐹) β†’ (((𝐹 β†Ύ 𝐡)β€˜π‘₯) < π‘Ž ↔ (πΉβ€˜π‘₯) < π‘Ž))
3433rabbiia 3410 . . . . . . . . . 10 {π‘₯ ∈ (𝐡 ∩ dom 𝐹) ∣ ((𝐹 β†Ύ 𝐡)β€˜π‘₯) < π‘Ž} = {π‘₯ ∈ (𝐡 ∩ dom 𝐹) ∣ (πΉβ€˜π‘₯) < π‘Ž}
3534a1i 11 . . . . . . . . 9 ({π‘₯ ∈ dom 𝐹 ∣ (πΉβ€˜π‘₯) < π‘Ž} = (𝑀 ∩ dom 𝐹) β†’ {π‘₯ ∈ (𝐡 ∩ dom 𝐹) ∣ ((𝐹 β†Ύ 𝐡)β€˜π‘₯) < π‘Ž} = {π‘₯ ∈ (𝐡 ∩ dom 𝐹) ∣ (πΉβ€˜π‘₯) < π‘Ž})
36 rabss2 4036 . . . . . . . . . . . . 13 ((𝐡 ∩ dom 𝐹) βŠ† dom 𝐹 β†’ {π‘₯ ∈ (𝐡 ∩ dom 𝐹) ∣ (πΉβ€˜π‘₯) < π‘Ž} βŠ† {π‘₯ ∈ dom 𝐹 ∣ (πΉβ€˜π‘₯) < π‘Ž})
373, 36ax-mp 5 . . . . . . . . . . . 12 {π‘₯ ∈ (𝐡 ∩ dom 𝐹) ∣ (πΉβ€˜π‘₯) < π‘Ž} βŠ† {π‘₯ ∈ dom 𝐹 ∣ (πΉβ€˜π‘₯) < π‘Ž}
38 id 22 . . . . . . . . . . . . 13 ({π‘₯ ∈ dom 𝐹 ∣ (πΉβ€˜π‘₯) < π‘Ž} = (𝑀 ∩ dom 𝐹) β†’ {π‘₯ ∈ dom 𝐹 ∣ (πΉβ€˜π‘₯) < π‘Ž} = (𝑀 ∩ dom 𝐹))
39 inss1 4189 . . . . . . . . . . . . . 14 (𝑀 ∩ dom 𝐹) βŠ† 𝑀
4039a1i 11 . . . . . . . . . . . . 13 ({π‘₯ ∈ dom 𝐹 ∣ (πΉβ€˜π‘₯) < π‘Ž} = (𝑀 ∩ dom 𝐹) β†’ (𝑀 ∩ dom 𝐹) βŠ† 𝑀)
4138, 40eqsstrd 3983 . . . . . . . . . . . 12 ({π‘₯ ∈ dom 𝐹 ∣ (πΉβ€˜π‘₯) < π‘Ž} = (𝑀 ∩ dom 𝐹) β†’ {π‘₯ ∈ dom 𝐹 ∣ (πΉβ€˜π‘₯) < π‘Ž} βŠ† 𝑀)
4237, 41sstrid 3956 . . . . . . . . . . 11 ({π‘₯ ∈ dom 𝐹 ∣ (πΉβ€˜π‘₯) < π‘Ž} = (𝑀 ∩ dom 𝐹) β†’ {π‘₯ ∈ (𝐡 ∩ dom 𝐹) ∣ (πΉβ€˜π‘₯) < π‘Ž} βŠ† 𝑀)
43 ssrab2 4038 . . . . . . . . . . . 12 {π‘₯ ∈ (𝐡 ∩ dom 𝐹) ∣ (πΉβ€˜π‘₯) < π‘Ž} βŠ† (𝐡 ∩ dom 𝐹)
4443a1i 11 . . . . . . . . . . 11 ({π‘₯ ∈ dom 𝐹 ∣ (πΉβ€˜π‘₯) < π‘Ž} = (𝑀 ∩ dom 𝐹) β†’ {π‘₯ ∈ (𝐡 ∩ dom 𝐹) ∣ (πΉβ€˜π‘₯) < π‘Ž} βŠ† (𝐡 ∩ dom 𝐹))
4542, 44ssind 4193 . . . . . . . . . 10 ({π‘₯ ∈ dom 𝐹 ∣ (πΉβ€˜π‘₯) < π‘Ž} = (𝑀 ∩ dom 𝐹) β†’ {π‘₯ ∈ (𝐡 ∩ dom 𝐹) ∣ (πΉβ€˜π‘₯) < π‘Ž} βŠ† (𝑀 ∩ (𝐡 ∩ dom 𝐹)))
46 nfrab1 3425 . . . . . . . . . . . . 13 β„²π‘₯{π‘₯ ∈ dom 𝐹 ∣ (πΉβ€˜π‘₯) < π‘Ž}
47 nfcv 2904 . . . . . . . . . . . . 13 β„²π‘₯(𝑀 ∩ dom 𝐹)
4846, 47nfeq 2917 . . . . . . . . . . . 12 β„²π‘₯{π‘₯ ∈ dom 𝐹 ∣ (πΉβ€˜π‘₯) < π‘Ž} = (𝑀 ∩ dom 𝐹)
49 elinel2 4157 . . . . . . . . . . . . . . . . . . 19 (π‘₯ ∈ (𝑀 ∩ (𝐡 ∩ dom 𝐹)) β†’ π‘₯ ∈ (𝐡 ∩ dom 𝐹))
5049adantl 483 . . . . . . . . . . . . . . . . . 18 (({π‘₯ ∈ dom 𝐹 ∣ (πΉβ€˜π‘₯) < π‘Ž} = (𝑀 ∩ dom 𝐹) ∧ π‘₯ ∈ (𝑀 ∩ (𝐡 ∩ dom 𝐹))) β†’ π‘₯ ∈ (𝐡 ∩ dom 𝐹))
51 elinel1 4156 . . . . . . . . . . . . . . . . . . . . . 22 (π‘₯ ∈ (𝑀 ∩ (𝐡 ∩ dom 𝐹)) β†’ π‘₯ ∈ 𝑀)
523sseli 3941 . . . . . . . . . . . . . . . . . . . . . . 23 (π‘₯ ∈ (𝐡 ∩ dom 𝐹) β†’ π‘₯ ∈ dom 𝐹)
5349, 52syl 17 . . . . . . . . . . . . . . . . . . . . . 22 (π‘₯ ∈ (𝑀 ∩ (𝐡 ∩ dom 𝐹)) β†’ π‘₯ ∈ dom 𝐹)
5451, 53elind 4155 . . . . . . . . . . . . . . . . . . . . 21 (π‘₯ ∈ (𝑀 ∩ (𝐡 ∩ dom 𝐹)) β†’ π‘₯ ∈ (𝑀 ∩ dom 𝐹))
5554adantl 483 . . . . . . . . . . . . . . . . . . . 20 (({π‘₯ ∈ dom 𝐹 ∣ (πΉβ€˜π‘₯) < π‘Ž} = (𝑀 ∩ dom 𝐹) ∧ π‘₯ ∈ (𝑀 ∩ (𝐡 ∩ dom 𝐹))) β†’ π‘₯ ∈ (𝑀 ∩ dom 𝐹))
5638eqcomd 2739 . . . . . . . . . . . . . . . . . . . . 21 ({π‘₯ ∈ dom 𝐹 ∣ (πΉβ€˜π‘₯) < π‘Ž} = (𝑀 ∩ dom 𝐹) β†’ (𝑀 ∩ dom 𝐹) = {π‘₯ ∈ dom 𝐹 ∣ (πΉβ€˜π‘₯) < π‘Ž})
5756adantr 482 . . . . . . . . . . . . . . . . . . . 20 (({π‘₯ ∈ dom 𝐹 ∣ (πΉβ€˜π‘₯) < π‘Ž} = (𝑀 ∩ dom 𝐹) ∧ π‘₯ ∈ (𝑀 ∩ (𝐡 ∩ dom 𝐹))) β†’ (𝑀 ∩ dom 𝐹) = {π‘₯ ∈ dom 𝐹 ∣ (πΉβ€˜π‘₯) < π‘Ž})
5855, 57eleqtrd 2836 . . . . . . . . . . . . . . . . . . 19 (({π‘₯ ∈ dom 𝐹 ∣ (πΉβ€˜π‘₯) < π‘Ž} = (𝑀 ∩ dom 𝐹) ∧ π‘₯ ∈ (𝑀 ∩ (𝐡 ∩ dom 𝐹))) β†’ π‘₯ ∈ {π‘₯ ∈ dom 𝐹 ∣ (πΉβ€˜π‘₯) < π‘Ž})
59 rabid 3426 . . . . . . . . . . . . . . . . . . . . 21 (π‘₯ ∈ {π‘₯ ∈ dom 𝐹 ∣ (πΉβ€˜π‘₯) < π‘Ž} ↔ (π‘₯ ∈ dom 𝐹 ∧ (πΉβ€˜π‘₯) < π‘Ž))
6059biimpi 215 . . . . . . . . . . . . . . . . . . . 20 (π‘₯ ∈ {π‘₯ ∈ dom 𝐹 ∣ (πΉβ€˜π‘₯) < π‘Ž} β†’ (π‘₯ ∈ dom 𝐹 ∧ (πΉβ€˜π‘₯) < π‘Ž))
6160simprd 497 . . . . . . . . . . . . . . . . . . 19 (π‘₯ ∈ {π‘₯ ∈ dom 𝐹 ∣ (πΉβ€˜π‘₯) < π‘Ž} β†’ (πΉβ€˜π‘₯) < π‘Ž)
6258, 61syl 17 . . . . . . . . . . . . . . . . . 18 (({π‘₯ ∈ dom 𝐹 ∣ (πΉβ€˜π‘₯) < π‘Ž} = (𝑀 ∩ dom 𝐹) ∧ π‘₯ ∈ (𝑀 ∩ (𝐡 ∩ dom 𝐹))) β†’ (πΉβ€˜π‘₯) < π‘Ž)
6350, 62jca 513 . . . . . . . . . . . . . . . . 17 (({π‘₯ ∈ dom 𝐹 ∣ (πΉβ€˜π‘₯) < π‘Ž} = (𝑀 ∩ dom 𝐹) ∧ π‘₯ ∈ (𝑀 ∩ (𝐡 ∩ dom 𝐹))) β†’ (π‘₯ ∈ (𝐡 ∩ dom 𝐹) ∧ (πΉβ€˜π‘₯) < π‘Ž))
64 rabid 3426 . . . . . . . . . . . . . . . . 17 (π‘₯ ∈ {π‘₯ ∈ (𝐡 ∩ dom 𝐹) ∣ (πΉβ€˜π‘₯) < π‘Ž} ↔ (π‘₯ ∈ (𝐡 ∩ dom 𝐹) ∧ (πΉβ€˜π‘₯) < π‘Ž))
6563, 64sylibr 233 . . . . . . . . . . . . . . . 16 (({π‘₯ ∈ dom 𝐹 ∣ (πΉβ€˜π‘₯) < π‘Ž} = (𝑀 ∩ dom 𝐹) ∧ π‘₯ ∈ (𝑀 ∩ (𝐡 ∩ dom 𝐹))) β†’ π‘₯ ∈ {π‘₯ ∈ (𝐡 ∩ dom 𝐹) ∣ (πΉβ€˜π‘₯) < π‘Ž})
6665ex 414 . . . . . . . . . . . . . . 15 ({π‘₯ ∈ dom 𝐹 ∣ (πΉβ€˜π‘₯) < π‘Ž} = (𝑀 ∩ dom 𝐹) β†’ (π‘₯ ∈ (𝑀 ∩ (𝐡 ∩ dom 𝐹)) β†’ π‘₯ ∈ {π‘₯ ∈ (𝐡 ∩ dom 𝐹) ∣ (πΉβ€˜π‘₯) < π‘Ž}))
6748, 66ralrimi 3239 . . . . . . . . . . . . . 14 ({π‘₯ ∈ dom 𝐹 ∣ (πΉβ€˜π‘₯) < π‘Ž} = (𝑀 ∩ dom 𝐹) β†’ βˆ€π‘₯ ∈ (𝑀 ∩ (𝐡 ∩ dom 𝐹))π‘₯ ∈ {π‘₯ ∈ (𝐡 ∩ dom 𝐹) ∣ (πΉβ€˜π‘₯) < π‘Ž})
68 nfcv 2904 . . . . . . . . . . . . . . 15 β„²π‘₯(𝑀 ∩ (𝐡 ∩ dom 𝐹))
69 nfrab1 3425 . . . . . . . . . . . . . . 15 β„²π‘₯{π‘₯ ∈ (𝐡 ∩ dom 𝐹) ∣ (πΉβ€˜π‘₯) < π‘Ž}
7068, 69dfss3f 3936 . . . . . . . . . . . . . 14 ((𝑀 ∩ (𝐡 ∩ dom 𝐹)) βŠ† {π‘₯ ∈ (𝐡 ∩ dom 𝐹) ∣ (πΉβ€˜π‘₯) < π‘Ž} ↔ βˆ€π‘₯ ∈ (𝑀 ∩ (𝐡 ∩ dom 𝐹))π‘₯ ∈ {π‘₯ ∈ (𝐡 ∩ dom 𝐹) ∣ (πΉβ€˜π‘₯) < π‘Ž})
7167, 70sylibr 233 . . . . . . . . . . . . 13 ({π‘₯ ∈ dom 𝐹 ∣ (πΉβ€˜π‘₯) < π‘Ž} = (𝑀 ∩ dom 𝐹) β†’ (𝑀 ∩ (𝐡 ∩ dom 𝐹)) βŠ† {π‘₯ ∈ (𝐡 ∩ dom 𝐹) ∣ (πΉβ€˜π‘₯) < π‘Ž})
7271sseld 3944 . . . . . . . . . . . 12 ({π‘₯ ∈ dom 𝐹 ∣ (πΉβ€˜π‘₯) < π‘Ž} = (𝑀 ∩ dom 𝐹) β†’ (π‘₯ ∈ (𝑀 ∩ (𝐡 ∩ dom 𝐹)) β†’ π‘₯ ∈ {π‘₯ ∈ (𝐡 ∩ dom 𝐹) ∣ (πΉβ€˜π‘₯) < π‘Ž}))
7348, 72ralrimi 3239 . . . . . . . . . . 11 ({π‘₯ ∈ dom 𝐹 ∣ (πΉβ€˜π‘₯) < π‘Ž} = (𝑀 ∩ dom 𝐹) β†’ βˆ€π‘₯ ∈ (𝑀 ∩ (𝐡 ∩ dom 𝐹))π‘₯ ∈ {π‘₯ ∈ (𝐡 ∩ dom 𝐹) ∣ (πΉβ€˜π‘₯) < π‘Ž})
7473, 70sylibr 233 . . . . . . . . . 10 ({π‘₯ ∈ dom 𝐹 ∣ (πΉβ€˜π‘₯) < π‘Ž} = (𝑀 ∩ dom 𝐹) β†’ (𝑀 ∩ (𝐡 ∩ dom 𝐹)) βŠ† {π‘₯ ∈ (𝐡 ∩ dom 𝐹) ∣ (πΉβ€˜π‘₯) < π‘Ž})
7545, 74eqssd 3962 . . . . . . . . 9 ({π‘₯ ∈ dom 𝐹 ∣ (πΉβ€˜π‘₯) < π‘Ž} = (𝑀 ∩ dom 𝐹) β†’ {π‘₯ ∈ (𝐡 ∩ dom 𝐹) ∣ (πΉβ€˜π‘₯) < π‘Ž} = (𝑀 ∩ (𝐡 ∩ dom 𝐹)))
7635, 75eqtrd 2773 . . . . . . . 8 ({π‘₯ ∈ dom 𝐹 ∣ (πΉβ€˜π‘₯) < π‘Ž} = (𝑀 ∩ dom 𝐹) β†’ {π‘₯ ∈ (𝐡 ∩ dom 𝐹) ∣ ((𝐹 β†Ύ 𝐡)β€˜π‘₯) < π‘Ž} = (𝑀 ∩ (𝐡 ∩ dom 𝐹)))
7776adantl 483 . . . . . . 7 (((πœ‘ ∧ π‘Ž ∈ ℝ) ∧ {π‘₯ ∈ dom 𝐹 ∣ (πΉβ€˜π‘₯) < π‘Ž} = (𝑀 ∩ dom 𝐹)) β†’ {π‘₯ ∈ (𝐡 ∩ dom 𝐹) ∣ ((𝐹 β†Ύ 𝐡)β€˜π‘₯) < π‘Ž} = (𝑀 ∩ (𝐡 ∩ dom 𝐹)))
78773adant2 1132 . . . . . 6 (((πœ‘ ∧ π‘Ž ∈ ℝ) ∧ 𝑀 ∈ 𝑆 ∧ {π‘₯ ∈ dom 𝐹 ∣ (πΉβ€˜π‘₯) < π‘Ž} = (𝑀 ∩ dom 𝐹)) β†’ {π‘₯ ∈ (𝐡 ∩ dom 𝐹) ∣ ((𝐹 β†Ύ 𝐡)β€˜π‘₯) < π‘Ž} = (𝑀 ∩ (𝐡 ∩ dom 𝐹)))
79223ad2ant1 1134 . . . . . . 7 (((πœ‘ ∧ π‘Ž ∈ ℝ) ∧ 𝑀 ∈ 𝑆 ∧ {π‘₯ ∈ dom 𝐹 ∣ (πΉβ€˜π‘₯) < π‘Ž} = (𝑀 ∩ dom 𝐹)) β†’ 𝑆 ∈ SAlg)
80 simp1l 1198 . . . . . . . 8 (((πœ‘ ∧ π‘Ž ∈ ℝ) ∧ 𝑀 ∈ 𝑆 ∧ {π‘₯ ∈ dom 𝐹 ∣ (πΉβ€˜π‘₯) < π‘Ž} = (𝑀 ∩ dom 𝐹)) β†’ πœ‘)
8126, 9ssexd 5282 . . . . . . . 8 (πœ‘ β†’ (𝐡 ∩ dom 𝐹) ∈ V)
8280, 81syl 17 . . . . . . 7 (((πœ‘ ∧ π‘Ž ∈ ℝ) ∧ 𝑀 ∈ 𝑆 ∧ {π‘₯ ∈ dom 𝐹 ∣ (πΉβ€˜π‘₯) < π‘Ž} = (𝑀 ∩ dom 𝐹)) β†’ (𝐡 ∩ dom 𝐹) ∈ V)
83 simp2 1138 . . . . . . 7 (((πœ‘ ∧ π‘Ž ∈ ℝ) ∧ 𝑀 ∈ 𝑆 ∧ {π‘₯ ∈ dom 𝐹 ∣ (πΉβ€˜π‘₯) < π‘Ž} = (𝑀 ∩ dom 𝐹)) β†’ 𝑀 ∈ 𝑆)
84 eqid 2733 . . . . . . 7 (𝑀 ∩ (𝐡 ∩ dom 𝐹)) = (𝑀 ∩ (𝐡 ∩ dom 𝐹))
8579, 82, 83, 84elrestd 43406 . . . . . 6 (((πœ‘ ∧ π‘Ž ∈ ℝ) ∧ 𝑀 ∈ 𝑆 ∧ {π‘₯ ∈ dom 𝐹 ∣ (πΉβ€˜π‘₯) < π‘Ž} = (𝑀 ∩ dom 𝐹)) β†’ (𝑀 ∩ (𝐡 ∩ dom 𝐹)) ∈ (𝑆 β†Ύt (𝐡 ∩ dom 𝐹)))
8678, 85eqeltrd 2834 . . . . 5 (((πœ‘ ∧ π‘Ž ∈ ℝ) ∧ 𝑀 ∈ 𝑆 ∧ {π‘₯ ∈ dom 𝐹 ∣ (πΉβ€˜π‘₯) < π‘Ž} = (𝑀 ∩ dom 𝐹)) β†’ {π‘₯ ∈ (𝐡 ∩ dom 𝐹) ∣ ((𝐹 β†Ύ 𝐡)β€˜π‘₯) < π‘Ž} ∈ (𝑆 β†Ύt (𝐡 ∩ dom 𝐹)))
87863exp 1120 . . . 4 ((πœ‘ ∧ π‘Ž ∈ ℝ) β†’ (𝑀 ∈ 𝑆 β†’ ({π‘₯ ∈ dom 𝐹 ∣ (πΉβ€˜π‘₯) < π‘Ž} = (𝑀 ∩ dom 𝐹) β†’ {π‘₯ ∈ (𝐡 ∩ dom 𝐹) ∣ ((𝐹 β†Ύ 𝐡)β€˜π‘₯) < π‘Ž} ∈ (𝑆 β†Ύt (𝐡 ∩ dom 𝐹)))))
8887rexlimdv 3147 . . 3 ((πœ‘ ∧ π‘Ž ∈ ℝ) β†’ (βˆƒπ‘€ ∈ 𝑆 {π‘₯ ∈ dom 𝐹 ∣ (πΉβ€˜π‘₯) < π‘Ž} = (𝑀 ∩ dom 𝐹) β†’ {π‘₯ ∈ (𝐡 ∩ dom 𝐹) ∣ ((𝐹 β†Ύ 𝐡)β€˜π‘₯) < π‘Ž} ∈ (𝑆 β†Ύt (𝐡 ∩ dom 𝐹))))
8930, 88mpd 15 . 2 ((πœ‘ ∧ π‘Ž ∈ ℝ) β†’ {π‘₯ ∈ (𝐡 ∩ dom 𝐹) ∣ ((𝐹 β†Ύ 𝐡)β€˜π‘₯) < π‘Ž} ∈ (𝑆 β†Ύt (𝐡 ∩ dom 𝐹)))
901, 2, 7, 21, 89issmfd 45062 1 (πœ‘ β†’ (𝐹 β†Ύ 𝐡) ∈ (SMblFnβ€˜π‘†))
Colors of variables: wff setvar class
Syntax hints:   β†’ wi 4   ↔ wb 205   ∧ wa 397   ∧ w3a 1088   = wceq 1542   ∈ wcel 2107  βˆ€wral 3061  βˆƒwrex 3070  {crab 3406  Vcvv 3444   ∩ cin 3910   βŠ† wss 3911  βˆͺ cuni 4866   class class class wbr 5106  dom cdm 5634   β†Ύ cres 5636  Rel wrel 5639  βŸΆwf 6493  β€˜cfv 6497  (class class class)co 7358  β„cr 11055   < clt 11194   β†Ύt crest 17307  SAlgcsalg 44635  SMblFncsmblfn 45022
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-pre-lttri 11130  ax-pre-lttrn 11131
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-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-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-id 5532  df-po 5546  df-so 5547  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-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-fv 6505  df-ov 7361  df-oprab 7362  df-mpo 7363  df-1st 7922  df-2nd 7923  df-er 8651  df-pm 8771  df-en 8887  df-dom 8888  df-sdom 8889  df-pnf 11196  df-mnf 11197  df-xr 11198  df-ltxr 11199  df-le 11200  df-ioo 13274  df-ico 13276  df-rest 17309  df-smblfn 45023
This theorem is referenced by:  sssmfmpt  45077
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