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Mirrors > Home > MPE Home > Th. List > cantnfrescl | Structured version Visualization version GIF version |
Description: A function is finitely supported from 𝐵 to 𝐴 iff the extended function is finitely supported from 𝐷 to 𝐴. (Contributed by Mario Carneiro, 25-May-2015.) |
Ref | Expression |
---|---|
cantnfs.s | ⊢ 𝑆 = dom (𝐴 CNF 𝐵) |
cantnfs.a | ⊢ (𝜑 → 𝐴 ∈ On) |
cantnfs.b | ⊢ (𝜑 → 𝐵 ∈ On) |
cantnfrescl.d | ⊢ (𝜑 → 𝐷 ∈ On) |
cantnfrescl.b | ⊢ (𝜑 → 𝐵 ⊆ 𝐷) |
cantnfrescl.x | ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐷 ∖ 𝐵)) → 𝑋 = ∅) |
cantnfrescl.a | ⊢ (𝜑 → ∅ ∈ 𝐴) |
cantnfrescl.t | ⊢ 𝑇 = dom (𝐴 CNF 𝐷) |
Ref | Expression |
---|---|
cantnfrescl | ⊢ (𝜑 → ((𝑛 ∈ 𝐵 ↦ 𝑋) ∈ 𝑆 ↔ (𝑛 ∈ 𝐷 ↦ 𝑋) ∈ 𝑇)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cantnfrescl.b | . . . . 5 ⊢ (𝜑 → 𝐵 ⊆ 𝐷) | |
2 | cantnfrescl.x | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐷 ∖ 𝐵)) → 𝑋 = ∅) | |
3 | cantnfrescl.a | . . . . . . . 8 ⊢ (𝜑 → ∅ ∈ 𝐴) | |
4 | 3 | adantr 484 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐷 ∖ 𝐵)) → ∅ ∈ 𝐴) |
5 | 2, 4 | eqeltrd 2890 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐷 ∖ 𝐵)) → 𝑋 ∈ 𝐴) |
6 | 5 | ralrimiva 3149 | . . . . 5 ⊢ (𝜑 → ∀𝑛 ∈ (𝐷 ∖ 𝐵)𝑋 ∈ 𝐴) |
7 | 1, 6 | raldifeq 4397 | . . . 4 ⊢ (𝜑 → (∀𝑛 ∈ 𝐵 𝑋 ∈ 𝐴 ↔ ∀𝑛 ∈ 𝐷 𝑋 ∈ 𝐴)) |
8 | eqid 2798 | . . . . 5 ⊢ (𝑛 ∈ 𝐵 ↦ 𝑋) = (𝑛 ∈ 𝐵 ↦ 𝑋) | |
9 | 8 | fmpt 6851 | . . . 4 ⊢ (∀𝑛 ∈ 𝐵 𝑋 ∈ 𝐴 ↔ (𝑛 ∈ 𝐵 ↦ 𝑋):𝐵⟶𝐴) |
10 | eqid 2798 | . . . . 5 ⊢ (𝑛 ∈ 𝐷 ↦ 𝑋) = (𝑛 ∈ 𝐷 ↦ 𝑋) | |
11 | 10 | fmpt 6851 | . . . 4 ⊢ (∀𝑛 ∈ 𝐷 𝑋 ∈ 𝐴 ↔ (𝑛 ∈ 𝐷 ↦ 𝑋):𝐷⟶𝐴) |
12 | 7, 9, 11 | 3bitr3g 316 | . . 3 ⊢ (𝜑 → ((𝑛 ∈ 𝐵 ↦ 𝑋):𝐵⟶𝐴 ↔ (𝑛 ∈ 𝐷 ↦ 𝑋):𝐷⟶𝐴)) |
13 | cantnfs.b | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ On) | |
14 | 13 | mptexd 6964 | . . . . 5 ⊢ (𝜑 → (𝑛 ∈ 𝐵 ↦ 𝑋) ∈ V) |
15 | funmpt 6362 | . . . . . 6 ⊢ Fun (𝑛 ∈ 𝐵 ↦ 𝑋) | |
16 | 15 | a1i 11 | . . . . 5 ⊢ (𝜑 → Fun (𝑛 ∈ 𝐵 ↦ 𝑋)) |
17 | cantnfrescl.d | . . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ On) | |
18 | 17 | mptexd 6964 | . . . . . 6 ⊢ (𝜑 → (𝑛 ∈ 𝐷 ↦ 𝑋) ∈ V) |
19 | funmpt 6362 | . . . . . 6 ⊢ Fun (𝑛 ∈ 𝐷 ↦ 𝑋) | |
20 | 18, 19 | jctir 524 | . . . . 5 ⊢ (𝜑 → ((𝑛 ∈ 𝐷 ↦ 𝑋) ∈ V ∧ Fun (𝑛 ∈ 𝐷 ↦ 𝑋))) |
21 | 14, 16, 20 | jca31 518 | . . . 4 ⊢ (𝜑 → (((𝑛 ∈ 𝐵 ↦ 𝑋) ∈ V ∧ Fun (𝑛 ∈ 𝐵 ↦ 𝑋)) ∧ ((𝑛 ∈ 𝐷 ↦ 𝑋) ∈ V ∧ Fun (𝑛 ∈ 𝐷 ↦ 𝑋)))) |
22 | 17, 1, 2 | extmptsuppeq 7837 | . . . 4 ⊢ (𝜑 → ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅) = ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅)) |
23 | suppeqfsuppbi 8831 | . . . 4 ⊢ ((((𝑛 ∈ 𝐵 ↦ 𝑋) ∈ V ∧ Fun (𝑛 ∈ 𝐵 ↦ 𝑋)) ∧ ((𝑛 ∈ 𝐷 ↦ 𝑋) ∈ V ∧ Fun (𝑛 ∈ 𝐷 ↦ 𝑋))) → (((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅) = ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅) → ((𝑛 ∈ 𝐵 ↦ 𝑋) finSupp ∅ ↔ (𝑛 ∈ 𝐷 ↦ 𝑋) finSupp ∅))) | |
24 | 21, 22, 23 | sylc 65 | . . 3 ⊢ (𝜑 → ((𝑛 ∈ 𝐵 ↦ 𝑋) finSupp ∅ ↔ (𝑛 ∈ 𝐷 ↦ 𝑋) finSupp ∅)) |
25 | 12, 24 | anbi12d 633 | . 2 ⊢ (𝜑 → (((𝑛 ∈ 𝐵 ↦ 𝑋):𝐵⟶𝐴 ∧ (𝑛 ∈ 𝐵 ↦ 𝑋) finSupp ∅) ↔ ((𝑛 ∈ 𝐷 ↦ 𝑋):𝐷⟶𝐴 ∧ (𝑛 ∈ 𝐷 ↦ 𝑋) finSupp ∅))) |
26 | cantnfs.s | . . 3 ⊢ 𝑆 = dom (𝐴 CNF 𝐵) | |
27 | cantnfs.a | . . 3 ⊢ (𝜑 → 𝐴 ∈ On) | |
28 | 26, 27, 13 | cantnfs 9113 | . 2 ⊢ (𝜑 → ((𝑛 ∈ 𝐵 ↦ 𝑋) ∈ 𝑆 ↔ ((𝑛 ∈ 𝐵 ↦ 𝑋):𝐵⟶𝐴 ∧ (𝑛 ∈ 𝐵 ↦ 𝑋) finSupp ∅))) |
29 | cantnfrescl.t | . . 3 ⊢ 𝑇 = dom (𝐴 CNF 𝐷) | |
30 | 29, 27, 17 | cantnfs 9113 | . 2 ⊢ (𝜑 → ((𝑛 ∈ 𝐷 ↦ 𝑋) ∈ 𝑇 ↔ ((𝑛 ∈ 𝐷 ↦ 𝑋):𝐷⟶𝐴 ∧ (𝑛 ∈ 𝐷 ↦ 𝑋) finSupp ∅))) |
31 | 25, 28, 30 | 3bitr4d 314 | 1 ⊢ (𝜑 → ((𝑛 ∈ 𝐵 ↦ 𝑋) ∈ 𝑆 ↔ (𝑛 ∈ 𝐷 ↦ 𝑋) ∈ 𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 = wceq 1538 ∈ wcel 2111 ∀wral 3106 Vcvv 3441 ∖ cdif 3878 ⊆ wss 3881 ∅c0 4243 class class class wbr 5030 ↦ cmpt 5110 dom cdm 5519 Oncon0 6159 Fun wfun 6318 ⟶wf 6320 (class class class)co 7135 supp csupp 7813 finSupp cfsupp 8817 CNF ccnf 9108 |
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 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-supp 7814 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-seqom 8067 df-map 8391 df-fsupp 8818 df-cnf 9109 |
This theorem is referenced by: cantnfres 9124 |
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