Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > wunstr | Structured version Visualization version GIF version |
Description: Closure of a structure index in a weak universe. (Contributed by Mario Carneiro, 12-Jan-2017.) |
Ref | Expression |
---|---|
ndxarg.1 | ⊢ 𝐸 = Slot 𝑁 |
wunstr.2 | ⊢ (𝜑 → 𝑈 ∈ WUni) |
wunstr.3 | ⊢ (𝜑 → 𝑆 ∈ 𝑈) |
Ref | Expression |
---|---|
wunstr | ⊢ (𝜑 → (𝐸‘𝑆) ∈ 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wunstr.2 | . 2 ⊢ (𝜑 → 𝑈 ∈ WUni) | |
2 | wunstr.3 | . . . 4 ⊢ (𝜑 → 𝑆 ∈ 𝑈) | |
3 | 1, 2 | wunrn 10140 | . . 3 ⊢ (𝜑 → ran 𝑆 ∈ 𝑈) |
4 | 1, 3 | wununi 10117 | . 2 ⊢ (𝜑 → ∪ ran 𝑆 ∈ 𝑈) |
5 | ndxarg.1 | . . . 4 ⊢ 𝐸 = Slot 𝑁 | |
6 | 5 | strfvss 16496 | . . 3 ⊢ (𝐸‘𝑆) ⊆ ∪ ran 𝑆 |
7 | 6 | a1i 11 | . 2 ⊢ (𝜑 → (𝐸‘𝑆) ⊆ ∪ ran 𝑆) |
8 | 1, 4, 7 | wunss 10123 | 1 ⊢ (𝜑 → (𝐸‘𝑆) ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1528 ∈ wcel 2105 ⊆ wss 3935 ∪ cuni 4832 ran crn 5550 ‘cfv 6349 WUnicwun 10111 Slot cslot 16472 |
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 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 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-rab 3147 df-v 3497 df-sbc 3772 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4833 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-iota 6308 df-fun 6351 df-fv 6357 df-wun 10113 df-slot 16477 |
This theorem is referenced by: wunress 16554 1strwun 16591 wunfunc 17159 wunnat 17216 catcoppccl 17358 catcfuccl 17359 estrcbasbas 17371 catcxpccl 17447 ringcbasbas 44203 ringcbasbasALTV 44227 |
Copyright terms: Public domain | W3C validator |