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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 14044 | . . . . 5 ℕ∞ | |
3 | eqid 2170 | . . . . 5 | |
4 | 2, 3 | fmptd 5650 | . . . 4 ℕ∞ |
5 | 2onn 6500 | . . . . . 6 | |
6 | 5 | a1i 9 | . . . . 5 ℕ∞ |
7 | omex 4577 | . . . . . 6 | |
8 | 7 | a1i 9 | . . . . 5 ℕ∞ |
9 | 6, 8 | elmapd 6640 | . . . 4 ℕ∞ |
10 | 4, 9 | mpbird 166 | . . 3 ℕ∞ |
11 | nninfsellemsuc 14045 | . . . . 5 ℕ∞ | |
12 | peano2 4579 | . . . . . 6 | |
13 | nninfsellemcl 14044 | . . . . . . 7 ℕ∞ | |
14 | 12, 13 | sylan2 284 | . . . . . 6 ℕ∞ |
15 | suceq 4387 | . . . . . . . . 9 | |
16 | 15 | raleqdv 2671 | . . . . . . . 8 |
17 | 16 | ifbid 3547 | . . . . . . 7 |
18 | 17, 3 | fvmptg 5572 | . . . . . 6 |
19 | 12, 14, 18 | syl2an2 589 | . . . . 5 ℕ∞ |
20 | simpr 109 | . . . . . 6 ℕ∞ | |
21 | nninfsellemcl 14044 | . . . . . 6 ℕ∞ | |
22 | suceq 4387 | . . . . . . . . 9 | |
23 | 22 | raleqdv 2671 | . . . . . . . 8 |
24 | 23 | ifbid 3547 | . . . . . . 7 |
25 | 24, 3 | fvmptg 5572 | . . . . . 6 |
26 | 20, 21, 25 | syl2anc 409 | . . . . 5 ℕ∞ |
27 | 11, 19, 26 | 3sstr4d 3192 | . . . 4 ℕ∞ |
28 | 27 | ralrimiva 2543 | . . 3 ℕ∞ |
29 | fveq1 5495 | . . . . . 6 | |
30 | fveq1 5495 | . . . . . 6 | |
31 | 29, 30 | sseq12d 3178 | . . . . 5 |
32 | 31 | ralbidv 2470 | . . . 4 |
33 | df-nninf 7097 | . . . 4 ℕ∞ | |
34 | 32, 33 | elrab2 2889 | . . 3 ℕ∞ |
35 | 10, 28, 34 | sylanbrc 415 | . 2 ℕ∞ ℕ∞ |
36 | 1, 35 | fmpti 5648 | 1 ℕ∞ℕ∞ |
Colors of variables: wff set class |
Syntax hints: wa 103 wceq 1348 wcel 2141 wral 2448 cvv 2730 wss 3121 c0 3414 cif 3526 cmpt 4050 csuc 4350 com 4574 wf 5194 cfv 5198 (class class class)co 5853 c1o 6388 c2o 6389 cmap 6626 ℕ∞xnninf 7096 |
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 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-nul 4115 ax-pow 4160 ax-pr 4194 ax-un 4418 ax-setind 4521 ax-iinf 4572 |
This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-eu 2022 df-mo 2023 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-ral 2453 df-rex 2454 df-rab 2457 df-v 2732 df-sbc 2956 df-dif 3123 df-un 3125 df-in 3127 df-ss 3134 df-nul 3415 df-if 3527 df-pw 3568 df-sn 3589 df-pr 3590 df-op 3592 df-uni 3797 df-int 3832 df-br 3990 df-opab 4051 df-mpt 4052 df-tr 4088 df-id 4278 df-iord 4351 df-on 4353 df-suc 4356 df-iom 4575 df-xp 4617 df-rel 4618 df-cnv 4619 df-co 4620 df-dm 4621 df-rn 4622 df-res 4623 df-ima 4624 df-iota 5160 df-fun 5200 df-fn 5201 df-f 5202 df-fv 5206 df-ov 5856 df-oprab 5857 df-mpo 5858 df-1o 6395 df-2o 6396 df-map 6628 df-nninf 7097 |
This theorem is referenced by: nninfsellemeq 14047 nninfsellemeqinf 14049 nninfomnilem 14051 |
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