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Mirrors > Home > MPE Home > Th. List > cantnflem1a | Structured version Visualization version GIF version |
Description: Lemma for cantnf 8473. (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 8464 | . . 3 ⊢ (𝜑 → (𝑋 ∈ 𝐵 ∧ (𝐹‘𝑋) ∈ (𝐺‘𝑋) ∧ ∀𝑤 ∈ 𝐵 (𝑋 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤)))) |
10 | 9 | simp1d 1066 | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝐵) |
11 | 9 | simp2d 1067 | . . 3 ⊢ (𝜑 → (𝐹‘𝑋) ∈ (𝐺‘𝑋)) |
12 | ne0i 3880 | . . 3 ⊢ ((𝐹‘𝑋) ∈ (𝐺‘𝑋) → (𝐺‘𝑋) ≠ ∅) | |
13 | 11, 12 | syl 17 | . 2 ⊢ (𝜑 → (𝐺‘𝑋) ≠ ∅) |
14 | 1, 2, 3 | cantnfs 8446 | . . . . . 6 ⊢ (𝜑 → (𝐺 ∈ 𝑆 ↔ (𝐺:𝐵⟶𝐴 ∧ 𝐺 finSupp ∅))) |
15 | 6, 14 | mpbid 221 | . . . . 5 ⊢ (𝜑 → (𝐺:𝐵⟶𝐴 ∧ 𝐺 finSupp ∅)) |
16 | 15 | simpld 474 | . . . 4 ⊢ (𝜑 → 𝐺:𝐵⟶𝐴) |
17 | ffn 5958 | . . . 4 ⊢ (𝐺:𝐵⟶𝐴 → 𝐺 Fn 𝐵) | |
18 | 16, 17 | syl 17 | . . 3 ⊢ (𝜑 → 𝐺 Fn 𝐵) |
19 | 0ex 4718 | . . . 4 ⊢ ∅ ∈ V | |
20 | 19 | a1i 11 | . . 3 ⊢ (𝜑 → ∅ ∈ V) |
21 | elsuppfn 7190 | . . 3 ⊢ ((𝐺 Fn 𝐵 ∧ 𝐵 ∈ On ∧ ∅ ∈ V) → (𝑋 ∈ (𝐺 supp ∅) ↔ (𝑋 ∈ 𝐵 ∧ (𝐺‘𝑋) ≠ ∅))) | |
22 | 18, 3, 20, 21 | syl3anc 1318 | . 2 ⊢ (𝜑 → (𝑋 ∈ (𝐺 supp ∅) ↔ (𝑋 ∈ 𝐵 ∧ (𝐺‘𝑋) ≠ ∅))) |
23 | 10, 13, 22 | mpbir2and 959 | 1 ⊢ (𝜑 → 𝑋 ∈ (𝐺 supp ∅)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 195 ∧ wa 383 = wceq 1475 ∈ wcel 1977 ≠ wne 2780 ∀wral 2896 ∃wrex 2897 {crab 2900 Vcvv 3173 ∅c0 3874 ∪ cuni 4372 class class class wbr 4583 {copab 4642 dom cdm 5038 Oncon0 5640 Fn wfn 5799 ⟶wf 5800 ‘cfv 5804 (class class class)co 6549 supp csupp 7182 finSupp cfsupp 8158 CNF ccnf 8441 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-fal 1481 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-pss 3556 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-tp 4130 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-tr 4681 df-eprel 4949 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-ord 5643 df-on 5644 df-lim 5645 df-suc 5646 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-ov 6552 df-oprab 6553 df-mpt2 6554 df-om 6958 df-supp 7183 df-wrecs 7294 df-recs 7355 df-rdg 7393 df-seqom 7430 df-1o 7447 df-er 7629 df-map 7746 df-en 7842 df-fin 7845 df-fsupp 8159 df-cnf 8442 |
This theorem is referenced by: cantnflem1b 8466 cantnflem1d 8468 cantnflem1 8469 |
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