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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 9165 | . . . 4 ⊢ ℕ = (ℕ0 ∖ {0}) | |
2 | invdif 3377 | . . . 4 ⊢ (ℕ0 ∩ (V ∖ {0})) = (ℕ0 ∖ {0}) | |
3 | 1, 2 | eqtr4i 2201 | . . 3 ⊢ ℕ = (ℕ0 ∩ (V ∖ {0})) |
4 | 3 | imaeq2i 4963 | . 2 ⊢ (◡𝐹 “ ℕ) = (◡𝐹 “ (ℕ0 ∩ (V ∖ {0}))) |
5 | ffun 5363 | . . . 4 ⊢ (𝐹:𝐼⟶ℕ0 → Fun 𝐹) | |
6 | inpreima 5637 | . . . 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 4986 | . . . . 5 ⊢ (◡𝐹 “ (V ∖ {0})) ⊆ dom 𝐹 | |
9 | fdm 5366 | . . . . . 6 ⊢ (𝐹:𝐼⟶ℕ0 → dom 𝐹 = 𝐼) | |
10 | fimacnv 5640 | . . . . . 6 ⊢ (𝐹:𝐼⟶ℕ0 → (◡𝐹 “ ℕ0) = 𝐼) | |
11 | 9, 10 | eqtr4d 2213 | . . . . 5 ⊢ (𝐹:𝐼⟶ℕ0 → dom 𝐹 = (◡𝐹 “ ℕ0)) |
12 | 8, 11 | sseqtrid 3205 | . . . 4 ⊢ (𝐹:𝐼⟶ℕ0 → (◡𝐹 “ (V ∖ {0})) ⊆ (◡𝐹 “ ℕ0)) |
13 | sseqin2 3354 | . . . 4 ⊢ ((◡𝐹 “ (V ∖ {0})) ⊆ (◡𝐹 “ ℕ0) ↔ ((◡𝐹 “ ℕ0) ∩ (◡𝐹 “ (V ∖ {0}))) = (◡𝐹 “ (V ∖ {0}))) | |
14 | 12, 13 | sylib 122 | . . 3 ⊢ (𝐹:𝐼⟶ℕ0 → ((◡𝐹 “ ℕ0) ∩ (◡𝐹 “ (V ∖ {0}))) = (◡𝐹 “ (V ∖ {0}))) |
15 | 7, 14 | eqtrd 2210 | . 2 ⊢ (𝐹:𝐼⟶ℕ0 → (◡𝐹 “ (ℕ0 ∩ (V ∖ {0}))) = (◡𝐹 “ (V ∖ {0}))) |
16 | 4, 15 | eqtr2id 2223 | 1 ⊢ (𝐹:𝐼⟶ℕ0 → (◡𝐹 “ (V ∖ {0})) = (◡𝐹 “ ℕ)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 = wceq 1353 Vcvv 2737 ∖ cdif 3126 ∩ cin 3128 ⊆ wss 3129 {csn 3591 ◡ccnv 4621 dom cdm 4622 “ cima 4625 Fun wfun 5205 ⟶wf 5207 0cc0 7789 ℕcn 8895 ℕ0cn0 9152 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 614 ax-in2 615 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-13 2150 ax-14 2151 ax-ext 2159 ax-sep 4118 ax-pow 4171 ax-pr 4205 ax-un 4429 ax-setind 4532 ax-cnex 7880 ax-resscn 7881 ax-1re 7883 ax-addrcl 7886 ax-0lt1 7895 ax-0id 7897 ax-rnegex 7898 ax-pre-ltirr 7901 ax-pre-lttrn 7903 ax-pre-ltadd 7905 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-fal 1359 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ne 2348 df-nel 2443 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2739 df-sbc 2963 df-dif 3131 df-un 3133 df-in 3135 df-ss 3142 df-nul 3423 df-pw 3576 df-sn 3597 df-pr 3598 df-op 3600 df-uni 3808 df-int 3843 df-br 4001 df-opab 4062 df-id 4289 df-xp 4628 df-rel 4629 df-cnv 4630 df-co 4631 df-dm 4632 df-rn 4633 df-res 4634 df-ima 4635 df-iota 5173 df-fun 5213 df-fn 5214 df-f 5215 df-fv 5219 df-ov 5871 df-pnf 7971 df-mnf 7972 df-xr 7973 df-ltxr 7974 df-le 7975 df-inn 8896 df-n0 9153 |
This theorem is referenced by: (None) |
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