Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > hasheqf1od | Structured version Visualization version GIF version |
Description: The size of two sets is equal if there is a bijection mapping one of the sets onto the other. (Contributed by AV, 4-May-2021.) |
Ref | Expression |
---|---|
hasheqf1od.a | ⊢ (𝜑 → 𝐴 ∈ 𝑈) |
hasheqf1od.f | ⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵) |
Ref | Expression |
---|---|
hasheqf1od | ⊢ (𝜑 → (♯‘𝐴) = (♯‘𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hasheqf1od.a | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑈) | |
2 | hasheqf1od.f | . . . . 5 ⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵) | |
3 | f1of 6621 | . . . . 5 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹:𝐴⟶𝐵) | |
4 | 2, 3 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐹:𝐴⟶𝐵) |
5 | 4, 1 | fexd 7003 | . . 3 ⊢ (𝜑 → 𝐹 ∈ V) |
6 | f1oeq1 6609 | . . 3 ⊢ (𝑓 = 𝐹 → (𝑓:𝐴–1-1-onto→𝐵 ↔ 𝐹:𝐴–1-1-onto→𝐵)) | |
7 | 5, 2, 6 | spcedv 3503 | . 2 ⊢ (𝜑 → ∃𝑓 𝑓:𝐴–1-1-onto→𝐵) |
8 | hasheqf1oi 13807 | . 2 ⊢ (𝐴 ∈ 𝑈 → (∃𝑓 𝑓:𝐴–1-1-onto→𝐵 → (♯‘𝐴) = (♯‘𝐵))) | |
9 | 1, 7, 8 | sylc 65 | 1 ⊢ (𝜑 → (♯‘𝐴) = (♯‘𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1542 ∃wex 1786 ∈ wcel 2114 Vcvv 3399 ⟶wf 6336 –1-1-onto→wf1o 6339 ‘cfv 6340 ♯chash 13785 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5155 ax-sep 5168 ax-nul 5175 ax-pow 5233 ax-pr 5297 ax-un 7482 ax-cnex 10674 ax-resscn 10675 ax-1cn 10676 ax-icn 10677 ax-addcl 10678 ax-addrcl 10679 ax-mulcl 10680 ax-mulrcl 10681 ax-mulcom 10682 ax-addass 10683 ax-mulass 10684 ax-distr 10685 ax-i2m1 10686 ax-1ne0 10687 ax-1rid 10688 ax-rnegex 10689 ax-rrecex 10690 ax-cnre 10691 ax-pre-lttri 10692 ax-pre-lttrn 10693 ax-pre-ltadd 10694 ax-pre-mulgt0 10695 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-nel 3040 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3401 df-sbc 3682 df-csb 3792 df-dif 3847 df-un 3849 df-in 3851 df-ss 3861 df-pss 3863 df-nul 4213 df-if 4416 df-pw 4491 df-sn 4518 df-pr 4520 df-tp 4522 df-op 4524 df-uni 4798 df-int 4838 df-iun 4884 df-br 5032 df-opab 5094 df-mpt 5112 df-tr 5138 df-id 5430 df-eprel 5435 df-po 5443 df-so 5444 df-fr 5484 df-we 5486 df-xp 5532 df-rel 5533 df-cnv 5534 df-co 5535 df-dm 5536 df-rn 5537 df-res 5538 df-ima 5539 df-pred 6130 df-ord 6176 df-on 6177 df-lim 6178 df-suc 6179 df-iota 6298 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-riota 7130 df-ov 7176 df-oprab 7177 df-mpo 7178 df-om 7603 df-wrecs 7979 df-recs 8040 df-rdg 8078 df-1o 8134 df-er 8323 df-en 8559 df-dom 8560 df-sdom 8561 df-fin 8562 df-card 9444 df-pnf 10758 df-mnf 10759 df-xr 10760 df-ltxr 10761 df-le 10762 df-sub 10953 df-neg 10954 df-nn 11720 df-n0 11980 df-z 12066 df-uz 12328 df-hash 13786 |
This theorem is referenced by: summolem2a 15168 prodmolem2a 15383 4sqlem11 16394 orbsta2 18565 sylow2a 18865 lsmhash 18952 gsumval3lem1 19147 gsumval3lem2 19148 gsumval3 19149 basellem5 25825 0sgmppw 25937 logfac2 25956 usgredgleordALT 27179 vtxdushgrfvedglem 27434 vtxdushgrfvedg 27435 clwlkssizeeq 28025 frgrncvvdeq 28249 numclwwlk2lem3 28320 subfacp1lem3 32718 subfacp1lem5 32720 |
Copyright terms: Public domain | W3C validator |