![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 16478 | . 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 11630 | . . . . 5 ⊢ ℕ ⊆ ℂ | |
6 | 5 | a1i 11 | . . . 4 ⊢ (𝜑 → ℕ ⊆ ℂ) |
7 | 2, 4, 6 | wunss 10123 | . . 3 ⊢ (𝜑 → ℕ ∈ 𝑈) |
8 | f1oi 6627 | . . . 4 ⊢ ( I ↾ ℕ):ℕ–1-1-onto→ℕ | |
9 | f1of 6590 | . . . 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 2894 | 1 ⊢ (𝜑 → ndx ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2111 ⊆ wss 3881 I cid 5424 ↾ cres 5521 ⟶wf 6320 –1-1-onto→wf1o 6323 ωcom 7560 WUnicwun 10111 ℂcc 10524 ℕcn 11625 ndxcnx 16472 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 ax-inf2 9088 ax-1cn 10584 ax-addcl 10586 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rmo 3114 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-pss 3900 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-tp 4530 df-op 4532 df-uni 4801 df-int 4839 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-tr 5137 df-id 5425 df-eprel 5430 df-po 5438 df-so 5439 df-fr 5478 df-we 5480 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-pred 6116 df-ord 6162 df-on 6163 df-lim 6164 df-suc 6165 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 df-ov 7138 df-oprab 7139 df-mpo 7140 df-om 7561 df-1st 7671 df-2nd 7672 df-wrecs 7930 df-recs 7991 df-rdg 8029 df-1o 8085 df-oadd 8089 df-omul 8090 df-er 8272 df-ec 8274 df-qs 8278 df-map 8391 df-pm 8392 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 11626 df-ndx 16478 |
This theorem is referenced by: wunress 16556 1strwun 16593 catcoppccl 17360 catcfuccl 17361 catcxpccl 17449 |
Copyright terms: Public domain | W3C validator |