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Mirrors > Home > MPE Home > Th. List > cantnfs | Structured version Visualization version GIF version |
Description: Elementhood in the set of finitely supported functions from 𝐵 to 𝐴. (Contributed by Mario Carneiro, 25-May-2015.) (Revised by AV, 28-Jun-2019.) |
Ref | Expression |
---|---|
cantnfs.s | ⊢ 𝑆 = dom (𝐴 CNF 𝐵) |
cantnfs.a | ⊢ (𝜑 → 𝐴 ∈ On) |
cantnfs.b | ⊢ (𝜑 → 𝐵 ∈ On) |
Ref | Expression |
---|---|
cantnfs | ⊢ (𝜑 → (𝐹 ∈ 𝑆 ↔ (𝐹:𝐵⟶𝐴 ∧ 𝐹 finSupp ∅))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cantnfs.s | . . . . 5 ⊢ 𝑆 = dom (𝐴 CNF 𝐵) | |
2 | eqid 2733 | . . . . . 6 ⊢ {𝑔 ∈ (𝐴 ↑m 𝐵) ∣ 𝑔 finSupp ∅} = {𝑔 ∈ (𝐴 ↑m 𝐵) ∣ 𝑔 finSupp ∅} | |
3 | cantnfs.a | . . . . . 6 ⊢ (𝜑 → 𝐴 ∈ On) | |
4 | cantnfs.b | . . . . . 6 ⊢ (𝜑 → 𝐵 ∈ On) | |
5 | 2, 3, 4 | cantnfdm 9608 | . . . . 5 ⊢ (𝜑 → dom (𝐴 CNF 𝐵) = {𝑔 ∈ (𝐴 ↑m 𝐵) ∣ 𝑔 finSupp ∅}) |
6 | 1, 5 | eqtrid 2785 | . . . 4 ⊢ (𝜑 → 𝑆 = {𝑔 ∈ (𝐴 ↑m 𝐵) ∣ 𝑔 finSupp ∅}) |
7 | 6 | eleq2d 2820 | . . 3 ⊢ (𝜑 → (𝐹 ∈ 𝑆 ↔ 𝐹 ∈ {𝑔 ∈ (𝐴 ↑m 𝐵) ∣ 𝑔 finSupp ∅})) |
8 | breq1 5112 | . . . 4 ⊢ (𝑔 = 𝐹 → (𝑔 finSupp ∅ ↔ 𝐹 finSupp ∅)) | |
9 | 8 | elrab 3649 | . . 3 ⊢ (𝐹 ∈ {𝑔 ∈ (𝐴 ↑m 𝐵) ∣ 𝑔 finSupp ∅} ↔ (𝐹 ∈ (𝐴 ↑m 𝐵) ∧ 𝐹 finSupp ∅)) |
10 | 7, 9 | bitrdi 287 | . 2 ⊢ (𝜑 → (𝐹 ∈ 𝑆 ↔ (𝐹 ∈ (𝐴 ↑m 𝐵) ∧ 𝐹 finSupp ∅))) |
11 | 3, 4 | elmapd 8785 | . . 3 ⊢ (𝜑 → (𝐹 ∈ (𝐴 ↑m 𝐵) ↔ 𝐹:𝐵⟶𝐴)) |
12 | 11 | anbi1d 631 | . 2 ⊢ (𝜑 → ((𝐹 ∈ (𝐴 ↑m 𝐵) ∧ 𝐹 finSupp ∅) ↔ (𝐹:𝐵⟶𝐴 ∧ 𝐹 finSupp ∅))) |
13 | 10, 12 | bitrd 279 | 1 ⊢ (𝜑 → (𝐹 ∈ 𝑆 ↔ (𝐹:𝐵⟶𝐴 ∧ 𝐹 finSupp ∅))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 {crab 3406 ∅c0 4286 class class class wbr 5109 dom cdm 5637 Oncon0 6321 ⟶wf 6496 (class class class)co 7361 ↑m cmap 8771 finSupp cfsupp 9311 CNF ccnf 9605 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-id 5535 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-ov 7364 df-oprab 7365 df-mpo 7366 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-seqom 8398 df-map 8773 df-cnf 9606 |
This theorem is referenced by: cantnfcl 9611 cantnfle 9615 cantnflt 9616 cantnff 9618 cantnf0 9619 cantnfrescl 9620 cantnfp1lem1 9622 cantnfp1lem2 9623 cantnfp1lem3 9624 cantnfp1 9625 oemapvali 9628 cantnflem1a 9629 cantnflem1b 9630 cantnflem1c 9631 cantnflem1d 9632 cantnflem1 9633 cantnflem3 9635 cantnf 9637 cnfcomlem 9643 cnfcom 9644 cnfcom2lem 9645 cnfcom3lem 9647 cnfcom3 9648 cantnfub 41703 cantnfresb 41706 cantnf2 41707 naddcnff 41725 naddcnffo 41727 naddcnfcom 41729 naddcnfid1 41730 naddcnfid2 41731 naddcnfass 41732 |
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