![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 4454 | . . . . . 6 ⊢ (𝑦 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) → 𝑦 ≠ ∅) | |
2 | intex 4957 | . . . . . 6 ⊢ (𝑦 ≠ ∅ ↔ ∩ 𝑦 ∈ V) | |
3 | 1, 2 | sylib 208 | . . . . 5 ⊢ (𝑦 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) → ∩ 𝑦 ∈ V) |
4 | 3 | rgen 3048 | . . . 4 ⊢ ∀𝑦 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅})∩ 𝑦 ∈ V |
5 | fifo.1 | . . . . 5 ⊢ 𝐹 = (𝑦 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ↦ ∩ 𝑦) | |
6 | 5 | fnmpt 6169 | . . . 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 6270 | . . 3 ⊢ (𝐹 Fn ((𝒫 𝐴 ∩ Fin) ∖ {∅}) ↔ 𝐹:((𝒫 𝐴 ∩ Fin) ∖ {∅})–onto→ran 𝐹) | |
9 | 7, 8 | sylib 208 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐹:((𝒫 𝐴 ∩ Fin) ∖ {∅})–onto→ran 𝐹) |
10 | elfi2 8473 | . . . . 5 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (fi‘𝐴) ↔ ∃𝑦 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅})𝑥 = ∩ 𝑦)) | |
11 | vex 3331 | . . . . . 6 ⊢ 𝑥 ∈ V | |
12 | 5 | elrnmpt 5515 | . . . . . 6 ⊢ (𝑥 ∈ V → (𝑥 ∈ ran 𝐹 ↔ ∃𝑦 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅})𝑥 = ∩ 𝑦)) |
13 | 11, 12 | ax-mp 5 | . . . . 5 ⊢ (𝑥 ∈ ran 𝐹 ↔ ∃𝑦 ∈ ((𝒫 𝐴 ∩ Fin) ∖ {∅})𝑥 = ∩ 𝑦) |
14 | 10, 13 | syl6bbr 278 | . . . 4 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ (fi‘𝐴) ↔ 𝑥 ∈ ran 𝐹)) |
15 | 14 | eqrdv 2746 | . . 3 ⊢ (𝐴 ∈ 𝑉 → (fi‘𝐴) = ran 𝐹) |
16 | foeq3 6262 | . . 3 ⊢ ((fi‘𝐴) = ran 𝐹 → (𝐹:((𝒫 𝐴 ∩ Fin) ∖ {∅})–onto→(fi‘𝐴) ↔ 𝐹:((𝒫 𝐴 ∩ Fin) ∖ {∅})–onto→ran 𝐹)) | |
17 | 15, 16 | syl 17 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝐹:((𝒫 𝐴 ∩ Fin) ∖ {∅})–onto→(fi‘𝐴) ↔ 𝐹:((𝒫 𝐴 ∩ Fin) ∖ {∅})–onto→ran 𝐹)) |
18 | 9, 17 | mpbird 247 | 1 ⊢ (𝐴 ∈ 𝑉 → 𝐹:((𝒫 𝐴 ∩ Fin) ∖ {∅})–onto→(fi‘𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 = wceq 1620 ∈ wcel 2127 ≠ wne 2920 ∀wral 3038 ∃wrex 3039 Vcvv 3328 ∖ cdif 3700 ∩ cin 3702 ∅c0 4046 𝒫 cpw 4290 {csn 4309 ∩ cint 4615 ↦ cmpt 4869 ran crn 5255 Fn wfn 6032 –onto→wfo 6035 ‘cfv 6037 Fincfn 8109 ficfi 8469 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1859 ax-4 1874 ax-5 1976 ax-6 2042 ax-7 2078 ax-8 2129 ax-9 2136 ax-10 2156 ax-11 2171 ax-12 2184 ax-13 2379 ax-ext 2728 ax-sep 4921 ax-nul 4929 ax-pow 4980 ax-pr 5043 ax-un 7102 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3an 1074 df-tru 1623 df-ex 1842 df-nf 1847 df-sb 2035 df-eu 2599 df-mo 2600 df-clab 2735 df-cleq 2741 df-clel 2744 df-nfc 2879 df-ne 2921 df-ral 3043 df-rex 3044 df-rab 3047 df-v 3330 df-sbc 3565 df-dif 3706 df-un 3708 df-in 3710 df-ss 3717 df-nul 4047 df-if 4219 df-pw 4292 df-sn 4310 df-pr 4312 df-op 4316 df-uni 4577 df-int 4616 df-br 4793 df-opab 4853 df-mpt 4870 df-id 5162 df-xp 5260 df-rel 5261 df-cnv 5262 df-co 5263 df-dm 5264 df-rn 5265 df-iota 6000 df-fun 6039 df-fn 6040 df-fo 6043 df-fv 6045 df-fi 8470 |
This theorem is referenced by: inffien 9047 fictb 9230 |
Copyright terms: Public domain | W3C validator |