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Mirrors > Home > ILE Home > Th. List > Mathboxes > nninfself | Unicode version |
Description: Domain and range of the selection function for ℕ∞. (Contributed by Jim Kingdon, 6-Aug-2022.) |
Ref | Expression |
---|---|
nninfsel.e | ℕ∞ |
Ref | Expression |
---|---|
nninfself | ℕ∞ℕ∞ |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nninfsel.e | . 2 ℕ∞ | |
2 | nninfsellemcl 13680 | . . . . 5 ℕ∞ | |
3 | eqid 2157 | . . . . 5 | |
4 | 2, 3 | fmptd 5624 | . . . 4 ℕ∞ |
5 | 2onn 6471 | . . . . . 6 | |
6 | 5 | a1i 9 | . . . . 5 ℕ∞ |
7 | omex 4555 | . . . . . 6 | |
8 | 7 | a1i 9 | . . . . 5 ℕ∞ |
9 | 6, 8 | elmapd 6610 | . . . 4 ℕ∞ |
10 | 4, 9 | mpbird 166 | . . 3 ℕ∞ |
11 | nninfsellemsuc 13681 | . . . . 5 ℕ∞ | |
12 | peano2 4557 | . . . . . 6 | |
13 | nninfsellemcl 13680 | . . . . . . 7 ℕ∞ | |
14 | 12, 13 | sylan2 284 | . . . . . 6 ℕ∞ |
15 | suceq 4365 | . . . . . . . . 9 | |
16 | 15 | raleqdv 2658 | . . . . . . . 8 |
17 | 16 | ifbid 3527 | . . . . . . 7 |
18 | 17, 3 | fvmptg 5547 | . . . . . 6 |
19 | 12, 14, 18 | syl2an2 584 | . . . . 5 ℕ∞ |
20 | simpr 109 | . . . . . 6 ℕ∞ | |
21 | nninfsellemcl 13680 | . . . . . 6 ℕ∞ | |
22 | suceq 4365 | . . . . . . . . 9 | |
23 | 22 | raleqdv 2658 | . . . . . . . 8 |
24 | 23 | ifbid 3527 | . . . . . . 7 |
25 | 24, 3 | fvmptg 5547 | . . . . . 6 |
26 | 20, 21, 25 | syl2anc 409 | . . . . 5 ℕ∞ |
27 | 11, 19, 26 | 3sstr4d 3173 | . . . 4 ℕ∞ |
28 | 27 | ralrimiva 2530 | . . 3 ℕ∞ |
29 | fveq1 5470 | . . . . . 6 | |
30 | fveq1 5470 | . . . . . 6 | |
31 | 29, 30 | sseq12d 3159 | . . . . 5 |
32 | 31 | ralbidv 2457 | . . . 4 |
33 | df-nninf 7067 | . . . 4 ℕ∞ | |
34 | 32, 33 | elrab2 2871 | . . 3 ℕ∞ |
35 | 10, 28, 34 | sylanbrc 414 | . 2 ℕ∞ ℕ∞ |
36 | 1, 35 | fmpti 5622 | 1 ℕ∞ℕ∞ |
Colors of variables: wff set class |
Syntax hints: wa 103 wceq 1335 wcel 2128 wral 2435 cvv 2712 wss 3102 c0 3395 cif 3506 cmpt 4028 csuc 4328 com 4552 wf 5169 cfv 5173 (class class class)co 5827 c1o 6359 c2o 6360 cmap 6596 ℕ∞xnninf 7066 |
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 604 ax-in2 605 ax-io 699 ax-5 1427 ax-7 1428 ax-gen 1429 ax-ie1 1473 ax-ie2 1474 ax-8 1484 ax-10 1485 ax-11 1486 ax-i12 1487 ax-bndl 1489 ax-4 1490 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-13 2130 ax-14 2131 ax-ext 2139 ax-sep 4085 ax-nul 4093 ax-pow 4138 ax-pr 4172 ax-un 4396 ax-setind 4499 ax-iinf 4550 |
This theorem depends on definitions: df-bi 116 df-dc 821 df-3or 964 df-3an 965 df-tru 1338 df-fal 1341 df-nf 1441 df-sb 1743 df-eu 2009 df-mo 2010 df-clab 2144 df-cleq 2150 df-clel 2153 df-nfc 2288 df-ne 2328 df-ral 2440 df-rex 2441 df-rab 2444 df-v 2714 df-sbc 2938 df-dif 3104 df-un 3106 df-in 3108 df-ss 3115 df-nul 3396 df-if 3507 df-pw 3546 df-sn 3567 df-pr 3568 df-op 3570 df-uni 3775 df-int 3810 df-br 3968 df-opab 4029 df-mpt 4030 df-tr 4066 df-id 4256 df-iord 4329 df-on 4331 df-suc 4334 df-iom 4553 df-xp 4595 df-rel 4596 df-cnv 4597 df-co 4598 df-dm 4599 df-rn 4600 df-res 4601 df-ima 4602 df-iota 5138 df-fun 5175 df-fn 5176 df-f 5177 df-fv 5181 df-ov 5830 df-oprab 5831 df-mpo 5832 df-1o 6366 df-2o 6367 df-map 6598 df-nninf 7067 |
This theorem is referenced by: nninfsellemeq 13683 nninfsellemeqinf 13685 nninfomnilem 13687 |
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