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Mirrors > Home > MPE Home > Th. List > hashf1rn | Structured version Visualization version 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 Alexander van der Vekens, 12-Jan-2018.) (Revised by AV, 4-May-2021.) |
Ref | Expression |
---|---|
hashf1rn | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐵) → (#‘𝐹) = (#‘ran 𝐹)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | f1f 6014 | . . . . 5 ⊢ (𝐹:𝐴–1-1→𝐵 → 𝐹:𝐴⟶𝐵) | |
2 | 1 | anim2i 591 | . . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐵) → (𝐴 ∈ 𝑉 ∧ 𝐹:𝐴⟶𝐵)) |
3 | 2 | ancomd 466 | . . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐵) → (𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉)) |
4 | fex 6394 | . . 3 ⊢ ((𝐹:𝐴⟶𝐵 ∧ 𝐴 ∈ 𝑉) → 𝐹 ∈ V) | |
5 | 3, 4 | syl 17 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐵) → 𝐹 ∈ V) |
6 | f1o2ndf1 7172 | . . . 4 ⊢ (𝐹:𝐴–1-1→𝐵 → (2nd ↾ 𝐹):𝐹–1-1-onto→ran 𝐹) | |
7 | df-2nd 7060 | . . . . . . . . 9 ⊢ 2nd = (𝑥 ∈ V ↦ ∪ ran {𝑥}) | |
8 | 7 | funmpt2 5841 | . . . . . . . 8 ⊢ Fun 2nd |
9 | resfunexg 6384 | . . . . . . . 8 ⊢ ((Fun 2nd ∧ 𝐹 ∈ V) → (2nd ↾ 𝐹) ∈ V) | |
10 | 8, 5, 9 | sylancr 694 | . . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐵) → (2nd ↾ 𝐹) ∈ V) |
11 | f1oeq1 6040 | . . . . . . . . . 10 ⊢ ((2nd ↾ 𝐹) = 𝑓 → ((2nd ↾ 𝐹):𝐹–1-1-onto→ran 𝐹 ↔ 𝑓:𝐹–1-1-onto→ran 𝐹)) | |
12 | 11 | biimpd 218 | . . . . . . . . 9 ⊢ ((2nd ↾ 𝐹) = 𝑓 → ((2nd ↾ 𝐹):𝐹–1-1-onto→ran 𝐹 → 𝑓:𝐹–1-1-onto→ran 𝐹)) |
13 | 12 | eqcoms 2618 | . . . . . . . 8 ⊢ (𝑓 = (2nd ↾ 𝐹) → ((2nd ↾ 𝐹):𝐹–1-1-onto→ran 𝐹 → 𝑓:𝐹–1-1-onto→ran 𝐹)) |
14 | 13 | adantl 481 | . . . . . . 7 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐵) ∧ 𝑓 = (2nd ↾ 𝐹)) → ((2nd ↾ 𝐹):𝐹–1-1-onto→ran 𝐹 → 𝑓:𝐹–1-1-onto→ran 𝐹)) |
15 | 10, 14 | spcimedv 3265 | . . . . . 6 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐵) → ((2nd ↾ 𝐹):𝐹–1-1-onto→ran 𝐹 → ∃𝑓 𝑓:𝐹–1-1-onto→ran 𝐹)) |
16 | 15 | ex 449 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝐹:𝐴–1-1→𝐵 → ((2nd ↾ 𝐹):𝐹–1-1-onto→ran 𝐹 → ∃𝑓 𝑓:𝐹–1-1-onto→ran 𝐹))) |
17 | 16 | com13 86 | . . . 4 ⊢ ((2nd ↾ 𝐹):𝐹–1-1-onto→ran 𝐹 → (𝐹:𝐴–1-1→𝐵 → (𝐴 ∈ 𝑉 → ∃𝑓 𝑓:𝐹–1-1-onto→ran 𝐹))) |
18 | 6, 17 | mpcom 37 | . . 3 ⊢ (𝐹:𝐴–1-1→𝐵 → (𝐴 ∈ 𝑉 → ∃𝑓 𝑓:𝐹–1-1-onto→ran 𝐹)) |
19 | 18 | impcom 445 | . 2 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐵) → ∃𝑓 𝑓:𝐹–1-1-onto→ran 𝐹) |
20 | hasheqf1oi 13002 | . 2 ⊢ (𝐹 ∈ V → (∃𝑓 𝑓:𝐹–1-1-onto→ran 𝐹 → (#‘𝐹) = (#‘ran 𝐹))) | |
21 | 5, 19, 20 | sylc 63 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐹:𝐴–1-1→𝐵) → (#‘𝐹) = (#‘ran 𝐹)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∃wex 1695 ∈ wcel 1977 Vcvv 3173 {csn 4125 ∪ cuni 4372 ran crn 5039 ↾ cres 5040 Fun wfun 5798 ⟶wf 5800 –1-1→wf1 5801 –1-1-onto→wf1o 5803 ‘cfv 5804 2nd c2nd 7058 #chash 12979 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-1cn 9873 ax-icn 9874 ax-addcl 9875 ax-addrcl 9876 ax-mulcl 9877 ax-mulrcl 9878 ax-mulcom 9879 ax-addass 9880 ax-mulass 9881 ax-distr 9882 ax-i2m1 9883 ax-1ne0 9884 ax-1rid 9885 ax-rnegex 9886 ax-rrecex 9887 ax-cnre 9888 ax-pre-lttri 9889 ax-pre-lttrn 9890 ax-pre-ltadd 9891 ax-pre-mulgt0 9892 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-int 4411 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-riota 6511 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-2nd 7060 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-1o 7447 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-fin 7845 df-card 8648 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959 df-sub 10147 df-neg 10148 df-nn 10898 df-n0 11170 df-z 11255 df-uz 11564 df-hash 12980 |
This theorem is referenced by: hashimarn 13085 sizeusglecusg 26014 usgrsizedg 40442 |
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