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| Mirrors > Home > MPE Home > Th. List > Mathboxes > fsuppinisegfi | Structured version Visualization version GIF version | ||
| Description: The initial segment (◡𝐹 “ {𝑌}) of a nonzero 𝑌 is finite if 𝐹 has finite support. (Contributed by Thierry Arnoux, 21-Jun-2024.) |
| Ref | Expression |
|---|---|
| fsuppinisegfi.1 | ⊢ (𝜑 → 𝐹 ∈ 𝑉) |
| fsuppinisegfi.2 | ⊢ (𝜑 → 0 ∈ 𝑊) |
| fsuppinisegfi.3 | ⊢ (𝜑 → 𝑌 ∈ (V ∖ { 0 })) |
| fsuppinisegfi.4 | ⊢ (𝜑 → 𝐹 finSupp 0 ) |
| Ref | Expression |
|---|---|
| fsuppinisegfi | ⊢ (𝜑 → (◡𝐹 “ {𝑌}) ∈ Fin) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | fsuppinisegfi.4 | . . 3 ⊢ (𝜑 → 𝐹 finSupp 0 ) | |
| 2 | 1 | fsuppimpd 9409 | . 2 ⊢ (𝜑 → (𝐹 supp 0 ) ∈ Fin) |
| 3 | fsuppinisegfi.3 | . . . . 5 ⊢ (𝜑 → 𝑌 ∈ (V ∖ { 0 })) | |
| 4 | 3 | snssd 4809 | . . . 4 ⊢ (𝜑 → {𝑌} ⊆ (V ∖ { 0 })) |
| 5 | imass2 6120 | . . . 4 ⊢ ({𝑌} ⊆ (V ∖ { 0 }) → (◡𝐹 “ {𝑌}) ⊆ (◡𝐹 “ (V ∖ { 0 }))) | |
| 6 | 4, 5 | syl 17 | . . 3 ⊢ (𝜑 → (◡𝐹 “ {𝑌}) ⊆ (◡𝐹 “ (V ∖ { 0 }))) |
| 7 | fsuppinisegfi.1 | . . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝑉) | |
| 8 | fsuppinisegfi.2 | . . . 4 ⊢ (𝜑 → 0 ∈ 𝑊) | |
| 9 | suppimacnvss 8198 | . . . 4 ⊢ ((𝐹 ∈ 𝑉 ∧ 0 ∈ 𝑊) → (◡𝐹 “ (V ∖ { 0 })) ⊆ (𝐹 supp 0 )) | |
| 10 | 7, 8, 9 | syl2anc 584 | . . 3 ⊢ (𝜑 → (◡𝐹 “ (V ∖ { 0 })) ⊆ (𝐹 supp 0 )) |
| 11 | 6, 10 | sstrd 3994 | . 2 ⊢ (𝜑 → (◡𝐹 “ {𝑌}) ⊆ (𝐹 supp 0 )) |
| 12 | 2, 11 | ssfid 9301 | 1 ⊢ (𝜑 → (◡𝐹 “ {𝑌}) ∈ Fin) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∈ wcel 2108 Vcvv 3480 ∖ cdif 3948 ⊆ wss 3951 {csn 4626 class class class wbr 5143 ◡ccnv 5684 “ cima 5688 (class class class)co 7431 supp csupp 8185 Fincfn 8985 finSupp cfsupp 9401 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3381 df-rab 3437 df-v 3482 df-sbc 3789 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-pss 3971 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-tr 5260 df-id 5578 df-eprel 5584 df-po 5592 df-so 5593 df-fr 5637 df-we 5639 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-ord 6387 df-on 6388 df-lim 6389 df-suc 6390 df-iota 6514 df-fun 6563 df-fn 6564 df-f 6565 df-f1 6566 df-fo 6567 df-f1o 6568 df-fv 6569 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-supp 8186 df-1o 8506 df-en 8986 df-fin 8989 df-fsupp 9402 |
| This theorem is referenced by: elrspunidl 33456 |
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