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Mirrors > Home > ILE Home > Th. List > isfinite4im | GIF version |
Description: A finite set is equinumerous to the range of integers from one up to the hash value of the set. (Contributed by Jim Kingdon, 22-Feb-2022.) |
Ref | Expression |
---|---|
isfinite4im | ⊢ (𝐴 ∈ Fin → (1...(♯‘𝐴)) ≈ 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hashcl 10022 | . . 3 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℕ0) | |
2 | hashfz1 10024 | . . 3 ⊢ ((♯‘𝐴) ∈ ℕ0 → (♯‘(1...(♯‘𝐴))) = (♯‘𝐴)) | |
3 | 1, 2 | syl 14 | . 2 ⊢ (𝐴 ∈ Fin → (♯‘(1...(♯‘𝐴))) = (♯‘𝐴)) |
4 | 1z 8670 | . . . 4 ⊢ 1 ∈ ℤ | |
5 | 1 | nn0zd 8760 | . . . 4 ⊢ (𝐴 ∈ Fin → (♯‘𝐴) ∈ ℤ) |
6 | fzfig 9724 | . . . 4 ⊢ ((1 ∈ ℤ ∧ (♯‘𝐴) ∈ ℤ) → (1...(♯‘𝐴)) ∈ Fin) | |
7 | 4, 5, 6 | sylancr 405 | . . 3 ⊢ (𝐴 ∈ Fin → (1...(♯‘𝐴)) ∈ Fin) |
8 | hashen 10025 | . . 3 ⊢ (((1...(♯‘𝐴)) ∈ Fin ∧ 𝐴 ∈ Fin) → ((♯‘(1...(♯‘𝐴))) = (♯‘𝐴) ↔ (1...(♯‘𝐴)) ≈ 𝐴)) | |
9 | 7, 8 | mpancom 413 | . 2 ⊢ (𝐴 ∈ Fin → ((♯‘(1...(♯‘𝐴))) = (♯‘𝐴) ↔ (1...(♯‘𝐴)) ≈ 𝐴)) |
10 | 3, 9 | mpbid 145 | 1 ⊢ (𝐴 ∈ Fin → (1...(♯‘𝐴)) ≈ 𝐴) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ↔ wb 103 = wceq 1285 ∈ wcel 1434 class class class wbr 3811 ‘cfv 4967 (class class class)co 5589 ≈ cen 6383 Fincfn 6385 1c1 7252 ℕ0cn0 8563 ℤcz 8644 ...cfz 9317 ♯chash 10016 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-coll 3919 ax-sep 3922 ax-nul 3930 ax-pow 3974 ax-pr 3999 ax-un 4223 ax-setind 4315 ax-iinf 4365 ax-cnex 7337 ax-resscn 7338 ax-1cn 7339 ax-1re 7340 ax-icn 7341 ax-addcl 7342 ax-addrcl 7343 ax-mulcl 7344 ax-addcom 7346 ax-addass 7348 ax-distr 7350 ax-i2m1 7351 ax-0lt1 7352 ax-0id 7354 ax-rnegex 7355 ax-cnre 7357 ax-pre-ltirr 7358 ax-pre-ltwlin 7359 ax-pre-lttrn 7360 ax-pre-apti 7361 ax-pre-ltadd 7362 |
This theorem depends on definitions: df-bi 115 df-dc 777 df-3or 921 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ne 2250 df-nel 2345 df-ral 2358 df-rex 2359 df-reu 2360 df-rab 2362 df-v 2614 df-sbc 2827 df-csb 2920 df-dif 2986 df-un 2988 df-in 2990 df-ss 2997 df-nul 3270 df-pw 3408 df-sn 3428 df-pr 3429 df-op 3431 df-uni 3628 df-int 3663 df-iun 3706 df-br 3812 df-opab 3866 df-mpt 3867 df-tr 3902 df-id 4083 df-iord 4156 df-on 4158 df-ilim 4159 df-suc 4161 df-iom 4368 df-xp 4405 df-rel 4406 df-cnv 4407 df-co 4408 df-dm 4409 df-rn 4410 df-res 4411 df-ima 4412 df-iota 4932 df-fun 4969 df-fn 4970 df-f 4971 df-f1 4972 df-fo 4973 df-f1o 4974 df-fv 4975 df-riota 5545 df-ov 5592 df-oprab 5593 df-mpt2 5594 df-1st 5844 df-2nd 5845 df-recs 6000 df-frec 6086 df-1o 6111 df-er 6220 df-en 6386 df-dom 6387 df-fin 6388 df-pnf 7425 df-mnf 7426 df-xr 7427 df-ltxr 7428 df-le 7429 df-sub 7556 df-neg 7557 df-inn 8315 df-n0 8564 df-z 8645 df-uz 8913 df-fz 9318 df-ihash 10017 |
This theorem is referenced by: fz1f1o 10570 |
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