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Mirrors > Home > ILE Home > Th. List > nn0supp | GIF version |
Description: Two ways to write the support of a function on ℕ0. (Contributed by Mario Carneiro, 29-Dec-2014.) |
Ref | Expression |
---|---|
nn0supp | ⊢ (𝐹:𝐼⟶ℕ0 → (◡𝐹 “ (V ∖ {0})) = (◡𝐹 “ ℕ)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | dfn2 8983 | . . . 4 ⊢ ℕ = (ℕ0 ∖ {0}) | |
2 | invdif 3313 | . . . 4 ⊢ (ℕ0 ∩ (V ∖ {0})) = (ℕ0 ∖ {0}) | |
3 | 1, 2 | eqtr4i 2161 | . . 3 ⊢ ℕ = (ℕ0 ∩ (V ∖ {0})) |
4 | 3 | imaeq2i 4874 | . 2 ⊢ (◡𝐹 “ ℕ) = (◡𝐹 “ (ℕ0 ∩ (V ∖ {0}))) |
5 | ffun 5270 | . . . 4 ⊢ (𝐹:𝐼⟶ℕ0 → Fun 𝐹) | |
6 | inpreima 5539 | . . . 4 ⊢ (Fun 𝐹 → (◡𝐹 “ (ℕ0 ∩ (V ∖ {0}))) = ((◡𝐹 “ ℕ0) ∩ (◡𝐹 “ (V ∖ {0})))) | |
7 | 5, 6 | syl 14 | . . 3 ⊢ (𝐹:𝐼⟶ℕ0 → (◡𝐹 “ (ℕ0 ∩ (V ∖ {0}))) = ((◡𝐹 “ ℕ0) ∩ (◡𝐹 “ (V ∖ {0})))) |
8 | cnvimass 4897 | . . . . 5 ⊢ (◡𝐹 “ (V ∖ {0})) ⊆ dom 𝐹 | |
9 | fdm 5273 | . . . . . 6 ⊢ (𝐹:𝐼⟶ℕ0 → dom 𝐹 = 𝐼) | |
10 | fimacnv 5542 | . . . . . 6 ⊢ (𝐹:𝐼⟶ℕ0 → (◡𝐹 “ ℕ0) = 𝐼) | |
11 | 9, 10 | eqtr4d 2173 | . . . . 5 ⊢ (𝐹:𝐼⟶ℕ0 → dom 𝐹 = (◡𝐹 “ ℕ0)) |
12 | 8, 11 | sseqtrid 3142 | . . . 4 ⊢ (𝐹:𝐼⟶ℕ0 → (◡𝐹 “ (V ∖ {0})) ⊆ (◡𝐹 “ ℕ0)) |
13 | sseqin2 3290 | . . . 4 ⊢ ((◡𝐹 “ (V ∖ {0})) ⊆ (◡𝐹 “ ℕ0) ↔ ((◡𝐹 “ ℕ0) ∩ (◡𝐹 “ (V ∖ {0}))) = (◡𝐹 “ (V ∖ {0}))) | |
14 | 12, 13 | sylib 121 | . . 3 ⊢ (𝐹:𝐼⟶ℕ0 → ((◡𝐹 “ ℕ0) ∩ (◡𝐹 “ (V ∖ {0}))) = (◡𝐹 “ (V ∖ {0}))) |
15 | 7, 14 | eqtrd 2170 | . 2 ⊢ (𝐹:𝐼⟶ℕ0 → (◡𝐹 “ (ℕ0 ∩ (V ∖ {0}))) = (◡𝐹 “ (V ∖ {0}))) |
16 | 4, 15 | syl5req 2183 | 1 ⊢ (𝐹:𝐼⟶ℕ0 → (◡𝐹 “ (V ∖ {0})) = (◡𝐹 “ ℕ)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 Vcvv 2681 ∖ cdif 3063 ∩ cin 3065 ⊆ wss 3066 {csn 3522 ◡ccnv 4533 dom cdm 4534 “ cima 4537 Fun wfun 5112 ⟶wf 5114 0cc0 7613 ℕcn 8713 ℕ0cn0 8970 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 603 ax-in2 604 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 ax-sep 4041 ax-pow 4093 ax-pr 4126 ax-un 4350 ax-setind 4447 ax-cnex 7704 ax-resscn 7705 ax-1re 7707 ax-addrcl 7710 ax-0lt1 7719 ax-0id 7721 ax-rnegex 7722 ax-pre-ltirr 7725 ax-pre-lttrn 7727 ax-pre-ltadd 7729 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-fal 1337 df-nf 1437 df-sb 1736 df-eu 2000 df-mo 2001 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-ne 2307 df-nel 2402 df-ral 2419 df-rex 2420 df-rab 2423 df-v 2683 df-sbc 2905 df-dif 3068 df-un 3070 df-in 3072 df-ss 3079 df-nul 3359 df-pw 3507 df-sn 3528 df-pr 3529 df-op 3531 df-uni 3732 df-int 3767 df-br 3925 df-opab 3985 df-id 4210 df-xp 4540 df-rel 4541 df-cnv 4542 df-co 4543 df-dm 4544 df-rn 4545 df-res 4546 df-ima 4547 df-iota 5083 df-fun 5120 df-fn 5121 df-f 5122 df-fv 5126 df-ov 5770 df-pnf 7795 df-mnf 7796 df-xr 7797 df-ltxr 7798 df-le 7799 df-inn 8714 df-n0 8971 |
This theorem is referenced by: (None) |
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