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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 8990 | . . . 4 ⊢ ℕ = (ℕ0 ∖ {0}) | |
2 | invdif 3318 | . . . 4 ⊢ (ℕ0 ∩ (V ∖ {0})) = (ℕ0 ∖ {0}) | |
3 | 1, 2 | eqtr4i 2163 | . . 3 ⊢ ℕ = (ℕ0 ∩ (V ∖ {0})) |
4 | 3 | imaeq2i 4879 | . 2 ⊢ (◡𝐹 “ ℕ) = (◡𝐹 “ (ℕ0 ∩ (V ∖ {0}))) |
5 | ffun 5275 | . . . 4 ⊢ (𝐹:𝐼⟶ℕ0 → Fun 𝐹) | |
6 | inpreima 5546 | . . . 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 4902 | . . . . 5 ⊢ (◡𝐹 “ (V ∖ {0})) ⊆ dom 𝐹 | |
9 | fdm 5278 | . . . . . 6 ⊢ (𝐹:𝐼⟶ℕ0 → dom 𝐹 = 𝐼) | |
10 | fimacnv 5549 | . . . . . 6 ⊢ (𝐹:𝐼⟶ℕ0 → (◡𝐹 “ ℕ0) = 𝐼) | |
11 | 9, 10 | eqtr4d 2175 | . . . . 5 ⊢ (𝐹:𝐼⟶ℕ0 → dom 𝐹 = (◡𝐹 “ ℕ0)) |
12 | 8, 11 | sseqtrid 3147 | . . . 4 ⊢ (𝐹:𝐼⟶ℕ0 → (◡𝐹 “ (V ∖ {0})) ⊆ (◡𝐹 “ ℕ0)) |
13 | sseqin2 3295 | . . . 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 2172 | . 2 ⊢ (𝐹:𝐼⟶ℕ0 → (◡𝐹 “ (ℕ0 ∩ (V ∖ {0}))) = (◡𝐹 “ (V ∖ {0}))) |
16 | 4, 15 | syl5req 2185 | 1 ⊢ (𝐹:𝐼⟶ℕ0 → (◡𝐹 “ (V ∖ {0})) = (◡𝐹 “ ℕ)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1331 Vcvv 2686 ∖ cdif 3068 ∩ cin 3070 ⊆ wss 3071 {csn 3527 ◡ccnv 4538 dom cdm 4539 “ cima 4542 Fun wfun 5117 ⟶wf 5119 0cc0 7620 ℕcn 8720 ℕ0cn0 8977 |
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 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 ax-setind 4452 ax-cnex 7711 ax-resscn 7712 ax-1re 7714 ax-addrcl 7717 ax-0lt1 7726 ax-0id 7728 ax-rnegex 7729 ax-pre-ltirr 7732 ax-pre-lttrn 7734 ax-pre-ltadd 7736 |
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 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ne 2309 df-nel 2404 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-sbc 2910 df-dif 3073 df-un 3075 df-in 3077 df-ss 3084 df-nul 3364 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-int 3772 df-br 3930 df-opab 3990 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fv 5131 df-ov 5777 df-pnf 7802 df-mnf 7803 df-xr 7804 df-ltxr 7805 df-le 7806 df-inn 8721 df-n0 8978 |
This theorem is referenced by: (None) |
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