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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 9003 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 9004 | . . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑅 Or 𝐴 ∧ 𝑆 Or 𝐵) ∧ 𝑍 ∈ V) → 𝑇 Or 𝑈) |
4 | 3 | expcom 414 | . 2 ⊢ (𝑍 ∈ V → ((𝐴 ∈ 𝑉 ∧ 𝑅 Or 𝐴 ∧ 𝑆 Or 𝐵) → 𝑇 Or 𝑈)) |
5 | so0 5502 | . . . 4 ⊢ 𝑇 Or ∅ | |
6 | relfsupp 8823 | . . . . . . . . . 10 ⊢ Rel finSupp | |
7 | 6 | brrelex2i 5602 | . . . . . . . . 9 ⊢ (𝑥 finSupp 𝑍 → 𝑍 ∈ V) |
8 | 7 | con3i 157 | . . . . . . . 8 ⊢ (¬ 𝑍 ∈ V → ¬ 𝑥 finSupp 𝑍) |
9 | 8 | ralrimivw 3180 | . . . . . . 7 ⊢ (¬ 𝑍 ∈ V → ∀𝑥 ∈ (𝐵 ↑m 𝐴) ¬ 𝑥 finSupp 𝑍) |
10 | rabeq0 4335 | . . . . . . 7 ⊢ ({𝑥 ∈ (𝐵 ↑m 𝐴) ∣ 𝑥 finSupp 𝑍} = ∅ ↔ ∀𝑥 ∈ (𝐵 ↑m 𝐴) ¬ 𝑥 finSupp 𝑍) | |
11 | 9, 10 | sylibr 235 | . . . . . 6 ⊢ (¬ 𝑍 ∈ V → {𝑥 ∈ (𝐵 ↑m 𝐴) ∣ 𝑥 finSupp 𝑍} = ∅) |
12 | 2, 11 | syl5eq 2865 | . . . . 5 ⊢ (¬ 𝑍 ∈ V → 𝑈 = ∅) |
13 | soeq2 5488 | . . . . 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 183 | 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 207 ∧ wa 396 ∧ w3a 1079 = wceq 1528 ∈ wcel 2105 ∀wral 3135 ∃wrex 3136 {crab 3139 Vcvv 3492 ∅c0 4288 class class class wbr 5057 {copab 5119 Or wor 5466 ‘cfv 6348 (class class class)co 7145 ↑m cmap 8395 finSupp cfsupp 8821 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pow 5257 ax-pr 5320 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-pss 3951 df-nul 4289 df-if 4464 df-pw 4537 df-sn 4558 df-pr 4560 df-tp 4562 df-op 4564 df-uni 4831 df-int 4868 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-tr 5164 df-id 5453 df-eprel 5458 df-po 5467 df-so 5468 df-fr 5507 df-we 5509 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-pred 6141 df-ord 6187 df-on 6188 df-lim 6189 df-suc 6190 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7570 df-1st 7678 df-2nd 7679 df-supp 7820 df-wrecs 7936 df-recs 7997 df-rdg 8035 df-1o 8091 df-oadd 8095 df-er 8278 df-map 8397 df-en 8498 df-fin 8501 df-fsupp 8822 |
This theorem is referenced by: oemapso 9133 |
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