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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 9589 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 9590 | . . 3 ⊢ (((𝐴 ∈ 𝑉 ∧ 𝑅 Or 𝐴 ∧ 𝑆 Or 𝐵) ∧ 𝑍 ∈ V) → 𝑇 Or 𝑈) |
4 | 3 | expcom 413 | . 2 ⊢ (𝑍 ∈ V → ((𝐴 ∈ 𝑉 ∧ 𝑅 Or 𝐴 ∧ 𝑆 Or 𝐵) → 𝑇 Or 𝑈)) |
5 | so0 5634 | . . . 4 ⊢ 𝑇 Or ∅ | |
6 | relfsupp 9401 | . . . . . . . . . 10 ⊢ Rel finSupp | |
7 | 6 | brrelex2i 5746 | . . . . . . . . 9 ⊢ (𝑥 finSupp 𝑍 → 𝑍 ∈ V) |
8 | 7 | con3i 154 | . . . . . . . 8 ⊢ (¬ 𝑍 ∈ V → ¬ 𝑥 finSupp 𝑍) |
9 | 8 | ralrimivw 3148 | . . . . . . 7 ⊢ (¬ 𝑍 ∈ V → ∀𝑥 ∈ (𝐵 ↑m 𝐴) ¬ 𝑥 finSupp 𝑍) |
10 | rabeq0 4394 | . . . . . . 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 2787 | . . . . 5 ⊢ (¬ 𝑍 ∈ V → 𝑈 = ∅) |
13 | soeq2 5619 | . . . . 5 ⊢ (𝑈 = ∅ → (𝑇 Or 𝑈 ↔ 𝑇 Or ∅)) | |
14 | 12, 13 | syl 17 | . . . 4 ⊢ (¬ 𝑍 ∈ V → (𝑇 Or 𝑈 ↔ 𝑇 Or ∅)) |
15 | 5, 14 | mpbiri 258 | . . 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 206 ∧ wa 395 ∧ w3a 1086 = wceq 1537 ∈ wcel 2106 ∀wral 3059 ∃wrex 3068 {crab 3433 Vcvv 3478 ∅c0 4339 class class class wbr 5148 {copab 5210 Or wor 5596 ‘cfv 6563 (class class class)co 7431 ↑m cmap 8865 finSupp cfsupp 9399 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-1o 8505 df-map 8867 df-en 8985 df-fin 8988 df-fsupp 9400 |
This theorem is referenced by: oemapso 9720 |
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