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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 9576 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 9577 | . . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑅 Or 𝐴 ∧ 𝑆 Or 𝐵) ∧ 𝑍 ∈ V) → 𝑇 Or 𝑈) |
4 | 3 | expcom 412 | . 2 ⊢ (𝑍 ∈ V → ((𝐴 ∈ 𝑉 ∧ 𝑅 Or 𝐴 ∧ 𝑆 Or 𝐵) → 𝑇 Or 𝑈)) |
5 | so0 5626 | . . . 4 ⊢ 𝑇 Or ∅ | |
6 | relfsupp 9389 | . . . . . . . . . 10 ⊢ Rel finSupp | |
7 | 6 | brrelex2i 5735 | . . . . . . . . 9 ⊢ (𝑥 finSupp 𝑍 → 𝑍 ∈ V) |
8 | 7 | con3i 154 | . . . . . . . 8 ⊢ (¬ 𝑍 ∈ V → ¬ 𝑥 finSupp 𝑍) |
9 | 8 | ralrimivw 3139 | . . . . . . 7 ⊢ (¬ 𝑍 ∈ V → ∀𝑥 ∈ (𝐵 ↑m 𝐴) ¬ 𝑥 finSupp 𝑍) |
10 | rabeq0 4386 | . . . . . . 7 ⊢ ({𝑥 ∈ (𝐵 ↑m 𝐴) ∣ 𝑥 finSupp 𝑍} = ∅ ↔ ∀𝑥 ∈ (𝐵 ↑m 𝐴) ¬ 𝑥 finSupp 𝑍) | |
11 | 9, 10 | sylibr 233 | . . . . . 6 ⊢ (¬ 𝑍 ∈ V → {𝑥 ∈ (𝐵 ↑m 𝐴) ∣ 𝑥 finSupp 𝑍} = ∅) |
12 | 2, 11 | eqtrid 2777 | . . . . 5 ⊢ (¬ 𝑍 ∈ V → 𝑈 = ∅) |
13 | soeq2 5612 | . . . . 5 ⊢ (𝑈 = ∅ → (𝑇 Or 𝑈 ↔ 𝑇 Or ∅)) | |
14 | 12, 13 | syl 17 | . . . 4 ⊢ (¬ 𝑍 ∈ V → (𝑇 Or 𝑈 ↔ 𝑇 Or ∅)) |
15 | 5, 14 | mpbiri 257 | . . 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 394 ∧ w3a 1084 = wceq 1533 ∈ wcel 2098 ∀wral 3050 ∃wrex 3059 {crab 3418 Vcvv 3461 ∅c0 4322 class class class wbr 5149 {copab 5211 Or wor 5589 ‘cfv 6549 (class class class)co 7419 ↑m cmap 8845 finSupp cfsupp 9387 |
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 2166 ax-ext 2696 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pow 5365 ax-pr 5429 ax-un 7741 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3or 1085 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 2703 df-cleq 2717 df-clel 2802 df-nfc 2877 df-ne 2930 df-ral 3051 df-rex 3060 df-reu 3364 df-rab 3419 df-v 3463 df-sbc 3774 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-pss 3964 df-nul 4323 df-if 4531 df-pw 4606 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-iun 4999 df-br 5150 df-opab 5212 df-mpt 5233 df-tr 5267 df-id 5576 df-eprel 5582 df-po 5590 df-so 5591 df-fr 5633 df-we 5635 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-ord 6374 df-on 6375 df-lim 6376 df-suc 6377 df-iota 6501 df-fun 6551 df-fn 6552 df-f 6553 df-f1 6554 df-fo 6555 df-f1o 6556 df-fv 6557 df-ov 7422 df-oprab 7423 df-mpo 7424 df-om 7872 df-1st 7994 df-2nd 7995 df-supp 8166 df-1o 8487 df-map 8847 df-en 8965 df-fin 8968 df-fsupp 9388 |
This theorem is referenced by: oemapso 9707 |
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