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| Description: Lemma for cantnf 9734. (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 9725 | . . 3 ⊢ (𝜑 → (𝑋 ∈ 𝐵 ∧ (𝐹‘𝑋) ∈ (𝐺‘𝑋) ∧ ∀𝑤 ∈ 𝐵 (𝑋 ∈ 𝑤 → (𝐹‘𝑤) = (𝐺‘𝑤)))) | 
| 10 | 9 | simp1d 1142 | . 2 ⊢ (𝜑 → 𝑋 ∈ 𝐵) | 
| 11 | 9 | simp2d 1143 | . . 3 ⊢ (𝜑 → (𝐹‘𝑋) ∈ (𝐺‘𝑋)) | 
| 12 | 11 | ne0d 4341 | . 2 ⊢ (𝜑 → (𝐺‘𝑋) ≠ ∅) | 
| 13 | 1, 2, 3 | cantnfs 9707 | . . . . . 6 ⊢ (𝜑 → (𝐺 ∈ 𝑆 ↔ (𝐺:𝐵⟶𝐴 ∧ 𝐺 finSupp ∅))) | 
| 14 | 6, 13 | mpbid 232 | . . . . 5 ⊢ (𝜑 → (𝐺:𝐵⟶𝐴 ∧ 𝐺 finSupp ∅)) | 
| 15 | 14 | simpld 494 | . . . 4 ⊢ (𝜑 → 𝐺:𝐵⟶𝐴) | 
| 16 | 15 | ffnd 6736 | . . 3 ⊢ (𝜑 → 𝐺 Fn 𝐵) | 
| 17 | 0ex 5306 | . . . 4 ⊢ ∅ ∈ V | |
| 18 | 17 | a1i 11 | . . 3 ⊢ (𝜑 → ∅ ∈ V) | 
| 19 | elsuppfn 8196 | . . 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 713 | 1 ⊢ (𝜑 → 𝑋 ∈ (𝐺 supp ∅)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1539 ∈ wcel 2107 ≠ wne 2939 ∀wral 3060 ∃wrex 3069 {crab 3435 Vcvv 3479 ∅c0 4332 ∪ cuni 4906 class class class wbr 5142 {copab 5204 dom cdm 5684 Oncon0 6383 Fn wfn 6555 ⟶wf 6556 ‘cfv 6560 (class class class)co 7432 supp csupp 8186 finSupp cfsupp 9402 CNF ccnf 9702 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pow 5364 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-pss 3970 df-nul 4333 df-if 4525 df-pw 4601 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-tr 5259 df-id 5577 df-eprel 5583 df-po 5591 df-so 5592 df-fr 5636 df-we 5638 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-pred 6320 df-ord 6386 df-on 6387 df-lim 6388 df-suc 6389 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 df-ov 7435 df-oprab 7436 df-mpo 7437 df-om 7889 df-supp 8187 df-frecs 8307 df-wrecs 8338 df-recs 8412 df-rdg 8451 df-seqom 8489 df-1o 8507 df-map 8869 df-en 8987 df-fin 8990 df-fsupp 9403 df-cnf 9703 | 
| This theorem is referenced by: cantnflem1b 9727 cantnflem1d 9729 cantnflem1 9730 | 
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