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Mirrors > Home > MPE Home > Th. List > cantnflem1a | Structured version Visualization version GIF version |
Description: Lemma for cantnf 9159. (Contributed by Mario Carneiro, 4-Jun-2015.) (Revised by AV, 2-Jul-2019.) |
Ref | Expression |
---|---|
cantnfs.s | ⊢ 𝑆 = dom (𝐴 CNF 𝐵) |
cantnfs.a | ⊢ (𝜑 → 𝐴 ∈ On) |
cantnfs.b | ⊢ (𝜑 → 𝐵 ∈ On) |
oemapval.t | ⊢ 𝑇 = {〈𝑥, 𝑦〉 ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))} |
oemapval.f | ⊢ (𝜑 → 𝐹 ∈ 𝑆) |
oemapval.g | ⊢ (𝜑 → 𝐺 ∈ 𝑆) |
oemapvali.r | ⊢ (𝜑 → 𝐹𝑇𝐺) |
oemapvali.x | ⊢ 𝑋 = ∪ {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} |
Ref | Expression |
---|---|
cantnflem1a | ⊢ (𝜑 → 𝑋 ∈ (𝐺 supp ∅)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cantnfs.s | . . . 4 ⊢ 𝑆 = dom (𝐴 CNF 𝐵) | |
2 | cantnfs.a | . . . 4 ⊢ (𝜑 → 𝐴 ∈ On) | |
3 | cantnfs.b | . . . 4 ⊢ (𝜑 → 𝐵 ∈ On) | |
4 | oemapval.t | . . . 4 ⊢ 𝑇 = {〈𝑥, 𝑦〉 ∣ ∃𝑧 ∈ 𝐵 ((𝑥‘𝑧) ∈ (𝑦‘𝑧) ∧ ∀𝑤 ∈ 𝐵 (𝑧 ∈ 𝑤 → (𝑥‘𝑤) = (𝑦‘𝑤)))} | |
5 | oemapval.f | . . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝑆) | |
6 | oemapval.g | . . . 4 ⊢ (𝜑 → 𝐺 ∈ 𝑆) | |
7 | oemapvali.r | . . . 4 ⊢ (𝜑 → 𝐹𝑇𝐺) | |
8 | oemapvali.x | . . . 4 ⊢ 𝑋 = ∪ {𝑐 ∈ 𝐵 ∣ (𝐹‘𝑐) ∈ (𝐺‘𝑐)} | |
9 | 1, 2, 3, 4, 5, 6, 7, 8 | oemapvali 9150 | . . 3 ⊢ (𝜑 → (𝑋 ∈ 𝐵 ∧ (𝐹‘𝑋) ∈ (𝐺‘𝑋) ∧ ∀𝑤 ∈ 𝐵 (𝑋 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤)))) |
10 | 9 | simp1d 1138 | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
11 | 9 | simp2d 1139 | . . 3 ⊢ (𝜑 → (𝐹‘𝑋) ∈ (𝐺‘𝑋)) |
12 | 11 | ne0d 4304 | . 2 ⊢ (𝜑 → (𝐺‘𝑋) ≠ ∅) |
13 | 1, 2, 3 | cantnfs 9132 | . . . . . 6 ⊢ (𝜑 → (𝐺 ∈ 𝑆 ↔ (𝐺:𝐵⟶𝐴 ∧ 𝐺 finSupp ∅))) |
14 | 6, 13 | mpbid 234 | . . . . 5 ⊢ (𝜑 → (𝐺:𝐵⟶𝐴 ∧ 𝐺 finSupp ∅)) |
15 | 14 | simpld 497 | . . . 4 ⊢ (𝜑 → 𝐺:𝐵⟶𝐴) |
16 | 15 | ffnd 6518 | . . 3 ⊢ (𝜑 → 𝐺 Fn 𝐵) |
17 | 0ex 5214 | . . . 4 ⊢ ∅ ∈ V | |
18 | 17 | a1i 11 | . . 3 ⊢ (𝜑 → ∅ ∈ V) |
19 | elsuppfn 7841 | . . 3 ⊢ ((𝐺 Fn 𝐵 ∧ 𝐵 ∈ On ∧ ∅ ∈ V) → (𝑋 ∈ (𝐺 supp ∅) ↔ (𝑋 ∈ 𝐵 ∧ (𝐺‘𝑋) ≠ ∅))) | |
20 | 16, 3, 18, 19 | syl3anc 1367 | . 2 ⊢ (𝜑 → (𝑋 ∈ (𝐺 supp ∅) ↔ (𝑋 ∈ 𝐵 ∧ (𝐺‘𝑋) ≠ ∅))) |
21 | 10, 12, 20 | mpbir2and 711 | 1 ⊢ (𝜑 → 𝑋 ∈ (𝐺 supp ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 = wceq 1536 ∈ wcel 2113 ≠ wne 3019 ∀wral 3141 ∃wrex 3142 {crab 3145 Vcvv 3497 ∅c0 4294 ∪ cuni 4841 class class class wbr 5069 {copab 5131 dom cdm 5558 Oncon0 6194 Fn wfn 6353 ⟶wf 6354 ‘cfv 6358 (class class class)co 7159 supp csupp 7833 finSupp cfsupp 8836 CNF ccnf 9127 |
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 1910 ax-6 1969 ax-7 2014 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2160 ax-12 2176 ax-ext 2796 ax-rep 5193 ax-sep 5206 ax-nul 5213 ax-pow 5269 ax-pr 5333 ax-un 7464 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1539 df-fal 1549 df-ex 1780 df-nf 1784 df-sb 2069 df-mo 2621 df-eu 2653 df-clab 2803 df-cleq 2817 df-clel 2896 df-nfc 2966 df-ne 3020 df-ral 3146 df-rex 3147 df-reu 3148 df-rab 3150 df-v 3499 df-sbc 3776 df-csb 3887 df-dif 3942 df-un 3944 df-in 3946 df-ss 3955 df-pss 3957 df-nul 4295 df-if 4471 df-pw 4544 df-sn 4571 df-pr 4573 df-tp 4575 df-op 4577 df-uni 4842 df-iun 4924 df-br 5070 df-opab 5132 df-mpt 5150 df-tr 5176 df-id 5463 df-eprel 5468 df-po 5477 df-so 5478 df-fr 5517 df-we 5519 df-xp 5564 df-rel 5565 df-cnv 5566 df-co 5567 df-dm 5568 df-rn 5569 df-res 5570 df-ima 5571 df-pred 6151 df-ord 6197 df-on 6198 df-lim 6199 df-suc 6200 df-iota 6317 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-ov 7162 df-oprab 7163 df-mpo 7164 df-om 7584 df-supp 7834 df-wrecs 7950 df-recs 8011 df-rdg 8049 df-seqom 8087 df-1o 8105 df-er 8292 df-map 8411 df-en 8513 df-fin 8516 df-fsupp 8837 df-cnf 9128 |
This theorem is referenced by: cantnflem1b 9152 cantnflem1d 9154 cantnflem1 9155 |
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