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Theorem infnninf 6988
Description: The point at infinity in (the constant sequence equal to 1o). (Contributed by Jim Kingdon, 14-Jul-2022.)
Assertion
Ref Expression
infnninf (ω × {1o}) ∈ ℕ

Proof of Theorem infnninf
Dummy variables 𝑓 𝑖 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 1oex 6287 . . . . . 6 1o ∈ V
21sucid 4307 . . . . 5 1o ∈ suc 1o
3 df-2o 6280 . . . . 5 2o = suc 1o
42, 3eleqtrri 2191 . . . 4 1o ∈ 2o
54fconst6 5290 . . 3 (ω × {1o}):ω⟶2o
6 2onn 6383 . . . . 5 2o ∈ ω
76elexi 2670 . . . 4 2o ∈ V
8 omex 4475 . . . 4 ω ∈ V
97, 8elmap 6537 . . 3 ((ω × {1o}) ∈ (2o𝑚 ω) ↔ (ω × {1o}):ω⟶2o)
105, 9mpbir 145 . 2 (ω × {1o}) ∈ (2o𝑚 ω)
11 peano2 4477 . . . . . 6 (𝑖 ∈ ω → suc 𝑖 ∈ ω)
121fvconst2 5602 . . . . . 6 (suc 𝑖 ∈ ω → ((ω × {1o})‘suc 𝑖) = 1o)
1311, 12syl 14 . . . . 5 (𝑖 ∈ ω → ((ω × {1o})‘suc 𝑖) = 1o)
141fvconst2 5602 . . . . 5 (𝑖 ∈ ω → ((ω × {1o})‘𝑖) = 1o)
1513, 14eqtr4d 2151 . . . 4 (𝑖 ∈ ω → ((ω × {1o})‘suc 𝑖) = ((ω × {1o})‘𝑖))
16 eqimss 3119 . . . 4 (((ω × {1o})‘suc 𝑖) = ((ω × {1o})‘𝑖) → ((ω × {1o})‘suc 𝑖) ⊆ ((ω × {1o})‘𝑖))
1715, 16syl 14 . . 3 (𝑖 ∈ ω → ((ω × {1o})‘suc 𝑖) ⊆ ((ω × {1o})‘𝑖))
1817rgen 2460 . 2 𝑖 ∈ ω ((ω × {1o})‘suc 𝑖) ⊆ ((ω × {1o})‘𝑖)
19 fveq1 5386 . . . . 5 (𝑓 = (ω × {1o}) → (𝑓‘suc 𝑖) = ((ω × {1o})‘suc 𝑖))
20 fveq1 5386 . . . . 5 (𝑓 = (ω × {1o}) → (𝑓𝑖) = ((ω × {1o})‘𝑖))
2119, 20sseq12d 3096 . . . 4 (𝑓 = (ω × {1o}) → ((𝑓‘suc 𝑖) ⊆ (𝑓𝑖) ↔ ((ω × {1o})‘suc 𝑖) ⊆ ((ω × {1o})‘𝑖)))
2221ralbidv 2412 . . 3 (𝑓 = (ω × {1o}) → (∀𝑖 ∈ ω (𝑓‘suc 𝑖) ⊆ (𝑓𝑖) ↔ ∀𝑖 ∈ ω ((ω × {1o})‘suc 𝑖) ⊆ ((ω × {1o})‘𝑖)))
23 df-nninf 6973 . . 3 = {𝑓 ∈ (2o𝑚 ω) ∣ ∀𝑖 ∈ ω (𝑓‘suc 𝑖) ⊆ (𝑓𝑖)}
2422, 23elrab2 2814 . 2 ((ω × {1o}) ∈ ℕ ↔ ((ω × {1o}) ∈ (2o𝑚 ω) ∧ ∀𝑖 ∈ ω ((ω × {1o})‘suc 𝑖) ⊆ ((ω × {1o})‘𝑖)))
2510, 18, 24mpbir2an 909 1 (ω × {1o}) ∈ ℕ
Colors of variables: wff set class
Syntax hints:   = wceq 1314  wcel 1463  wral 2391  wss 3039  {csn 3495  suc csuc 4255  ωcom 4472   × cxp 4505  wf 5087  cfv 5091  (class class class)co 5740  1oc1o 6272  2oc2o 6273  𝑚 cmap 6508  xnninf 6971
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-iinf 4470
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-rab 2400  df-v 2660  df-sbc 2881  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-id 4183  df-iord 4256  df-on 4258  df-suc 4261  df-iom 4473  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-fv 5099  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1o 6279  df-2o 6280  df-map 6510  df-nninf 6973
This theorem is referenced by:  fxnn0nninf  10151  nninffeq  13018
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