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Mirrors > Home > MPE Home > Th. List > wunndx | Structured version Visualization version GIF version |
Description: Closure of the index extractor in an infinite weak universe. (Contributed by Mario Carneiro, 12-Jan-2017.) |
Ref | Expression |
---|---|
wunndx.1 | ⊢ (𝜑 → 𝑈 ∈ WUni) |
wunndx.2 | ⊢ (𝜑 → ω ∈ 𝑈) |
Ref | Expression |
---|---|
wunndx | ⊢ (𝜑 → ndx ∈ 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-ndx 16476 | . 2 ⊢ ndx = ( I ↾ ℕ) | |
2 | wunndx.1 | . . 3 ⊢ (𝜑 → 𝑈 ∈ WUni) | |
3 | wunndx.2 | . . . . 5 ⊢ (𝜑 → ω ∈ 𝑈) | |
4 | 2, 3 | wuncn 10581 | . . . 4 ⊢ (𝜑 → ℂ ∈ 𝑈) |
5 | nnsscn 11632 | . . . . 5 ⊢ ℕ ⊆ ℂ | |
6 | 5 | a1i 11 | . . . 4 ⊢ (𝜑 → ℕ ⊆ ℂ) |
7 | 2, 4, 6 | wunss 10123 | . . 3 ⊢ (𝜑 → ℕ ∈ 𝑈) |
8 | f1oi 6646 | . . . 4 ⊢ ( I ↾ ℕ):ℕ–1-1-onto→ℕ | |
9 | f1of 6609 | . . . 4 ⊢ (( I ↾ ℕ):ℕ–1-1-onto→ℕ → ( I ↾ ℕ):ℕ⟶ℕ) | |
10 | 8, 9 | mp1i 13 | . . 3 ⊢ (𝜑 → ( I ↾ ℕ):ℕ⟶ℕ) |
11 | 2, 7, 7, 10 | wunf 10138 | . 2 ⊢ (𝜑 → ( I ↾ ℕ) ∈ 𝑈) |
12 | 1, 11 | eqeltrid 2917 | 1 ⊢ (𝜑 → ndx ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 ⊆ wss 3935 I cid 5453 ↾ cres 5551 ⟶wf 6345 –1-1-onto→wf1o 6348 ωcom 7568 WUnicwun 10111 ℂcc 10524 ℕcn 11627 ndxcnx 16470 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 ax-inf2 9093 ax-1cn 10584 ax-addcl 10586 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3or 1080 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-pss 3953 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-tp 4564 df-op 4566 df-uni 4833 df-int 4870 df-iun 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-eprel 5459 df-po 5468 df-so 5469 df-fr 5508 df-we 5510 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-pred 6142 df-ord 6188 df-on 6189 df-lim 6190 df-suc 6191 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 df-ov 7148 df-oprab 7149 df-mpo 7150 df-om 7569 df-1st 7680 df-2nd 7681 df-wrecs 7938 df-recs 7999 df-rdg 8037 df-1o 8093 df-oadd 8097 df-omul 8098 df-er 8279 df-ec 8281 df-qs 8285 df-map 8398 df-pm 8399 df-wun 10113 df-ni 10283 df-pli 10284 df-mi 10285 df-lti 10286 df-plpq 10319 df-mpq 10320 df-ltpq 10321 df-enq 10322 df-nq 10323 df-erq 10324 df-plq 10325 df-mq 10326 df-1nq 10327 df-rq 10328 df-ltnq 10329 df-np 10392 df-plp 10394 df-ltp 10396 df-enr 10466 df-nr 10467 df-c 10532 df-nn 11628 df-ndx 16476 |
This theorem is referenced by: wunress 16554 1strwun 16591 catcoppccl 17358 catcfuccl 17359 catcxpccl 17447 |
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