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Mirrors > Home > MPE Home > Th. List > wemapso2 | Structured version Visualization version GIF version |
Description: An alternative to having a well-order on 𝑅 in wemapso 9350 is to restrict the function set to finitely-supported functions. (Contributed by Mario Carneiro, 8-Feb-2015.) (Revised by AV, 1-Jul-2019.) |
Ref | Expression |
---|---|
wemapso.t | ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐴 ((𝑥‘𝑧)𝑆(𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤)))} |
wemapso2.u | ⊢ 𝑈 = {𝑥 ∈ (𝐵 ↑m 𝐴) ∣ 𝑥 finSupp 𝑍} |
Ref | Expression |
---|---|
wemapso2 | ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 Or 𝐴 ∧ 𝑆 Or 𝐵) → 𝑇 Or 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wemapso.t | . . . 4 ⊢ 𝑇 = {⟨𝑥, 𝑦⟩ ∣ ∃𝑧 ∈ 𝐴 ((𝑥‘𝑧)𝑆(𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐴 (𝑤𝑅𝑧 → (𝑥‘𝑤) = (𝑦‘𝑤)))} | |
2 | wemapso2.u | . . . 4 ⊢ 𝑈 = {𝑥 ∈ (𝐵 ↑m 𝐴) ∣ 𝑥 finSupp 𝑍} | |
3 | 1, 2 | wemapso2lem 9351 | . . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑅 Or 𝐴 ∧ 𝑆 Or 𝐵) ∧ 𝑍 ∈ V) → 𝑇 Or 𝑈) |
4 | 3 | expcom 415 | . 2 ⊢ (𝑍 ∈ V → ((𝐴 ∈ 𝑉 ∧ 𝑅 Or 𝐴 ∧ 𝑆 Or 𝐵) → 𝑇 Or 𝑈)) |
5 | so0 5546 | . . . 4 ⊢ 𝑇 Or ∅ | |
6 | relfsupp 9170 | . . . . . . . . . 10 ⊢ Rel finSupp | |
7 | 6 | brrelex2i 5651 | . . . . . . . . 9 ⊢ (𝑥 finSupp 𝑍 → 𝑍 ∈ V) |
8 | 7 | con3i 154 | . . . . . . . 8 ⊢ (¬ 𝑍 ∈ V → ¬ 𝑥 finSupp 𝑍) |
9 | 8 | ralrimivw 3144 | . . . . . . 7 ⊢ (¬ 𝑍 ∈ V → ∀𝑥 ∈ (𝐵 ↑m 𝐴) ¬ 𝑥 finSupp 𝑍) |
10 | rabeq0 4324 | . . . . . . 7 ⊢ ({𝑥 ∈ (𝐵 ↑m 𝐴) ∣ 𝑥 finSupp 𝑍} = ∅ ↔ ∀𝑥 ∈ (𝐵 ↑m 𝐴) ¬ 𝑥 finSupp 𝑍) | |
11 | 9, 10 | sylibr 234 | . . . . . 6 ⊢ (¬ 𝑍 ∈ V → {𝑥 ∈ (𝐵 ↑m 𝐴) ∣ 𝑥 finSupp 𝑍} = ∅) |
12 | 2, 11 | eqtrid 2788 | . . . . 5 ⊢ (¬ 𝑍 ∈ V → 𝑈 = ∅) |
13 | soeq2 5532 | . . . . 5 ⊢ (𝑈 = ∅ → (𝑇 Or 𝑈 ↔ 𝑇 Or ∅)) | |
14 | 12, 13 | syl 17 | . . . 4 ⊢ (¬ 𝑍 ∈ V → (𝑇 Or 𝑈 ↔ 𝑇 Or ∅)) |
15 | 5, 14 | mpbiri 259 | . . 3 ⊢ (¬ 𝑍 ∈ V → 𝑇 Or 𝑈) |
16 | 15 | a1d 25 | . 2 ⊢ (¬ 𝑍 ∈ V → ((𝐴 ∈ 𝑉 ∧ 𝑅 Or 𝐴 ∧ 𝑆 Or 𝐵) → 𝑇 Or 𝑈)) |
17 | 4, 16 | pm2.61i 182 | 1 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝑅 Or 𝐴 ∧ 𝑆 Or 𝐵) → 𝑇 Or 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 397 ∧ w3a 1087 = wceq 1539 ∈ wcel 2104 ∀wral 3062 ∃wrex 3071 {crab 3284 Vcvv 3437 ∅c0 4262 class class class wbr 5081 {copab 5143 Or wor 5509 ‘cfv 6454 (class class class)co 7303 ↑m cmap 8642 finSupp cfsupp 9168 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2707 ax-rep 5218 ax-sep 5232 ax-nul 5239 ax-pow 5297 ax-pr 5361 ax-un 7616 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 846 df-3or 1088 df-3an 1089 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2887 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3286 df-rab 3287 df-v 3439 df-sbc 3722 df-csb 3838 df-dif 3895 df-un 3897 df-in 3899 df-ss 3909 df-pss 3911 df-nul 4263 df-if 4466 df-pw 4541 df-sn 4566 df-pr 4568 df-op 4572 df-uni 4845 df-iun 4933 df-br 5082 df-opab 5144 df-mpt 5165 df-tr 5199 df-id 5496 df-eprel 5502 df-po 5510 df-so 5511 df-fr 5551 df-we 5553 df-xp 5602 df-rel 5603 df-cnv 5604 df-co 5605 df-dm 5606 df-rn 5607 df-res 5608 df-ima 5609 df-ord 6280 df-on 6281 df-lim 6282 df-suc 6283 df-iota 6406 df-fun 6456 df-fn 6457 df-f 6458 df-f1 6459 df-fo 6460 df-f1o 6461 df-fv 6462 df-ov 7306 df-oprab 7307 df-mpo 7308 df-om 7741 df-1st 7859 df-2nd 7860 df-supp 8005 df-1o 8324 df-map 8644 df-en 8761 df-fin 8764 df-fsupp 9169 |
This theorem is referenced by: oemapso 9480 |
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