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Mirrors > Home > MPE Home > Th. List > hashf | Structured version Visualization version GIF version |
Description: The size function maps all finite sets to their cardinality, as members of ℕ0, and infinite sets to +∞. TODO-AV: mark as OBSOLETE and replace it by hashfxnn0 13552? (Contributed by Mario Carneiro, 13-Sep-2013.) (Revised by Mario Carneiro, 13-Jul-2014.) (Proof shortened by AV, 24-Oct-2021.) |
Ref | Expression |
---|---|
hashf | ⊢ ♯:V⟶(ℕ0 ∪ {+∞}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hashfxnn0 13552 | . 2 ⊢ ♯:V⟶ℕ0* | |
2 | eqid 2795 | . . 3 ⊢ V = V | |
3 | df-xnn0 11821 | . . . 4 ⊢ ℕ0* = (ℕ0 ∪ {+∞}) | |
4 | 3 | eqcomi 2804 | . . 3 ⊢ (ℕ0 ∪ {+∞}) = ℕ0* |
5 | 2, 4 | feq23i 6381 | . 2 ⊢ (♯:V⟶(ℕ0 ∪ {+∞}) ↔ ♯:V⟶ℕ0*) |
6 | 1, 5 | mpbir 232 | 1 ⊢ ♯:V⟶(ℕ0 ∪ {+∞}) |
Colors of variables: wff setvar class |
Syntax hints: Vcvv 3437 ∪ cun 3861 {csn 4476 ⟶wf 6226 +∞cpnf 10523 ℕ0cn0 11750 ℕ0*cxnn0 11820 ♯chash 13545 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1777 ax-4 1791 ax-5 1888 ax-6 1947 ax-7 1992 ax-8 2083 ax-9 2091 ax-10 2112 ax-11 2126 ax-12 2141 ax-13 2344 ax-ext 2769 ax-sep 5099 ax-nul 5106 ax-pow 5162 ax-pr 5226 ax-un 7324 ax-cnex 10444 ax-resscn 10445 ax-1cn 10446 ax-icn 10447 ax-addcl 10448 ax-addrcl 10449 ax-mulcl 10450 ax-mulrcl 10451 ax-mulcom 10452 ax-addass 10453 ax-mulass 10454 ax-distr 10455 ax-i2m1 10456 ax-1ne0 10457 ax-1rid 10458 ax-rnegex 10459 ax-rrecex 10460 ax-cnre 10461 ax-pre-lttri 10462 ax-pre-lttrn 10463 ax-pre-ltadd 10464 ax-pre-mulgt0 10465 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 843 df-3or 1081 df-3an 1082 df-tru 1525 df-ex 1762 df-nf 1766 df-sb 2043 df-mo 2576 df-eu 2612 df-clab 2776 df-cleq 2788 df-clel 2863 df-nfc 2935 df-ne 2985 df-nel 3091 df-ral 3110 df-rex 3111 df-reu 3112 df-rab 3114 df-v 3439 df-sbc 3710 df-csb 3816 df-dif 3866 df-un 3868 df-in 3870 df-ss 3878 df-pss 3880 df-nul 4216 df-if 4386 df-pw 4459 df-sn 4477 df-pr 4479 df-tp 4481 df-op 4483 df-uni 4750 df-int 4787 df-iun 4831 df-br 4967 df-opab 5029 df-mpt 5046 df-tr 5069 df-id 5353 df-eprel 5358 df-po 5367 df-so 5368 df-fr 5407 df-we 5409 df-xp 5454 df-rel 5455 df-cnv 5456 df-co 5457 df-dm 5458 df-rn 5459 df-res 5460 df-ima 5461 df-pred 6028 df-ord 6074 df-on 6075 df-lim 6076 df-suc 6077 df-iota 6194 df-fun 6232 df-fn 6233 df-f 6234 df-f1 6235 df-fo 6236 df-f1o 6237 df-fv 6238 df-riota 6982 df-ov 7024 df-oprab 7025 df-mpo 7026 df-om 7442 df-wrecs 7803 df-recs 7865 df-rdg 7903 df-er 8144 df-en 8363 df-dom 8364 df-sdom 8365 df-fin 8366 df-card 9219 df-pnf 10528 df-mnf 10529 df-xr 10530 df-ltxr 10531 df-le 10532 df-sub 10724 df-neg 10725 df-nn 11492 df-n0 11751 df-xnn0 11821 df-z 11835 df-uz 12099 df-hash 13546 |
This theorem is referenced by: hashresfn 13555 dmhashres 13556 hashnn0pnf 13557 hashxrcl 13573 hashgt1 30219 s3clhash 30309 tocyc01 30412 cyc3evpm 30435 cycpmconjslem2 30440 cyc3conja 30442 dimval 30610 dimvalfi 30611 esumcst 30944 hashf2 30965 sseqmw 31271 sseqf 31272 sseqp1 31275 fiblem 31278 fibp1 31281 coinflippv 31363 erdszelem2 32054 erdszelem5 32057 erdszelem7 32059 erdszelem8 32060 |
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