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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 8578 | . . . 4 ⊢ ℕ = (ℕ0 ∖ {0}) | |
2 | invdif 3224 | . . . 4 ⊢ (ℕ0 ∩ (V ∖ {0})) = (ℕ0 ∖ {0}) | |
3 | 1, 2 | eqtr4i 2106 | . . 3 ⊢ ℕ = (ℕ0 ∩ (V ∖ {0})) |
4 | 3 | imaeq2i 4727 | . 2 ⊢ (◡𝐹 “ ℕ) = (◡𝐹 “ (ℕ0 ∩ (V ∖ {0}))) |
5 | ffun 5117 | . . . 4 ⊢ (𝐹:𝐼⟶ℕ0 → Fun 𝐹) | |
6 | inpreima 5370 | . . . 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 4750 | . . . . 5 ⊢ (◡𝐹 “ (V ∖ {0})) ⊆ dom 𝐹 | |
9 | fdm 5119 | . . . . . 6 ⊢ (𝐹:𝐼⟶ℕ0 → dom 𝐹 = 𝐼) | |
10 | fimacnv 5373 | . . . . . 6 ⊢ (𝐹:𝐼⟶ℕ0 → (◡𝐹 “ ℕ0) = 𝐼) | |
11 | 9, 10 | eqtr4d 2118 | . . . . 5 ⊢ (𝐹:𝐼⟶ℕ0 → dom 𝐹 = (◡𝐹 “ ℕ0)) |
12 | 8, 11 | syl5sseq 3058 | . . . 4 ⊢ (𝐹:𝐼⟶ℕ0 → (◡𝐹 “ (V ∖ {0})) ⊆ (◡𝐹 “ ℕ0)) |
13 | sseqin2 3203 | . . . 4 ⊢ ((◡𝐹 “ (V ∖ {0})) ⊆ (◡𝐹 “ ℕ0) ↔ ((◡𝐹 “ ℕ0) ∩ (◡𝐹 “ (V ∖ {0}))) = (◡𝐹 “ (V ∖ {0}))) | |
14 | 12, 13 | sylib 120 | . . 3 ⊢ (𝐹:𝐼⟶ℕ0 → ((◡𝐹 “ ℕ0) ∩ (◡𝐹 “ (V ∖ {0}))) = (◡𝐹 “ (V ∖ {0}))) |
15 | 7, 14 | eqtrd 2115 | . 2 ⊢ (𝐹:𝐼⟶ℕ0 → (◡𝐹 “ (ℕ0 ∩ (V ∖ {0}))) = (◡𝐹 “ (V ∖ {0}))) |
16 | 4, 15 | syl5req 2128 | 1 ⊢ (𝐹:𝐼⟶ℕ0 → (◡𝐹 “ (V ∖ {0})) = (◡𝐹 “ ℕ)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1285 Vcvv 2612 ∖ cdif 2981 ∩ cin 2983 ⊆ wss 2984 {csn 3422 ◡ccnv 4400 dom cdm 4401 “ cima 4404 Fun wfun 4963 ⟶wf 4965 0cc0 7253 ℕcn 8316 ℕ0cn0 8565 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-in1 577 ax-in2 578 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-13 1445 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-sep 3922 ax-pow 3974 ax-pr 4000 ax-un 4224 ax-setind 4316 ax-cnex 7339 ax-resscn 7340 ax-1re 7342 ax-addrcl 7345 ax-0lt1 7354 ax-0id 7356 ax-rnegex 7357 ax-pre-ltirr 7360 ax-pre-lttrn 7362 ax-pre-ltadd 7364 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-fal 1291 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ne 2250 df-nel 2345 df-ral 2358 df-rex 2359 df-rab 2362 df-v 2614 df-sbc 2827 df-dif 2986 df-un 2988 df-in 2990 df-ss 2997 df-nul 3270 df-pw 3408 df-sn 3428 df-pr 3429 df-op 3431 df-uni 3628 df-int 3663 df-br 3812 df-opab 3866 df-id 4084 df-xp 4407 df-rel 4408 df-cnv 4409 df-co 4410 df-dm 4411 df-rn 4412 df-res 4413 df-ima 4414 df-iota 4934 df-fun 4971 df-fn 4972 df-f 4973 df-fv 4977 df-ov 5594 df-pnf 7427 df-mnf 7428 df-xr 7429 df-ltxr 7430 df-le 7431 df-inn 8317 df-n0 8566 |
This theorem is referenced by: (None) |
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