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Theorem nnsf 14757
Description: Domain and range of 𝑆. Part of Definition 3.3 of [PradicBrown2022], p. 5. (Contributed by Jim Kingdon, 30-Jul-2022.)
Hypothesis
Ref Expression
nns.s 𝑆 = (𝑝 ∈ β„•βˆž ↦ (𝑖 ∈ Ο‰ ↦ if(𝑖 = βˆ…, 1o, (π‘β€˜βˆͺ 𝑖))))
Assertion
Ref Expression
nnsf 𝑆:β„•βˆžβŸΆβ„•βˆž
Distinct variable group:   𝑖,𝑝
Allowed substitution hints:   𝑆(𝑖,𝑝)

Proof of Theorem nnsf
Dummy variables 𝑓 𝑗 π‘˜ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nns.s . 2 𝑆 = (𝑝 ∈ β„•βˆž ↦ (𝑖 ∈ Ο‰ ↦ if(𝑖 = βˆ…, 1o, (π‘β€˜βˆͺ 𝑖))))
2 1lt2o 6443 . . . . . . 7 1o ∈ 2o
32a1i 9 . . . . . 6 ((𝑝 ∈ β„•βˆž ∧ 𝑖 ∈ Ο‰) β†’ 1o ∈ 2o)
4 nninff 7121 . . . . . . . 8 (𝑝 ∈ β„•βˆž β†’ 𝑝:Ο‰βŸΆ2o)
54adantr 276 . . . . . . 7 ((𝑝 ∈ β„•βˆž ∧ 𝑖 ∈ Ο‰) β†’ 𝑝:Ο‰βŸΆ2o)
6 nnpredcl 4623 . . . . . . . 8 (𝑖 ∈ Ο‰ β†’ βˆͺ 𝑖 ∈ Ο‰)
76adantl 277 . . . . . . 7 ((𝑝 ∈ β„•βˆž ∧ 𝑖 ∈ Ο‰) β†’ βˆͺ 𝑖 ∈ Ο‰)
85, 7ffvelcdmd 5653 . . . . . 6 ((𝑝 ∈ β„•βˆž ∧ 𝑖 ∈ Ο‰) β†’ (π‘β€˜βˆͺ 𝑖) ∈ 2o)
9 nndceq0 4618 . . . . . . 7 (𝑖 ∈ Ο‰ β†’ DECID 𝑖 = βˆ…)
109adantl 277 . . . . . 6 ((𝑝 ∈ β„•βˆž ∧ 𝑖 ∈ Ο‰) β†’ DECID 𝑖 = βˆ…)
113, 8, 10ifcldcd 3571 . . . . 5 ((𝑝 ∈ β„•βˆž ∧ 𝑖 ∈ Ο‰) β†’ if(𝑖 = βˆ…, 1o, (π‘β€˜βˆͺ 𝑖)) ∈ 2o)
12 eqid 2177 . . . . 5 (𝑖 ∈ Ο‰ ↦ if(𝑖 = βˆ…, 1o, (π‘β€˜βˆͺ 𝑖))) = (𝑖 ∈ Ο‰ ↦ if(𝑖 = βˆ…, 1o, (π‘β€˜βˆͺ 𝑖)))
1311, 12fmptd 5671 . . . 4 (𝑝 ∈ β„•βˆž β†’ (𝑖 ∈ Ο‰ ↦ if(𝑖 = βˆ…, 1o, (π‘β€˜βˆͺ 𝑖))):Ο‰βŸΆ2o)
14 2onn 6522 . . . . 5 2o ∈ Ο‰
15 omex 4593 . . . . 5 Ο‰ ∈ V
16 elmapg 6661 . . . . 5 ((2o ∈ Ο‰ ∧ Ο‰ ∈ V) β†’ ((𝑖 ∈ Ο‰ ↦ if(𝑖 = βˆ…, 1o, (π‘β€˜βˆͺ 𝑖))) ∈ (2o β†‘π‘š Ο‰) ↔ (𝑖 ∈ Ο‰ ↦ if(𝑖 = βˆ…, 1o, (π‘β€˜βˆͺ 𝑖))):Ο‰βŸΆ2o))
1714, 15, 16mp2an 426 . . . 4 ((𝑖 ∈ Ο‰ ↦ if(𝑖 = βˆ…, 1o, (π‘β€˜βˆͺ 𝑖))) ∈ (2o β†‘π‘š Ο‰) ↔ (𝑖 ∈ Ο‰ ↦ if(𝑖 = βˆ…, 1o, (π‘β€˜βˆͺ 𝑖))):Ο‰βŸΆ2o)
1813, 17sylibr 134 . . 3 (𝑝 ∈ β„•βˆž β†’ (𝑖 ∈ Ο‰ ↦ if(𝑖 = βˆ…, 1o, (π‘β€˜βˆͺ 𝑖))) ∈ (2o β†‘π‘š Ο‰))
19 1on 6424 . . . . . . . . 9 1o ∈ On
2019ontrci 4428 . . . . . . . 8 Tr 1o
212a1i 9 . . . . . . . . . . 11 ((𝑝 ∈ β„•βˆž ∧ 𝑗 ∈ Ο‰) β†’ 1o ∈ 2o)
224adantr 276 . . . . . . . . . . . 12 ((𝑝 ∈ β„•βˆž ∧ 𝑗 ∈ Ο‰) β†’ 𝑝:Ο‰βŸΆ2o)
23 peano2 4595 . . . . . . . . . . . . . 14 (𝑗 ∈ Ο‰ β†’ suc 𝑗 ∈ Ο‰)
2423adantl 277 . . . . . . . . . . . . 13 ((𝑝 ∈ β„•βˆž ∧ 𝑗 ∈ Ο‰) β†’ suc 𝑗 ∈ Ο‰)
25 nnpredcl 4623 . . . . . . . . . . . . 13 (suc 𝑗 ∈ Ο‰ β†’ βˆͺ suc 𝑗 ∈ Ο‰)
2624, 25syl 14 . . . . . . . . . . . 12 ((𝑝 ∈ β„•βˆž ∧ 𝑗 ∈ Ο‰) β†’ βˆͺ suc 𝑗 ∈ Ο‰)
2722, 26ffvelcdmd 5653 . . . . . . . . . . 11 ((𝑝 ∈ β„•βˆž ∧ 𝑗 ∈ Ο‰) β†’ (π‘β€˜βˆͺ suc 𝑗) ∈ 2o)
28 nndceq0 4618 . . . . . . . . . . . 12 (suc 𝑗 ∈ Ο‰ β†’ DECID suc 𝑗 = βˆ…)
2924, 28syl 14 . . . . . . . . . . 11 ((𝑝 ∈ β„•βˆž ∧ 𝑗 ∈ Ο‰) β†’ DECID suc 𝑗 = βˆ…)
3021, 27, 29ifcldcd 3571 . . . . . . . . . 10 ((𝑝 ∈ β„•βˆž ∧ 𝑗 ∈ Ο‰) β†’ if(suc 𝑗 = βˆ…, 1o, (π‘β€˜βˆͺ suc 𝑗)) ∈ 2o)
3130adantr 276 . . . . . . . . 9 (((𝑝 ∈ β„•βˆž ∧ 𝑗 ∈ Ο‰) ∧ 𝑗 = βˆ…) β†’ if(suc 𝑗 = βˆ…, 1o, (π‘β€˜βˆͺ suc 𝑗)) ∈ 2o)
32 df-2o 6418 . . . . . . . . 9 2o = suc 1o
3331, 32eleqtrdi 2270 . . . . . . . 8 (((𝑝 ∈ β„•βˆž ∧ 𝑗 ∈ Ο‰) ∧ 𝑗 = βˆ…) β†’ if(suc 𝑗 = βˆ…, 1o, (π‘β€˜βˆͺ suc 𝑗)) ∈ suc 1o)
34 trsucss 4424 . . . . . . . 8 (Tr 1o β†’ (if(suc 𝑗 = βˆ…, 1o, (π‘β€˜βˆͺ suc 𝑗)) ∈ suc 1o β†’ if(suc 𝑗 = βˆ…, 1o, (π‘β€˜βˆͺ suc 𝑗)) βŠ† 1o))
3520, 33, 34mpsyl 65 . . . . . . 7 (((𝑝 ∈ β„•βˆž ∧ 𝑗 ∈ Ο‰) ∧ 𝑗 = βˆ…) β†’ if(suc 𝑗 = βˆ…, 1o, (π‘β€˜βˆͺ suc 𝑗)) βŠ† 1o)
36 iftrue 3540 . . . . . . . 8 (𝑗 = βˆ… β†’ if(𝑗 = βˆ…, 1o, (π‘β€˜βˆͺ 𝑗)) = 1o)
3736adantl 277 . . . . . . 7 (((𝑝 ∈ β„•βˆž ∧ 𝑗 ∈ Ο‰) ∧ 𝑗 = βˆ…) β†’ if(𝑗 = βˆ…, 1o, (π‘β€˜βˆͺ 𝑗)) = 1o)
3835, 37sseqtrrd 3195 . . . . . 6 (((𝑝 ∈ β„•βˆž ∧ 𝑗 ∈ Ο‰) ∧ 𝑗 = βˆ…) β†’ if(suc 𝑗 = βˆ…, 1o, (π‘β€˜βˆͺ suc 𝑗)) βŠ† if(𝑗 = βˆ…, 1o, (π‘β€˜βˆͺ 𝑗)))
39 simpr 110 . . . . . . . . . . . 12 ((𝑝 ∈ β„•βˆž ∧ 𝑗 ∈ Ο‰) β†’ 𝑗 ∈ Ο‰)
4039adantr 276 . . . . . . . . . . 11 (((𝑝 ∈ β„•βˆž ∧ 𝑗 ∈ Ο‰) ∧ Β¬ 𝑗 = βˆ…) β†’ 𝑗 ∈ Ο‰)
41 nnord 4612 . . . . . . . . . . 11 (𝑗 ∈ Ο‰ β†’ Ord 𝑗)
42 ordtr 4379 . . . . . . . . . . 11 (Ord 𝑗 β†’ Tr 𝑗)
4340, 41, 423syl 17 . . . . . . . . . 10 (((𝑝 ∈ β„•βˆž ∧ 𝑗 ∈ Ο‰) ∧ Β¬ 𝑗 = βˆ…) β†’ Tr 𝑗)
44 unisucg 4415 . . . . . . . . . . 11 (𝑗 ∈ Ο‰ β†’ (Tr 𝑗 ↔ βˆͺ suc 𝑗 = 𝑗))
4540, 44syl 14 . . . . . . . . . 10 (((𝑝 ∈ β„•βˆž ∧ 𝑗 ∈ Ο‰) ∧ Β¬ 𝑗 = βˆ…) β†’ (Tr 𝑗 ↔ βˆͺ suc 𝑗 = 𝑗))
4643, 45mpbid 147 . . . . . . . . 9 (((𝑝 ∈ β„•βˆž ∧ 𝑗 ∈ Ο‰) ∧ Β¬ 𝑗 = βˆ…) β†’ βˆͺ suc 𝑗 = 𝑗)
4746fveq2d 5520 . . . . . . . 8 (((𝑝 ∈ β„•βˆž ∧ 𝑗 ∈ Ο‰) ∧ Β¬ 𝑗 = βˆ…) β†’ (π‘β€˜βˆͺ suc 𝑗) = (π‘β€˜π‘—))
48 simpr 110 . . . . . . . . . . . 12 (((𝑝 ∈ β„•βˆž ∧ 𝑗 ∈ Ο‰) ∧ Β¬ 𝑗 = βˆ…) β†’ Β¬ 𝑗 = βˆ…)
4948neqned 2354 . . . . . . . . . . 11 (((𝑝 ∈ β„•βˆž ∧ 𝑗 ∈ Ο‰) ∧ Β¬ 𝑗 = βˆ…) β†’ 𝑗 β‰  βˆ…)
50 nnsucpred 4617 . . . . . . . . . . 11 ((𝑗 ∈ Ο‰ ∧ 𝑗 β‰  βˆ…) β†’ suc βˆͺ 𝑗 = 𝑗)
5140, 49, 50syl2anc 411 . . . . . . . . . 10 (((𝑝 ∈ β„•βˆž ∧ 𝑗 ∈ Ο‰) ∧ Β¬ 𝑗 = βˆ…) β†’ suc βˆͺ 𝑗 = 𝑗)
5251fveq2d 5520 . . . . . . . . 9 (((𝑝 ∈ β„•βˆž ∧ 𝑗 ∈ Ο‰) ∧ Β¬ 𝑗 = βˆ…) β†’ (π‘β€˜suc βˆͺ 𝑗) = (π‘β€˜π‘—))
53 suceq 4403 . . . . . . . . . . . 12 (π‘˜ = βˆͺ 𝑗 β†’ suc π‘˜ = suc βˆͺ 𝑗)
5453fveq2d 5520 . . . . . . . . . . 11 (π‘˜ = βˆͺ 𝑗 β†’ (π‘β€˜suc π‘˜) = (π‘β€˜suc βˆͺ 𝑗))
55 fveq2 5516 . . . . . . . . . . 11 (π‘˜ = βˆͺ 𝑗 β†’ (π‘β€˜π‘˜) = (π‘β€˜βˆͺ 𝑗))
5654, 55sseq12d 3187 . . . . . . . . . 10 (π‘˜ = βˆͺ 𝑗 β†’ ((π‘β€˜suc π‘˜) βŠ† (π‘β€˜π‘˜) ↔ (π‘β€˜suc βˆͺ 𝑗) βŠ† (π‘β€˜βˆͺ 𝑗)))
57 fveq1 5515 . . . . . . . . . . . . . . . 16 (𝑓 = 𝑝 β†’ (π‘“β€˜suc 𝑗) = (π‘β€˜suc 𝑗))
58 fveq1 5515 . . . . . . . . . . . . . . . 16 (𝑓 = 𝑝 β†’ (π‘“β€˜π‘—) = (π‘β€˜π‘—))
5957, 58sseq12d 3187 . . . . . . . . . . . . . . 15 (𝑓 = 𝑝 β†’ ((π‘“β€˜suc 𝑗) βŠ† (π‘“β€˜π‘—) ↔ (π‘β€˜suc 𝑗) βŠ† (π‘β€˜π‘—)))
6059ralbidv 2477 . . . . . . . . . . . . . 14 (𝑓 = 𝑝 β†’ (βˆ€π‘— ∈ Ο‰ (π‘“β€˜suc 𝑗) βŠ† (π‘“β€˜π‘—) ↔ βˆ€π‘— ∈ Ο‰ (π‘β€˜suc 𝑗) βŠ† (π‘β€˜π‘—)))
61 df-nninf 7119 . . . . . . . . . . . . . 14 β„•βˆž = {𝑓 ∈ (2o β†‘π‘š Ο‰) ∣ βˆ€π‘— ∈ Ο‰ (π‘“β€˜suc 𝑗) βŠ† (π‘“β€˜π‘—)}
6260, 61elrab2 2897 . . . . . . . . . . . . 13 (𝑝 ∈ β„•βˆž ↔ (𝑝 ∈ (2o β†‘π‘š Ο‰) ∧ βˆ€π‘— ∈ Ο‰ (π‘β€˜suc 𝑗) βŠ† (π‘β€˜π‘—)))
6362simprbi 275 . . . . . . . . . . . 12 (𝑝 ∈ β„•βˆž β†’ βˆ€π‘— ∈ Ο‰ (π‘β€˜suc 𝑗) βŠ† (π‘β€˜π‘—))
64 suceq 4403 . . . . . . . . . . . . . . 15 (𝑗 = π‘˜ β†’ suc 𝑗 = suc π‘˜)
6564fveq2d 5520 . . . . . . . . . . . . . 14 (𝑗 = π‘˜ β†’ (π‘β€˜suc 𝑗) = (π‘β€˜suc π‘˜))
66 fveq2 5516 . . . . . . . . . . . . . 14 (𝑗 = π‘˜ β†’ (π‘β€˜π‘—) = (π‘β€˜π‘˜))
6765, 66sseq12d 3187 . . . . . . . . . . . . 13 (𝑗 = π‘˜ β†’ ((π‘β€˜suc 𝑗) βŠ† (π‘β€˜π‘—) ↔ (π‘β€˜suc π‘˜) βŠ† (π‘β€˜π‘˜)))
6867cbvralv 2704 . . . . . . . . . . . 12 (βˆ€π‘— ∈ Ο‰ (π‘β€˜suc 𝑗) βŠ† (π‘β€˜π‘—) ↔ βˆ€π‘˜ ∈ Ο‰ (π‘β€˜suc π‘˜) βŠ† (π‘β€˜π‘˜))
6963, 68sylib 122 . . . . . . . . . . 11 (𝑝 ∈ β„•βˆž β†’ βˆ€π‘˜ ∈ Ο‰ (π‘β€˜suc π‘˜) βŠ† (π‘β€˜π‘˜))
7069ad2antrr 488 . . . . . . . . . 10 (((𝑝 ∈ β„•βˆž ∧ 𝑗 ∈ Ο‰) ∧ Β¬ 𝑗 = βˆ…) β†’ βˆ€π‘˜ ∈ Ο‰ (π‘β€˜suc π‘˜) βŠ† (π‘β€˜π‘˜))
71 nnpredcl 4623 . . . . . . . . . . . 12 (𝑗 ∈ Ο‰ β†’ βˆͺ 𝑗 ∈ Ο‰)
7271adantl 277 . . . . . . . . . . 11 ((𝑝 ∈ β„•βˆž ∧ 𝑗 ∈ Ο‰) β†’ βˆͺ 𝑗 ∈ Ο‰)
7372adantr 276 . . . . . . . . . 10 (((𝑝 ∈ β„•βˆž ∧ 𝑗 ∈ Ο‰) ∧ Β¬ 𝑗 = βˆ…) β†’ βˆͺ 𝑗 ∈ Ο‰)
7456, 70, 73rspcdva 2847 . . . . . . . . 9 (((𝑝 ∈ β„•βˆž ∧ 𝑗 ∈ Ο‰) ∧ Β¬ 𝑗 = βˆ…) β†’ (π‘β€˜suc βˆͺ 𝑗) βŠ† (π‘β€˜βˆͺ 𝑗))
7552, 74eqsstrrd 3193 . . . . . . . 8 (((𝑝 ∈ β„•βˆž ∧ 𝑗 ∈ Ο‰) ∧ Β¬ 𝑗 = βˆ…) β†’ (π‘β€˜π‘—) βŠ† (π‘β€˜βˆͺ 𝑗))
7647, 75eqsstrd 3192 . . . . . . 7 (((𝑝 ∈ β„•βˆž ∧ 𝑗 ∈ Ο‰) ∧ Β¬ 𝑗 = βˆ…) β†’ (π‘β€˜βˆͺ suc 𝑗) βŠ† (π‘β€˜βˆͺ 𝑗))
77 peano3 4596 . . . . . . . . . 10 (𝑗 ∈ Ο‰ β†’ suc 𝑗 β‰  βˆ…)
7877neneqd 2368 . . . . . . . . 9 (𝑗 ∈ Ο‰ β†’ Β¬ suc 𝑗 = βˆ…)
7978ad2antlr 489 . . . . . . . 8 (((𝑝 ∈ β„•βˆž ∧ 𝑗 ∈ Ο‰) ∧ Β¬ 𝑗 = βˆ…) β†’ Β¬ suc 𝑗 = βˆ…)
8079iffalsed 3545 . . . . . . 7 (((𝑝 ∈ β„•βˆž ∧ 𝑗 ∈ Ο‰) ∧ Β¬ 𝑗 = βˆ…) β†’ if(suc 𝑗 = βˆ…, 1o, (π‘β€˜βˆͺ suc 𝑗)) = (π‘β€˜βˆͺ suc 𝑗))
8148iffalsed 3545 . . . . . . 7 (((𝑝 ∈ β„•βˆž ∧ 𝑗 ∈ Ο‰) ∧ Β¬ 𝑗 = βˆ…) β†’ if(𝑗 = βˆ…, 1o, (π‘β€˜βˆͺ 𝑗)) = (π‘β€˜βˆͺ 𝑗))
8276, 80, 813sstr4d 3201 . . . . . 6 (((𝑝 ∈ β„•βˆž ∧ 𝑗 ∈ Ο‰) ∧ Β¬ 𝑗 = βˆ…) β†’ if(suc 𝑗 = βˆ…, 1o, (π‘β€˜βˆͺ suc 𝑗)) βŠ† if(𝑗 = βˆ…, 1o, (π‘β€˜βˆͺ 𝑗)))
83 nndceq0 4618 . . . . . . . 8 (𝑗 ∈ Ο‰ β†’ DECID 𝑗 = βˆ…)
8483adantl 277 . . . . . . 7 ((𝑝 ∈ β„•βˆž ∧ 𝑗 ∈ Ο‰) β†’ DECID 𝑗 = βˆ…)
85 exmiddc 836 . . . . . . 7 (DECID 𝑗 = βˆ… β†’ (𝑗 = βˆ… ∨ Β¬ 𝑗 = βˆ…))
8684, 85syl 14 . . . . . 6 ((𝑝 ∈ β„•βˆž ∧ 𝑗 ∈ Ο‰) β†’ (𝑗 = βˆ… ∨ Β¬ 𝑗 = βˆ…))
8738, 82, 86mpjaodan 798 . . . . 5 ((𝑝 ∈ β„•βˆž ∧ 𝑗 ∈ Ο‰) β†’ if(suc 𝑗 = βˆ…, 1o, (π‘β€˜βˆͺ suc 𝑗)) βŠ† if(𝑗 = βˆ…, 1o, (π‘β€˜βˆͺ 𝑗)))
88 eqeq1 2184 . . . . . . . 8 (𝑖 = suc 𝑗 β†’ (𝑖 = βˆ… ↔ suc 𝑗 = βˆ…))
89 unieq 3819 . . . . . . . . 9 (𝑖 = suc 𝑗 β†’ βˆͺ 𝑖 = βˆͺ suc 𝑗)
9089fveq2d 5520 . . . . . . . 8 (𝑖 = suc 𝑗 β†’ (π‘β€˜βˆͺ 𝑖) = (π‘β€˜βˆͺ suc 𝑗))
9188, 90ifbieq2d 3559 . . . . . . 7 (𝑖 = suc 𝑗 β†’ if(𝑖 = βˆ…, 1o, (π‘β€˜βˆͺ 𝑖)) = if(suc 𝑗 = βˆ…, 1o, (π‘β€˜βˆͺ suc 𝑗)))
9291, 12fvmptg 5593 . . . . . 6 ((suc 𝑗 ∈ Ο‰ ∧ if(suc 𝑗 = βˆ…, 1o, (π‘β€˜βˆͺ suc 𝑗)) ∈ 2o) β†’ ((𝑖 ∈ Ο‰ ↦ if(𝑖 = βˆ…, 1o, (π‘β€˜βˆͺ 𝑖)))β€˜suc 𝑗) = if(suc 𝑗 = βˆ…, 1o, (π‘β€˜βˆͺ suc 𝑗)))
9324, 30, 92syl2anc 411 . . . . 5 ((𝑝 ∈ β„•βˆž ∧ 𝑗 ∈ Ο‰) β†’ ((𝑖 ∈ Ο‰ ↦ if(𝑖 = βˆ…, 1o, (π‘β€˜βˆͺ 𝑖)))β€˜suc 𝑗) = if(suc 𝑗 = βˆ…, 1o, (π‘β€˜βˆͺ suc 𝑗)))
9422, 72ffvelcdmd 5653 . . . . . . 7 ((𝑝 ∈ β„•βˆž ∧ 𝑗 ∈ Ο‰) β†’ (π‘β€˜βˆͺ 𝑗) ∈ 2o)
9521, 94, 84ifcldcd 3571 . . . . . 6 ((𝑝 ∈ β„•βˆž ∧ 𝑗 ∈ Ο‰) β†’ if(𝑗 = βˆ…, 1o, (π‘β€˜βˆͺ 𝑗)) ∈ 2o)
96 eqeq1 2184 . . . . . . . 8 (𝑖 = 𝑗 β†’ (𝑖 = βˆ… ↔ 𝑗 = βˆ…))
97 unieq 3819 . . . . . . . . 9 (𝑖 = 𝑗 β†’ βˆͺ 𝑖 = βˆͺ 𝑗)
9897fveq2d 5520 . . . . . . . 8 (𝑖 = 𝑗 β†’ (π‘β€˜βˆͺ 𝑖) = (π‘β€˜βˆͺ 𝑗))
9996, 98ifbieq2d 3559 . . . . . . 7 (𝑖 = 𝑗 β†’ if(𝑖 = βˆ…, 1o, (π‘β€˜βˆͺ 𝑖)) = if(𝑗 = βˆ…, 1o, (π‘β€˜βˆͺ 𝑗)))
10099, 12fvmptg 5593 . . . . . 6 ((𝑗 ∈ Ο‰ ∧ if(𝑗 = βˆ…, 1o, (π‘β€˜βˆͺ 𝑗)) ∈ 2o) β†’ ((𝑖 ∈ Ο‰ ↦ if(𝑖 = βˆ…, 1o, (π‘β€˜βˆͺ 𝑖)))β€˜π‘—) = if(𝑗 = βˆ…, 1o, (π‘β€˜βˆͺ 𝑗)))
10139, 95, 100syl2anc 411 . . . . 5 ((𝑝 ∈ β„•βˆž ∧ 𝑗 ∈ Ο‰) β†’ ((𝑖 ∈ Ο‰ ↦ if(𝑖 = βˆ…, 1o, (π‘β€˜βˆͺ 𝑖)))β€˜π‘—) = if(𝑗 = βˆ…, 1o, (π‘β€˜βˆͺ 𝑗)))
10287, 93, 1013sstr4d 3201 . . . 4 ((𝑝 ∈ β„•βˆž ∧ 𝑗 ∈ Ο‰) β†’ ((𝑖 ∈ Ο‰ ↦ if(𝑖 = βˆ…, 1o, (π‘β€˜βˆͺ 𝑖)))β€˜suc 𝑗) βŠ† ((𝑖 ∈ Ο‰ ↦ if(𝑖 = βˆ…, 1o, (π‘β€˜βˆͺ 𝑖)))β€˜π‘—))
103102ralrimiva 2550 . . 3 (𝑝 ∈ β„•βˆž β†’ βˆ€π‘— ∈ Ο‰ ((𝑖 ∈ Ο‰ ↦ if(𝑖 = βˆ…, 1o, (π‘β€˜βˆͺ 𝑖)))β€˜suc 𝑗) βŠ† ((𝑖 ∈ Ο‰ ↦ if(𝑖 = βˆ…, 1o, (π‘β€˜βˆͺ 𝑖)))β€˜π‘—))
104 fveq1 5515 . . . . . 6 (𝑓 = (𝑖 ∈ Ο‰ ↦ if(𝑖 = βˆ…, 1o, (π‘β€˜βˆͺ 𝑖))) β†’ (π‘“β€˜suc 𝑗) = ((𝑖 ∈ Ο‰ ↦ if(𝑖 = βˆ…, 1o, (π‘β€˜βˆͺ 𝑖)))β€˜suc 𝑗))
105 fveq1 5515 . . . . . 6 (𝑓 = (𝑖 ∈ Ο‰ ↦ if(𝑖 = βˆ…, 1o, (π‘β€˜βˆͺ 𝑖))) β†’ (π‘“β€˜π‘—) = ((𝑖 ∈ Ο‰ ↦ if(𝑖 = βˆ…, 1o, (π‘β€˜βˆͺ 𝑖)))β€˜π‘—))
106104, 105sseq12d 3187 . . . . 5 (𝑓 = (𝑖 ∈ Ο‰ ↦ if(𝑖 = βˆ…, 1o, (π‘β€˜βˆͺ 𝑖))) β†’ ((π‘“β€˜suc 𝑗) βŠ† (π‘“β€˜π‘—) ↔ ((𝑖 ∈ Ο‰ ↦ if(𝑖 = βˆ…, 1o, (π‘β€˜βˆͺ 𝑖)))β€˜suc 𝑗) βŠ† ((𝑖 ∈ Ο‰ ↦ if(𝑖 = βˆ…, 1o, (π‘β€˜βˆͺ 𝑖)))β€˜π‘—)))
107106ralbidv 2477 . . . 4 (𝑓 = (𝑖 ∈ Ο‰ ↦ if(𝑖 = βˆ…, 1o, (π‘β€˜βˆͺ 𝑖))) β†’ (βˆ€π‘— ∈ Ο‰ (π‘“β€˜suc 𝑗) βŠ† (π‘“β€˜π‘—) ↔ βˆ€π‘— ∈ Ο‰ ((𝑖 ∈ Ο‰ ↦ if(𝑖 = βˆ…, 1o, (π‘β€˜βˆͺ 𝑖)))β€˜suc 𝑗) βŠ† ((𝑖 ∈ Ο‰ ↦ if(𝑖 = βˆ…, 1o, (π‘β€˜βˆͺ 𝑖)))β€˜π‘—)))
108107, 61elrab2 2897 . . 3 ((𝑖 ∈ Ο‰ ↦ if(𝑖 = βˆ…, 1o, (π‘β€˜βˆͺ 𝑖))) ∈ β„•βˆž ↔ ((𝑖 ∈ Ο‰ ↦ if(𝑖 = βˆ…, 1o, (π‘β€˜βˆͺ 𝑖))) ∈ (2o β†‘π‘š Ο‰) ∧ βˆ€π‘— ∈ Ο‰ ((𝑖 ∈ Ο‰ ↦ if(𝑖 = βˆ…, 1o, (π‘β€˜βˆͺ 𝑖)))β€˜suc 𝑗) βŠ† ((𝑖 ∈ Ο‰ ↦ if(𝑖 = βˆ…, 1o, (π‘β€˜βˆͺ 𝑖)))β€˜π‘—)))
10918, 103, 108sylanbrc 417 . 2 (𝑝 ∈ β„•βˆž β†’ (𝑖 ∈ Ο‰ ↦ if(𝑖 = βˆ…, 1o, (π‘β€˜βˆͺ 𝑖))) ∈ β„•βˆž)
1101, 109fmpti 5669 1 𝑆:β„•βˆžβŸΆβ„•βˆž
Colors of variables: wff set class
Syntax hints:  Β¬ wn 3   ∧ wa 104   ↔ wb 105   ∨ wo 708  DECID wdc 834   = wceq 1353   ∈ wcel 2148   β‰  wne 2347  βˆ€wral 2455  Vcvv 2738   βŠ† wss 3130  βˆ…c0 3423  ifcif 3535  βˆͺ cuni 3810   ↦ cmpt 4065  Tr wtr 4102  Ord word 4363  suc csuc 4366  Ο‰com 4590  βŸΆwf 5213  β€˜cfv 5217  (class class class)co 5875  1oc1o 6410  2oc2o 6411   β†‘π‘š cmap 6648  β„•βˆžxnninf 7118
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4122  ax-nul 4130  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537  ax-iinf 4588
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2740  df-sbc 2964  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-if 3536  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-int 3846  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-id 4294  df-iord 4367  df-on 4369  df-suc 4372  df-iom 4591  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-fv 5225  df-ov 5878  df-oprab 5879  df-mpo 5880  df-1o 6417  df-2o 6418  df-map 6650  df-nninf 7119
This theorem is referenced by:  peano4nninf  14758
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