![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > fihashf1rn | GIF version |
Description: The size of a finite set which is a one-to-one function is equal to the size of the function's range. (Contributed by Jim Kingdon, 21-Feb-2022.) |
Ref | Expression |
---|---|
fihashf1rn | ⊢ ((𝐴 ∈ Fin ∧ 𝐹:𝐴–1-1→𝐵) → (♯‘𝐹) = (♯‘ran 𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1fn 5166 | . . 3 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹 Fn 𝐴) | |
2 | simpl 107 | . . 3 ⊢ ((𝐴 ∈ Fin ∧ 𝐹:𝐴–1-1→𝐵) → 𝐴 ∈ Fin) | |
3 | fnfi 6571 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ Fin) → 𝐹 ∈ Fin) | |
4 | 1, 2, 3 | syl2an2 559 | . 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐹:𝐴–1-1→𝐵) → 𝐹 ∈ Fin) |
5 | f1o2ndf1 5928 | . . . 4 ⊢ (𝐹:𝐴–1-1→𝐵 → (2nd ↾ 𝐹):𝐹–1-1-onto→ran 𝐹) | |
6 | df-2nd 5847 | . . . . . . . . 9 ⊢ 2nd = (𝑥 ∈ V ↦ ∪ ran {𝑥}) | |
7 | 6 | funmpt2 5006 | . . . . . . . 8 ⊢ Fun 2nd |
8 | f1f 5164 | . . . . . . . . . . 11 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴⟶𝐵) | |
9 | 8 | anim2i 334 | . . . . . . . . . 10 ⊢ ((𝐴 ∈ Fin ∧ 𝐹:𝐴–1-1→𝐵) → (𝐴 ∈ Fin ∧ 𝐹:𝐴⟶𝐵)) |
10 | 9 | ancomd 263 | . . . . . . . . 9 ⊢ ((𝐴 ∈ Fin ∧ 𝐹:𝐴–1-1→𝐵) → (𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ Fin)) |
11 | fex 5464 | . . . . . . . . 9 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ Fin) → 𝐹 ∈ V) | |
12 | 10, 11 | syl 14 | . . . . . . . 8 ⊢ ((𝐴 ∈ Fin ∧ 𝐹:𝐴–1-1→𝐵) → 𝐹 ∈ V) |
13 | resfunexg 5458 | . . . . . . . 8 ⊢ ((Fun 2nd ∧ 𝐹 ∈ V) → (2nd ↾ 𝐹) ∈ V) | |
14 | 7, 12, 13 | sylancr 405 | . . . . . . 7 ⊢ ((𝐴 ∈ Fin ∧ 𝐹:𝐴–1-1→𝐵) → (2nd ↾ 𝐹) ∈ V) |
15 | f1oeq1 5192 | . . . . . . . . . 10 ⊢ ((2nd ↾ 𝐹) = 𝑓 → ((2nd ↾ 𝐹):𝐹–1-1-onto→ran 𝐹 ↔ 𝑓:𝐹–1-1-onto→ran 𝐹)) | |
16 | 15 | biimpd 142 | . . . . . . . . 9 ⊢ ((2nd ↾ 𝐹) = 𝑓 → ((2nd ↾ 𝐹):𝐹–1-1-onto→ran 𝐹 → 𝑓:𝐹–1-1-onto→ran 𝐹)) |
17 | 16 | eqcoms 2086 | . . . . . . . 8 ⊢ (𝑓 = (2nd ↾ 𝐹) → ((2nd ↾ 𝐹):𝐹–1-1-onto→ran 𝐹 → 𝑓:𝐹–1-1-onto→ran 𝐹)) |
18 | 17 | adantl 271 | . . . . . . 7 ⊢ (((𝐴 ∈ Fin ∧ 𝐹:𝐴–1-1→𝐵) ∧ 𝑓 = (2nd ↾ 𝐹)) → ((2nd ↾ 𝐹):𝐹–1-1-onto→ran 𝐹 → 𝑓:𝐹–1-1-onto→ran 𝐹)) |
19 | 14, 18 | spcimedv 2695 | . . . . . 6 ⊢ ((𝐴 ∈ Fin ∧ 𝐹:𝐴–1-1→𝐵) → ((2nd ↾ 𝐹):𝐹–1-1-onto→ran 𝐹 → ∃𝑓 𝑓:𝐹–1-1-onto→ran 𝐹)) |
20 | 19 | ex 113 | . . . . 5 ⊢ (𝐴 ∈ Fin → (𝐹:𝐴–1-1→𝐵 → ((2nd ↾ 𝐹):𝐹–1-1-onto→ran 𝐹 → ∃𝑓 𝑓:𝐹–1-1-onto→ran 𝐹))) |
21 | 20 | com13 79 | . . . 4 ⊢ ((2nd ↾ 𝐹):𝐹–1-1-onto→ran 𝐹 → (𝐹:𝐴–1-1→𝐵 → (𝐴 ∈ Fin → ∃𝑓 𝑓:𝐹–1-1-onto→ran 𝐹))) |
22 | 5, 21 | mpcom 36 | . . 3 ⊢ (𝐹:𝐴–1-1→𝐵 → (𝐴 ∈ Fin → ∃𝑓 𝑓:𝐹–1-1-onto→ran 𝐹)) |
23 | 22 | impcom 123 | . 2 ⊢ ((𝐴 ∈ Fin ∧ 𝐹:𝐴–1-1→𝐵) → ∃𝑓 𝑓:𝐹–1-1-onto→ran 𝐹) |
24 | fihasheqf1oi 10031 | . . . 4 ⊢ ((𝐹 ∈ Fin ∧ 𝑓:𝐹–1-1-onto→ran 𝐹) → (♯‘𝐹) = (♯‘ran 𝐹)) | |
25 | 24 | ex 113 | . . 3 ⊢ (𝐹 ∈ Fin → (𝑓:𝐹–1-1-onto→ran 𝐹 → (♯‘𝐹) = (♯‘ran 𝐹))) |
26 | 25 | exlimdv 1742 | . 2 ⊢ (𝐹 ∈ Fin → (∃𝑓 𝑓:𝐹–1-1-onto→ran 𝐹 → (♯‘𝐹) = (♯‘ran 𝐹))) |
27 | 4, 23, 26 | sylc 61 | 1 ⊢ ((𝐴 ∈ Fin ∧ 𝐹:𝐴–1-1→𝐵) → (♯‘𝐹) = (♯‘ran 𝐹)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 102 = wceq 1285 ∃wex 1422 ∈ wcel 1434 Vcvv 2612 {csn 3422 ∪ cuni 3627 ran crn 4402 ↾ cres 4403 Fun wfun 4963 Fn wfn 4964 ⟶wf 4965 –1-1→wf1 4966 –1-1-onto→wf1o 4968 ‘cfv 4969 2nd c2nd 5845 Fincfn 6387 ♯chash 10018 |
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 4000 ax-un 4224 ax-setind 4316 ax-iinf 4366 ax-cnex 7339 ax-resscn 7340 ax-1cn 7341 ax-1re 7342 ax-icn 7343 ax-addcl 7344 ax-addrcl 7345 ax-mulcl 7346 ax-addcom 7348 ax-addass 7350 ax-distr 7352 ax-i2m1 7353 ax-0lt1 7354 ax-0id 7356 ax-rnegex 7357 ax-cnre 7359 ax-pre-ltirr 7360 ax-pre-ltwlin 7361 ax-pre-lttrn 7362 ax-pre-ltadd 7364 |
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-if 3374 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 4084 df-iord 4157 df-on 4159 df-ilim 4160 df-suc 4162 df-iom 4369 df-xp 4407 df-rel 4408 df-cnv 4409 df-co 4410 df-dm 4411 df-rn 4412 df-res 4413 df-ima 4414 df-iota 4934 df-fun 4971 df-fn 4972 df-f 4973 df-f1 4974 df-fo 4975 df-f1o 4976 df-fv 4977 df-riota 5547 df-ov 5594 df-oprab 5595 df-mpt2 5596 df-2nd 5847 df-recs 6002 df-frec 6088 df-1o 6113 df-er 6222 df-en 6388 df-dom 6389 df-fin 6390 df-pnf 7427 df-mnf 7428 df-xr 7429 df-ltxr 7430 df-le 7431 df-sub 7558 df-neg 7559 df-inn 8317 df-n0 8566 df-z 8647 df-uz 8915 df-ihash 10019 |
This theorem is referenced by: (None) |
Copyright terms: Public domain | W3C validator |