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Mirrors > Home > MPE Home > Th. List > hashresfn | Structured version Visualization version GIF version |
Description: Restriction of the domain of the size function. (Contributed by Thierry Arnoux, 31-Jan-2017.) |
Ref | Expression |
---|---|
hashresfn | ⊢ (♯ ↾ 𝐴) Fn 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | hashf 13548 | . . 3 ⊢ ♯:V⟶(ℕ0 ∪ {+∞}) | |
2 | ffn 6385 | . . 3 ⊢ (♯:V⟶(ℕ0 ∪ {+∞}) → ♯ Fn V) | |
3 | fnresin2 6346 | . . 3 ⊢ (♯ Fn V → (♯ ↾ (𝐴 ∩ V)) Fn (𝐴 ∩ V)) | |
4 | 1, 2, 3 | mp2b 10 | . 2 ⊢ (♯ ↾ (𝐴 ∩ V)) Fn (𝐴 ∩ V) |
5 | inv1 4270 | . . . 4 ⊢ (𝐴 ∩ V) = 𝐴 | |
6 | 5 | reseq2i 5734 | . . 3 ⊢ (♯ ↾ (𝐴 ∩ V)) = (♯ ↾ 𝐴) |
7 | fneq12 6322 | . . 3 ⊢ (((♯ ↾ (𝐴 ∩ V)) = (♯ ↾ 𝐴) ∧ (𝐴 ∩ V) = 𝐴) → ((♯ ↾ (𝐴 ∩ V)) Fn (𝐴 ∩ V) ↔ (♯ ↾ 𝐴) Fn 𝐴)) | |
8 | 6, 5, 7 | mp2an 688 | . 2 ⊢ ((♯ ↾ (𝐴 ∩ V)) Fn (𝐴 ∩ V) ↔ (♯ ↾ 𝐴) Fn 𝐴) |
9 | 4, 8 | mpbi 231 | 1 ⊢ (♯ ↾ 𝐴) Fn 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 207 = wceq 1522 Vcvv 3436 ∪ cun 3859 ∩ cin 3860 {csn 4474 ↾ cres 5448 Fn wfn 6223 ⟶wf 6224 +∞cpnf 10521 ℕ0cn0 11747 ♯chash 13540 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1778 ax-4 1792 ax-5 1889 ax-6 1948 ax-7 1993 ax-8 2082 ax-9 2090 ax-10 2111 ax-11 2125 ax-12 2140 ax-13 2343 ax-ext 2768 ax-sep 5097 ax-nul 5104 ax-pow 5160 ax-pr 5224 ax-un 7322 ax-cnex 10442 ax-resscn 10443 ax-1cn 10444 ax-icn 10445 ax-addcl 10446 ax-addrcl 10447 ax-mulcl 10448 ax-mulrcl 10449 ax-mulcom 10450 ax-addass 10451 ax-mulass 10452 ax-distr 10453 ax-i2m1 10454 ax-1ne0 10455 ax-1rid 10456 ax-rnegex 10457 ax-rrecex 10458 ax-cnre 10459 ax-pre-lttri 10460 ax-pre-lttrn 10461 ax-pre-ltadd 10462 ax-pre-mulgt0 10463 |
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 1763 df-nf 1767 df-sb 2042 df-mo 2575 df-eu 2611 df-clab 2775 df-cleq 2787 df-clel 2862 df-nfc 2934 df-ne 2984 df-nel 3090 df-ral 3109 df-rex 3110 df-reu 3111 df-rab 3113 df-v 3438 df-sbc 3708 df-csb 3814 df-dif 3864 df-un 3866 df-in 3868 df-ss 3876 df-pss 3878 df-nul 4214 df-if 4384 df-pw 4457 df-sn 4475 df-pr 4477 df-tp 4479 df-op 4481 df-uni 4748 df-int 4785 df-iun 4829 df-br 4965 df-opab 5027 df-mpt 5044 df-tr 5067 df-id 5351 df-eprel 5356 df-po 5365 df-so 5366 df-fr 5405 df-we 5407 df-xp 5452 df-rel 5453 df-cnv 5454 df-co 5455 df-dm 5456 df-rn 5457 df-res 5458 df-ima 5459 df-pred 6026 df-ord 6072 df-on 6073 df-lim 6074 df-suc 6075 df-iota 6192 df-fun 6230 df-fn 6231 df-f 6232 df-f1 6233 df-fo 6234 df-f1o 6235 df-fv 6236 df-riota 6980 df-ov 7022 df-oprab 7023 df-mpo 7024 df-om 7440 df-wrecs 7801 df-recs 7863 df-rdg 7901 df-er 8142 df-en 8361 df-dom 8362 df-sdom 8363 df-fin 8364 df-card 9217 df-pnf 10526 df-mnf 10527 df-xr 10528 df-ltxr 10529 df-le 10530 df-sub 10721 df-neg 10722 df-nn 11489 df-n0 11748 df-xnn0 11818 df-z 11832 df-uz 12094 df-hash 13541 |
This theorem is referenced by: hashgval2 13587 coinfliplem 31345 coinflipspace 31347 |
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