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Mirrors > Home > MPE Home > Th. List > Mathboxes > erdszelem6 | Structured version Visualization version GIF version |
Description: Lemma for erdsze 31783. (Contributed by Mario Carneiro, 22-Jan-2015.) |
Ref | Expression |
---|---|
erdsze.n | ⊢ (𝜑 → 𝑁 ∈ ℕ) |
erdsze.f | ⊢ (𝜑 → 𝐹:(1...𝑁)–1-1→ℝ) |
erdszelem.k | ⊢ 𝐾 = (𝑥 ∈ (1...𝑁) ↦ sup((♯ “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝑥 ∈ 𝑦)}), ℝ, < )) |
erdszelem.o | ⊢ 𝑂 Or ℝ |
Ref | Expression |
---|---|
erdszelem6 | ⊢ (𝜑 → 𝐾:(1...𝑁)⟶ℕ) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ltso 10457 | . . . 4 ⊢ < Or ℝ | |
2 | 1 | supex 8657 | . . 3 ⊢ sup((♯ “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝑥 ∈ 𝑦)}), ℝ, < ) ∈ V |
3 | 2 | a1i 11 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ (1...𝑁)) → sup((♯ “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝑥 ∈ 𝑦)}), ℝ, < ) ∈ V) |
4 | erdszelem.k | . . 3 ⊢ 𝐾 = (𝑥 ∈ (1...𝑁) ↦ sup((♯ “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝑥 ∈ 𝑦)}), ℝ, < )) | |
5 | 4 | a1i 11 | . 2 ⊢ (𝜑 → 𝐾 = (𝑥 ∈ (1...𝑁) ↦ sup((♯ “ {𝑦 ∈ 𝒫 (1...𝑥) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝑥 ∈ 𝑦)}), ℝ, < ))) |
6 | eqid 2777 | . . . . 5 ⊢ {𝑦 ∈ 𝒫 (1...𝑧) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝑧 ∈ 𝑦)} = {𝑦 ∈ 𝒫 (1...𝑧) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝑧 ∈ 𝑦)} | |
7 | 6 | erdszelem2 31773 | . . . 4 ⊢ ((♯ “ {𝑦 ∈ 𝒫 (1...𝑧) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝑧 ∈ 𝑦)}) ∈ Fin ∧ (♯ “ {𝑦 ∈ 𝒫 (1...𝑧) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝑧 ∈ 𝑦)}) ⊆ ℕ) |
8 | 7 | simpri 481 | . . 3 ⊢ (♯ “ {𝑦 ∈ 𝒫 (1...𝑧) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝑧 ∈ 𝑦)}) ⊆ ℕ |
9 | erdsze.n | . . . 4 ⊢ (𝜑 → 𝑁 ∈ ℕ) | |
10 | erdsze.f | . . . 4 ⊢ (𝜑 → 𝐹:(1...𝑁)–1-1→ℝ) | |
11 | erdszelem.o | . . . 4 ⊢ 𝑂 Or ℝ | |
12 | 9, 10, 4, 11 | erdszelem5 31776 | . . 3 ⊢ ((𝜑 ∧ 𝑧 ∈ (1...𝑁)) → (𝐾‘𝑧) ∈ (♯ “ {𝑦 ∈ 𝒫 (1...𝑧) ∣ ((𝐹 ↾ 𝑦) Isom < , 𝑂 (𝑦, (𝐹 “ 𝑦)) ∧ 𝑧 ∈ 𝑦)})) |
13 | 8, 12 | sseldi 3818 | . 2 ⊢ ((𝜑 ∧ 𝑧 ∈ (1...𝑁)) → (𝐾‘𝑧) ∈ ℕ) |
14 | 3, 5, 13 | fmpt2d 6657 | 1 ⊢ (𝜑 → 𝐾:(1...𝑁)⟶ℕ) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1601 ∈ wcel 2106 {crab 3093 Vcvv 3397 ⊆ wss 3791 𝒫 cpw 4378 ↦ cmpt 4965 Or wor 5273 ↾ cres 5357 “ cima 5358 ⟶wf 6131 –1-1→wf1 6132 ‘cfv 6135 Isom wiso 6136 (class class class)co 6922 Fincfn 8241 supcsup 8634 ℝcr 10271 1c1 10273 < clt 10411 ℕcn 11374 ...cfz 12643 ♯chash 13435 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2054 ax-8 2108 ax-9 2115 ax-10 2134 ax-11 2149 ax-12 2162 ax-13 2333 ax-ext 2753 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 ax-cnex 10328 ax-resscn 10329 ax-1cn 10330 ax-icn 10331 ax-addcl 10332 ax-addrcl 10333 ax-mulcl 10334 ax-mulrcl 10335 ax-mulcom 10336 ax-addass 10337 ax-mulass 10338 ax-distr 10339 ax-i2m1 10340 ax-1ne0 10341 ax-1rid 10342 ax-rnegex 10343 ax-rrecex 10344 ax-cnre 10345 ax-pre-lttri 10346 ax-pre-lttrn 10347 ax-pre-ltadd 10348 ax-pre-mulgt0 10349 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3or 1072 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2550 df-eu 2586 df-clab 2763 df-cleq 2769 df-clel 2773 df-nfc 2920 df-ne 2969 df-nel 3075 df-ral 3094 df-rex 3095 df-reu 3096 df-rmo 3097 df-rab 3098 df-v 3399 df-sbc 3652 df-csb 3751 df-dif 3794 df-un 3796 df-in 3798 df-ss 3805 df-pss 3807 df-nul 4141 df-if 4307 df-pw 4380 df-sn 4398 df-pr 4400 df-tp 4402 df-op 4404 df-uni 4672 df-int 4711 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-tr 4988 df-id 5261 df-eprel 5266 df-po 5274 df-so 5275 df-fr 5314 df-we 5316 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-pred 5933 df-ord 5979 df-on 5980 df-lim 5981 df-suc 5982 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 df-isom 6144 df-riota 6883 df-ov 6925 df-oprab 6926 df-mpt2 6927 df-om 7344 df-1st 7445 df-2nd 7446 df-wrecs 7689 df-recs 7751 df-rdg 7789 df-1o 7843 df-2o 7844 df-oadd 7847 df-er 8026 df-map 8142 df-en 8242 df-dom 8243 df-sdom 8244 df-fin 8245 df-sup 8636 df-card 9098 df-pnf 10413 df-mnf 10414 df-xr 10415 df-ltxr 10416 df-le 10417 df-sub 10608 df-neg 10609 df-nn 11375 df-n0 11643 df-xnn0 11715 df-z 11729 df-uz 11993 df-fz 12644 df-hash 13436 |
This theorem is referenced by: erdszelem7 31778 erdszelem8 31779 erdszelem9 31780 |
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