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Mirrors > Home > MPE Home > Th. List > frnnn0fsuppg | Structured version Visualization version GIF version |
Description: Version of frnnn0fsupp 12004 avoiding ax-rep 5160 by assuming 𝐹 is a set rather than its domain 𝐼. (Contributed by SN, 5-Aug-2024.) |
Ref | Expression |
---|---|
frnnn0fsuppg | ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐼⟶ℕ0) → (𝐹 finSupp 0 ↔ (◡𝐹 “ ℕ) ∈ Fin)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ffun 6506 | . . 3 ⊢ (𝐹:𝐼⟶ℕ0 → Fun 𝐹) | |
2 | simpl 486 | . . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐼⟶ℕ0) → 𝐹 ∈ 𝑉) | |
3 | c0ex 10686 | . . . 4 ⊢ 0 ∈ V | |
4 | funisfsupp 8884 | . . . 4 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉 ∧ 0 ∈ V) → (𝐹 finSupp 0 ↔ (𝐹 supp 0) ∈ Fin)) | |
5 | 3, 4 | mp3an3 1447 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝐹 ∈ 𝑉) → (𝐹 finSupp 0 ↔ (𝐹 supp 0) ∈ Fin)) |
6 | 1, 2, 5 | syl2an2 685 | . 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐼⟶ℕ0) → (𝐹 finSupp 0 ↔ (𝐹 supp 0) ∈ Fin)) |
7 | frnnn0suppg 12005 | . . 3 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐼⟶ℕ0) → (𝐹 supp 0) = (◡𝐹 “ ℕ)) | |
8 | 7 | eleq1d 2836 | . 2 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐼⟶ℕ0) → ((𝐹 supp 0) ∈ Fin ↔ (◡𝐹 “ ℕ) ∈ Fin)) |
9 | 6, 8 | bitrd 282 | 1 ⊢ ((𝐹 ∈ 𝑉 ∧ 𝐹:𝐼⟶ℕ0) → (𝐹 finSupp 0 ↔ (◡𝐹 “ ℕ) ∈ Fin)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∈ wcel 2111 Vcvv 3409 class class class wbr 5036 ◡ccnv 5527 “ cima 5531 Fun wfun 6334 ⟶wf 6336 (class class class)co 7156 supp csupp 7841 Fincfn 8540 finSupp cfsupp 8879 0cc0 10588 ℕcn 11687 ℕ0cn0 11947 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2729 ax-sep 5173 ax-nul 5180 ax-pow 5238 ax-pr 5302 ax-un 7465 ax-resscn 10645 ax-1cn 10646 ax-icn 10647 ax-addcl 10648 ax-addrcl 10649 ax-mulcl 10650 ax-mulrcl 10651 ax-mulcom 10652 ax-addass 10653 ax-mulass 10654 ax-distr 10655 ax-i2m1 10656 ax-1ne0 10657 ax-1rid 10658 ax-rnegex 10659 ax-rrecex 10660 ax-cnre 10661 ax-pre-lttri 10662 ax-pre-lttrn 10663 ax-pre-ltadd 10664 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2557 df-eu 2588 df-clab 2736 df-cleq 2750 df-clel 2830 df-nfc 2901 df-ne 2952 df-nel 3056 df-ral 3075 df-rex 3076 df-reu 3077 df-rab 3079 df-v 3411 df-sbc 3699 df-csb 3808 df-dif 3863 df-un 3865 df-in 3867 df-ss 3877 df-pss 3879 df-nul 4228 df-if 4424 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4802 df-iun 4888 df-br 5037 df-opab 5099 df-mpt 5117 df-tr 5143 df-id 5434 df-eprel 5439 df-po 5447 df-so 5448 df-fr 5487 df-we 5489 df-xp 5534 df-rel 5535 df-cnv 5536 df-co 5537 df-dm 5538 df-rn 5539 df-res 5540 df-ima 5541 df-pred 6131 df-ord 6177 df-on 6178 df-lim 6179 df-suc 6180 df-iota 6299 df-fun 6342 df-fn 6343 df-f 6344 df-f1 6345 df-fo 6346 df-f1o 6347 df-fv 6348 df-ov 7159 df-oprab 7160 df-mpo 7161 df-om 7586 df-supp 7842 df-wrecs 7963 df-recs 8024 df-rdg 8062 df-er 8305 df-en 8541 df-dom 8542 df-sdom 8543 df-fsupp 8880 df-pnf 10728 df-mnf 10729 df-xr 10730 df-ltxr 10731 df-le 10732 df-nn 11688 df-n0 11948 |
This theorem is referenced by: psrbagfsupp 20695 |
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