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Mirrors > Home > MPE Home > Th. List > hashfn | Structured version Visualization version GIF version |
Description: A function is equinumerous to its domain. (Contributed by Mario Carneiro, 12-Mar-2015.) |
Ref | Expression |
---|---|
hashfn | ⊢ (𝐹 Fn 𝐴 → (♯‘𝐹) = (♯‘𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fndmeng 9035 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ V) → 𝐴 ≈ 𝐹) | |
2 | ensym 8999 | . . 3 ⊢ (𝐴 ≈ 𝐹 → 𝐹 ≈ 𝐴) | |
3 | hasheni 14308 | . . 3 ⊢ (𝐹 ≈ 𝐴 → (♯‘𝐹) = (♯‘𝐴)) | |
4 | 1, 2, 3 | 3syl 18 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ 𝐴 ∈ V) → (♯‘𝐹) = (♯‘𝐴)) |
5 | dmexg 7894 | . . . . . 6 ⊢ (𝐹 ∈ V → dom 𝐹 ∈ V) | |
6 | fndm 6653 | . . . . . . 7 ⊢ (𝐹 Fn 𝐴 → dom 𝐹 = 𝐴) | |
7 | 6 | eleq1d 2819 | . . . . . 6 ⊢ (𝐹 Fn 𝐴 → (dom 𝐹 ∈ V ↔ 𝐴 ∈ V)) |
8 | 5, 7 | imbitrid 243 | . . . . 5 ⊢ (𝐹 Fn 𝐴 → (𝐹 ∈ V → 𝐴 ∈ V)) |
9 | 8 | con3dimp 410 | . . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ ¬ 𝐴 ∈ V) → ¬ 𝐹 ∈ V) |
10 | fvprc 6884 | . . . 4 ⊢ (¬ 𝐹 ∈ V → (♯‘𝐹) = ∅) | |
11 | 9, 10 | syl 17 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ ¬ 𝐴 ∈ V) → (♯‘𝐹) = ∅) |
12 | fvprc 6884 | . . . 4 ⊢ (¬ 𝐴 ∈ V → (♯‘𝐴) = ∅) | |
13 | 12 | adantl 483 | . . 3 ⊢ ((𝐹 Fn 𝐴 ∧ ¬ 𝐴 ∈ V) → (♯‘𝐴) = ∅) |
14 | 11, 13 | eqtr4d 2776 | . 2 ⊢ ((𝐹 Fn 𝐴 ∧ ¬ 𝐴 ∈ V) → (♯‘𝐹) = (♯‘𝐴)) |
15 | 4, 14 | pm2.61dan 812 | 1 ⊢ (𝐹 Fn 𝐴 → (♯‘𝐹) = (♯‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 397 = wceq 1542 ∈ wcel 2107 Vcvv 3475 ∅c0 4323 class class class wbr 5149 dom cdm 5677 Fn wfn 6539 ‘cfv 6544 ≈ cen 8936 ♯chash 14290 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5364 ax-pr 5428 ax-un 7725 ax-cnex 11166 ax-resscn 11167 ax-1cn 11168 ax-icn 11169 ax-addcl 11170 ax-addrcl 11171 ax-mulcl 11172 ax-mulrcl 11173 ax-mulcom 11174 ax-addass 11175 ax-mulass 11176 ax-distr 11177 ax-i2m1 11178 ax-1ne0 11179 ax-1rid 11180 ax-rnegex 11181 ax-rrecex 11182 ax-cnre 11183 ax-pre-lttri 11184 ax-pre-lttrn 11185 ax-pre-ltadd 11186 ax-pre-mulgt0 11187 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-nel 3048 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-pss 3968 df-nul 4324 df-if 4530 df-pw 4605 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-int 4952 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5575 df-eprel 5581 df-po 5589 df-so 5590 df-fr 5632 df-we 5634 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-pred 6301 df-ord 6368 df-on 6369 df-lim 6370 df-suc 6371 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 df-riota 7365 df-ov 7412 df-oprab 7413 df-mpo 7414 df-om 7856 df-2nd 7976 df-frecs 8266 df-wrecs 8297 df-recs 8371 df-rdg 8410 df-1o 8466 df-er 8703 df-en 8940 df-dom 8941 df-sdom 8942 df-fin 8943 df-card 9934 df-pnf 11250 df-mnf 11251 df-xr 11252 df-ltxr 11253 df-le 11254 df-sub 11446 df-neg 11447 df-nn 12213 df-n0 12473 df-z 12559 df-uz 12823 df-hash 14291 |
This theorem is referenced by: fseq1hash 14336 hashfun 14397 hashimarn 14400 fnfz0hash 14405 fnfz0hashnn0 14407 ffzo0hash 14408 fnfzo0hashnn0 14410 wrdred1hash 14511 ccatlen 14525 swrdlen 14597 swrdwrdsymb 14612 pfxlen 14633 revlen 14712 repswlen 14726 lenco 14783 ofccat 14916 pmtrdifwrdellem2 19350 frlmdim 32727 ply1degltdim 32739 subiwrdlen 33416 signstlen 33609 signsvtn0 33612 signstres 33617 signshlen 33632 sticksstones2 41011 frlmvscadiccat 41128 |
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