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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 480 | . . . . . . 7 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐷 ∖ 𝐵)) → ∅ ∈ 𝐴) |
5 | 2, 4 | eqeltrd 2829 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐷 ∖ 𝐵)) → 𝑋 ∈ 𝐴) |
6 | 5 | ralrimiva 3142 | . . . . 5 ⊢ (𝜑 → ∀𝑛 ∈ (𝐷 ∖ 𝐵)𝑋 ∈ 𝐴) |
7 | 1, 6 | raldifeq 4490 | . . . 4 ⊢ (𝜑 → (∀𝑛 ∈ 𝐵 𝑋 ∈ 𝐴 ↔ ∀𝑛 ∈ 𝐷 𝑋 ∈ 𝐴)) |
8 | eqid 2728 | . . . . 5 ⊢ (𝑛 ∈ 𝐵 ↦ 𝑋) = (𝑛 ∈ 𝐵 ↦ 𝑋) | |
9 | 8 | fmpt 7115 | . . . 4 ⊢ (∀𝑛 ∈ 𝐵 𝑋 ∈ 𝐴 ↔ (𝑛 ∈ 𝐵 ↦ 𝑋):𝐵⟶𝐴) |
10 | eqid 2728 | . . . . 5 ⊢ (𝑛 ∈ 𝐷 ↦ 𝑋) = (𝑛 ∈ 𝐷 ↦ 𝑋) | |
11 | 10 | fmpt 7115 | . . . 4 ⊢ (∀𝑛 ∈ 𝐷 𝑋 ∈ 𝐴 ↔ (𝑛 ∈ 𝐷 ↦ 𝑋):𝐷⟶𝐴) |
12 | 7, 9, 11 | 3bitr3g 313 | . . 3 ⊢ (𝜑 → ((𝑛 ∈ 𝐵 ↦ 𝑋):𝐵⟶𝐴 ↔ (𝑛 ∈ 𝐷 ↦ 𝑋):𝐷⟶𝐴)) |
13 | cantnfs.b | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ On) | |
14 | 13 | mptexd 7231 | . . . . 5 ⊢ (𝜑 → (𝑛 ∈ 𝐵 ↦ 𝑋) ∈ V) |
15 | funmpt 6586 | . . . . . 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 7231 | . . . . . 6 ⊢ (𝜑 → (𝑛 ∈ 𝐷 ↦ 𝑋) ∈ V) |
19 | funmpt 6586 | . . . . . 6 ⊢ Fun (𝑛 ∈ 𝐷 ↦ 𝑋) | |
20 | 18, 19 | jctir 520 | . . . . 5 ⊢ (𝜑 → ((𝑛 ∈ 𝐷 ↦ 𝑋) ∈ V ∧ Fun (𝑛 ∈ 𝐷 ↦ 𝑋))) |
21 | 14, 16, 20 | jca31 514 | . . . 4 ⊢ (𝜑 → (((𝑛 ∈ 𝐵 ↦ 𝑋) ∈ V ∧ Fun (𝑛 ∈ 𝐵 ↦ 𝑋)) ∧ ((𝑛 ∈ 𝐷 ↦ 𝑋) ∈ V ∧ Fun (𝑛 ∈ 𝐷 ↦ 𝑋)))) |
22 | 17, 1, 2 | extmptsuppeq 8187 | . . . 4 ⊢ (𝜑 → ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅) = ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅)) |
23 | suppeqfsuppbi 9397 | . . . 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 631 | . 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 9684 | . 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 9684 | . 2 ⊢ (𝜑 → ((𝑛 ∈ 𝐷 ↦ 𝑋) ∈ 𝑇 ↔ ((𝑛 ∈ 𝐷 ↦ 𝑋):𝐷⟶𝐴 ∧ (𝑛 ∈ 𝐷 ↦ 𝑋) finSupp ∅))) |
31 | 25, 28, 30 | 3bitr4d 311 | 1 ⊢ (𝜑 → ((𝑛 ∈ 𝐵 ↦ 𝑋) ∈ 𝑆 ↔ (𝑛 ∈ 𝐷 ↦ 𝑋) ∈ 𝑇)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ∀wral 3057 Vcvv 3470 ∖ cdif 3942 ⊆ wss 3945 ∅c0 4319 class class class wbr 5143 ↦ cmpt 5226 dom cdm 5673 Oncon0 6364 Fun wfun 6537 ⟶wf 6539 (class class class)co 7415 supp csupp 8160 finSupp cfsupp 9380 CNF ccnf 9679 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pow 5360 ax-pr 5424 ax-un 7735 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-reu 3373 df-rab 3429 df-v 3472 df-sbc 3776 df-csb 3891 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4320 df-if 4526 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-iun 4994 df-br 5144 df-opab 5206 df-mpt 5227 df-id 5571 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-pred 6300 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-ov 7418 df-oprab 7419 df-mpo 7420 df-supp 8161 df-frecs 8281 df-wrecs 8312 df-recs 8386 df-rdg 8425 df-seqom 8463 df-map 8841 df-fsupp 9381 df-cnf 9680 |
This theorem is referenced by: cantnfres 9695 |
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