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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 9145 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 9146 | . . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑅 Or 𝐴 ∧ 𝑆 Or 𝐵) ∧ 𝑍 ∈ V) → 𝑇 Or 𝑈) |
4 | 3 | expcom 417 | . 2 ⊢ (𝑍 ∈ V → ((𝐴 ∈ 𝑉 ∧ 𝑅 Or 𝐴 ∧ 𝑆 Or 𝐵) → 𝑇 Or 𝑈)) |
5 | so0 5489 | . . . 4 ⊢ 𝑇 Or ∅ | |
6 | relfsupp 8965 | . . . . . . . . . 10 ⊢ Rel finSupp | |
7 | 6 | brrelex2i 5591 | . . . . . . . . 9 ⊢ (𝑥 finSupp 𝑍 → 𝑍 ∈ V) |
8 | 7 | con3i 157 | . . . . . . . 8 ⊢ (¬ 𝑍 ∈ V → ¬ 𝑥 finSupp 𝑍) |
9 | 8 | ralrimivw 3096 | . . . . . . 7 ⊢ (¬ 𝑍 ∈ V → ∀𝑥 ∈ (𝐵 ↑m 𝐴) ¬ 𝑥 finSupp 𝑍) |
10 | rabeq0 4285 | . . . . . . 7 ⊢ ({𝑥 ∈ (𝐵 ↑m 𝐴) ∣ 𝑥 finSupp 𝑍} = ∅ ↔ ∀𝑥 ∈ (𝐵 ↑m 𝐴) ¬ 𝑥 finSupp 𝑍) | |
11 | 9, 10 | sylibr 237 | . . . . . 6 ⊢ (¬ 𝑍 ∈ V → {𝑥 ∈ (𝐵 ↑m 𝐴) ∣ 𝑥 finSupp 𝑍} = ∅) |
12 | 2, 11 | syl5eq 2783 | . . . . 5 ⊢ (¬ 𝑍 ∈ V → 𝑈 = ∅) |
13 | soeq2 5475 | . . . . 5 ⊢ (𝑈 = ∅ → (𝑇 Or 𝑈 ↔ 𝑇 Or ∅)) | |
14 | 12, 13 | syl 17 | . . . 4 ⊢ (¬ 𝑍 ∈ V → (𝑇 Or 𝑈 ↔ 𝑇 Or ∅)) |
15 | 5, 14 | mpbiri 261 | . . 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 185 | 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 209 ∧ wa 399 ∧ w3a 1089 = wceq 1543 ∈ wcel 2112 ∀wral 3051 ∃wrex 3052 {crab 3055 Vcvv 3398 ∅c0 4223 class class class wbr 5039 {copab 5101 Or wor 5452 ‘cfv 6358 (class class class)co 7191 ↑m cmap 8486 finSupp cfsupp 8963 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pow 5243 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7194 df-oprab 7195 df-mpo 7196 df-om 7623 df-1st 7739 df-2nd 7740 df-supp 7882 df-1o 8180 df-map 8488 df-en 8605 df-fin 8608 df-fsupp 8964 |
This theorem is referenced by: oemapso 9275 |
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