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| Mirrors > Home > MPE Home > Th. List > fifo | Structured version Visualization version GIF version | ||
| Description: Describe a surjection from nonempty finite sets to finite intersections. (Contributed by Mario Carneiro, 18-May-2015.) | 
| Ref | Expression | 
|---|---|
| fifo.1 | ⊢ 𝐹 = (𝑦 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ↦ ∩ 𝑦) | 
| Ref | Expression | 
|---|---|
| fifo | ⊢ (𝐴 ∈ 𝑉 → 𝐹:((𝒫 𝐴 ∩ Fin) ∖ {∅})–onto→(fi‘𝐴)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | eldifsni 4789 | . . . . . 6 ⊢ (𝑦 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) → 𝑦 ≠ ∅) | |
| 2 | intex 5343 | . . . . . 6 ⊢ (𝑦 ≠ ∅ ↔ ∩ 𝑦 ∈ V) | |
| 3 | 1, 2 | sylib 218 | . . . . 5 ⊢ (𝑦 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) → ∩ 𝑦 ∈ V) | 
| 4 | 3 | rgen 3062 | . . . 4 ⊢ ∀𝑦 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅})∩ 𝑦 ∈ V | 
| 5 | fifo.1 | . . . . 5 ⊢ 𝐹 = (𝑦 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ↦ ∩ 𝑦) | |
| 6 | 5 | fnmpt 6707 | . . . 4 ⊢ (∀𝑦 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅})∩ 𝑦 ∈ V → 𝐹 Fn ((𝒫 𝐴 ∩ Fin) ∖ {∅})) | 
| 7 | 4, 6 | mp1i 13 | . . 3 ⊢ (𝐴 ∈ 𝑉 → 𝐹 Fn ((𝒫 𝐴 ∩ Fin) ∖ {∅})) | 
| 8 | dffn4 6825 | . . 3 ⊢ (𝐹 Fn ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ↔ 𝐹:((𝒫 𝐴 ∩ Fin) ∖ {∅})–onto→ran 𝐹) | |
| 9 | 7, 8 | sylib 218 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐹:((𝒫 𝐴 ∩ Fin) ∖ {∅})–onto→ran 𝐹) | 
| 10 | elfi2 9455 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (fi‘𝐴) ↔ ∃𝑦 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅})𝑥 = ∩ 𝑦)) | |
| 11 | 5 | elrnmpt 5968 | . . . . . 6 ⊢ (𝑥 ∈ V → (𝑥 ∈ ran 𝐹 ↔ ∃𝑦 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅})𝑥 = ∩ 𝑦)) | 
| 12 | 11 | elv 3484 | . . . . 5 ⊢ (𝑥 ∈ ran 𝐹 ↔ ∃𝑦 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅})𝑥 = ∩ 𝑦) | 
| 13 | 10, 12 | bitr4di 289 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (fi‘𝐴) ↔ 𝑥 ∈ ran 𝐹)) | 
| 14 | 13 | eqrdv 2734 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (fi‘𝐴) = ran 𝐹) | 
| 15 | foeq3 6817 | . . 3 ⊢ ((fi‘𝐴) = ran 𝐹 → (𝐹:((𝒫 𝐴 ∩ Fin) ∖ {∅})–onto→(fi‘𝐴) ↔ 𝐹:((𝒫 𝐴 ∩ Fin) ∖ {∅})–onto→ran 𝐹)) | |
| 16 | 14, 15 | syl 17 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐹:((𝒫 𝐴 ∩ Fin) ∖ {∅})–onto→(fi‘𝐴) ↔ 𝐹:((𝒫 𝐴 ∩ Fin) ∖ {∅})–onto→ran 𝐹)) | 
| 17 | 9, 16 | mpbird 257 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝐹:((𝒫 𝐴 ∩ Fin) ∖ {∅})–onto→(fi‘𝐴)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 = wceq 1539 ∈ wcel 2107 ≠ wne 2939 ∀wral 3060 ∃wrex 3069 Vcvv 3479 ∖ cdif 3947 ∩ cin 3949 ∅c0 4332 𝒫 cpw 4599 {csn 4625 ∩ cint 4945 ↦ cmpt 5224 ran crn 5685 Fn wfn 6555 –onto→wfo 6558 ‘cfv 6560 Fincfn 8986 ficfi 9451 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-int 4946 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-iota 6513 df-fun 6562 df-fn 6563 df-fo 6566 df-fv 6568 df-fi 9452 | 
| This theorem is referenced by: inffien 10104 fictb 10285 | 
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