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Mirrors > Home > MPE Home > Th. List > cantnflem1a | Structured version Visualization version GIF version |
Description: Lemma for cantnf 9637. (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 9628 | . . 3 ⊢ (𝜑 → (𝑋 ∈ 𝐵 ∧ (𝐹‘𝑋) ∈ (𝐺‘𝑋) ∧ ∀𝑤 ∈ 𝐵 (𝑋 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤)))) |
10 | 9 | simp1d 1143 | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
11 | 9 | simp2d 1144 | . . 3 ⊢ (𝜑 → (𝐹‘𝑋) ∈ (𝐺‘𝑋)) |
12 | 11 | ne0d 4299 | . 2 ⊢ (𝜑 → (𝐺‘𝑋) ≠ ∅) |
13 | 1, 2, 3 | cantnfs 9610 | . . . . . 6 ⊢ (𝜑 → (𝐺 ∈ 𝑆 ↔ (𝐺:𝐵⟶𝐴 ∧ 𝐺 finSupp ∅))) |
14 | 6, 13 | mpbid 231 | . . . . 5 ⊢ (𝜑 → (𝐺:𝐵⟶𝐴 ∧ 𝐺 finSupp ∅)) |
15 | 14 | simpld 496 | . . . 4 ⊢ (𝜑 → 𝐺:𝐵⟶𝐴) |
16 | 15 | ffnd 6673 | . . 3 ⊢ (𝜑 → 𝐺 Fn 𝐵) |
17 | 0ex 5268 | . . . 4 ⊢ ∅ ∈ V | |
18 | 17 | a1i 11 | . . 3 ⊢ (𝜑 → ∅ ∈ V) |
19 | elsuppfn 8106 | . . 3 ⊢ ((𝐺 Fn 𝐵 ∧ 𝐵 ∈ On ∧ ∅ ∈ V) → (𝑋 ∈ (𝐺 supp ∅) ↔ (𝑋 ∈ 𝐵 ∧ (𝐺‘𝑋) ≠ ∅))) | |
20 | 16, 3, 18, 19 | syl3anc 1372 | . 2 ⊢ (𝜑 → (𝑋 ∈ (𝐺 supp ∅) ↔ (𝑋 ∈ 𝐵 ∧ (𝐺‘𝑋) ≠ ∅))) |
21 | 10, 12, 20 | mpbir2and 712 | 1 ⊢ (𝜑 → 𝑋 ∈ (𝐺 supp ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 397 = wceq 1542 ∈ wcel 2107 ≠ wne 2940 ∀wral 3061 ∃wrex 3070 {crab 3406 Vcvv 3447 ∅c0 4286 ∪ cuni 4869 class class class wbr 5109 {copab 5171 dom cdm 5637 Oncon0 6321 Fn wfn 6495 ⟶wf 6496 ‘cfv 6500 (class class class)co 7361 supp csupp 8096 finSupp cfsupp 9311 CNF ccnf 9605 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5246 ax-sep 5260 ax-nul 5267 ax-pow 5324 ax-pr 5388 ax-un 7676 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3or 1089 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3353 df-rab 3407 df-v 3449 df-sbc 3744 df-csb 3860 df-dif 3917 df-un 3919 df-in 3921 df-ss 3931 df-pss 3933 df-nul 4287 df-if 4491 df-pw 4566 df-sn 4591 df-pr 4593 df-op 4597 df-uni 4870 df-iun 4960 df-br 5110 df-opab 5172 df-mpt 5193 df-tr 5227 df-id 5535 df-eprel 5541 df-po 5549 df-so 5550 df-fr 5592 df-we 5594 df-xp 5643 df-rel 5644 df-cnv 5645 df-co 5646 df-dm 5647 df-rn 5648 df-res 5649 df-ima 5650 df-pred 6257 df-ord 6324 df-on 6325 df-lim 6326 df-suc 6327 df-iota 6452 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-ov 7364 df-oprab 7365 df-mpo 7366 df-om 7807 df-supp 8097 df-frecs 8216 df-wrecs 8247 df-recs 8321 df-rdg 8360 df-seqom 8398 df-1o 8416 df-map 8773 df-en 8890 df-fin 8893 df-fsupp 9312 df-cnf 9606 |
This theorem is referenced by: cantnflem1b 9630 cantnflem1d 9632 cantnflem1 9633 |
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