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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 2827 | . . . . . 6 ⊢ ((𝜑 ∧ 𝑛 ∈ (𝐷 ∖ 𝐵)) → 𝑋 ∈ 𝐴) |
6 | 5 | ralrimiva 3140 | . . . . 5 ⊢ (𝜑 → ∀𝑛 ∈ (𝐷 ∖ 𝐵)𝑋 ∈ 𝐴) |
7 | 1, 6 | raldifeq 4488 | . . . 4 ⊢ (𝜑 → (∀𝑛 ∈ 𝐵 𝑋 ∈ 𝐴 ↔ ∀𝑛 ∈ 𝐷 𝑋 ∈ 𝐴)) |
8 | eqid 2726 | . . . . 5 ⊢ (𝑛 ∈ 𝐵 ↦ 𝑋) = (𝑛 ∈ 𝐵 ↦ 𝑋) | |
9 | 8 | fmpt 7105 | . . . 4 ⊢ (∀𝑛 ∈ 𝐵 𝑋 ∈ 𝐴 ↔ (𝑛 ∈ 𝐵 ↦ 𝑋):𝐵⟶𝐴) |
10 | eqid 2726 | . . . . 5 ⊢ (𝑛 ∈ 𝐷 ↦ 𝑋) = (𝑛 ∈ 𝐷 ↦ 𝑋) | |
11 | 10 | fmpt 7105 | . . . 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 7221 | . . . . 5 ⊢ (𝜑 → (𝑛 ∈ 𝐵 ↦ 𝑋) ∈ V) |
15 | funmpt 6580 | . . . . . 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 7221 | . . . . . 6 ⊢ (𝜑 → (𝑛 ∈ 𝐷 ↦ 𝑋) ∈ V) |
19 | funmpt 6580 | . . . . . 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 8173 | . . . 4 ⊢ (𝜑 → ((𝑛 ∈ 𝐵 ↦ 𝑋) supp ∅) = ((𝑛 ∈ 𝐷 ↦ 𝑋) supp ∅)) |
23 | suppeqfsuppbi 9379 | . . . 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 630 | . 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 9663 | . 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 9663 | . 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 1533 ∈ wcel 2098 ∀wral 3055 Vcvv 3468 ∖ cdif 3940 ⊆ wss 3943 ∅c0 4317 class class class wbr 5141 ↦ cmpt 5224 dom cdm 5669 Oncon0 6358 Fun wfun 6531 ⟶wf 6533 (class class class)co 7405 supp csupp 8146 finSupp cfsupp 9363 CNF ccnf 9658 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2697 ax-rep 5278 ax-sep 5292 ax-nul 5299 ax-pow 5356 ax-pr 5420 ax-un 7722 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2704 df-cleq 2718 df-clel 2804 df-nfc 2879 df-ne 2935 df-ral 3056 df-rex 3065 df-reu 3371 df-rab 3427 df-v 3470 df-sbc 3773 df-csb 3889 df-dif 3946 df-un 3948 df-in 3950 df-ss 3960 df-nul 4318 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-iun 4992 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-pred 6294 df-iota 6489 df-fun 6539 df-fn 6540 df-f 6541 df-f1 6542 df-fo 6543 df-f1o 6544 df-fv 6545 df-ov 7408 df-oprab 7409 df-mpo 7410 df-supp 8147 df-frecs 8267 df-wrecs 8298 df-recs 8372 df-rdg 8411 df-seqom 8449 df-map 8824 df-fsupp 9364 df-cnf 9659 |
This theorem is referenced by: cantnfres 9674 |
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