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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 |
---|---|
strfvss.e | ⊢ 𝐸 = Slot 𝑁 |
wunstr.u | ⊢ (𝜑 → 𝑈 ∈ WUni) |
wunstr.s | ⊢ (𝜑 → 𝑆 ∈ 𝑈) |
Ref | Expression |
---|---|
wunstr | ⊢ (𝜑 → (𝐸‘𝑆) ∈ 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wunstr.u | . 2 ⊢ (𝜑 → 𝑈 ∈ WUni) | |
2 | wunstr.s | . . . 4 ⊢ (𝜑 → 𝑆 ∈ 𝑈) | |
3 | 1, 2 | wunrn 10367 | . . 3 ⊢ (𝜑 → ran 𝑆 ∈ 𝑈) |
4 | 1, 3 | wununi 10344 | . 2 ⊢ (𝜑 → ∪ ran 𝑆 ∈ 𝑈) |
5 | strfvss.e | . . . 4 ⊢ 𝐸 = Slot 𝑁 | |
6 | 5 | strfvss 16764 | . . 3 ⊢ (𝐸‘𝑆) ⊆ ∪ ran 𝑆 |
7 | 6 | a1i 11 | . 2 ⊢ (𝜑 → (𝐸‘𝑆) ⊆ ∪ ran 𝑆) |
8 | 1, 4, 7 | wunss 10350 | 1 ⊢ (𝜑 → (𝐸‘𝑆) ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1543 ∈ wcel 2111 ⊆ wss 3880 ∪ cuni 4833 ran crn 5566 ‘cfv 6397 WUnicwun 10338 Slot cslot 16758 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2159 ax-12 2176 ax-ext 2709 ax-sep 5206 ax-nul 5213 ax-pr 5336 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2072 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2887 df-ne 2942 df-ral 3067 df-rex 3068 df-rab 3071 df-v 3422 df-dif 3883 df-un 3885 df-in 3887 df-ss 3897 df-nul 4252 df-if 4454 df-pw 4529 df-sn 4556 df-pr 4558 df-op 4562 df-uni 4834 df-br 5068 df-opab 5130 df-mpt 5150 df-tr 5176 df-id 5469 df-xp 5571 df-rel 5572 df-cnv 5573 df-co 5574 df-dm 5575 df-rn 5576 df-iota 6355 df-fun 6399 df-fv 6405 df-wun 10340 df-slot 16759 |
This theorem is referenced by: 1strwun 16801 wunress 16825 wunfunc 17429 wunfuncOLD 17430 wunnat 17487 wunnatOLD 17488 catcslotelcl 17643 catcoppccl 17647 catcoppcclOLD 17648 catcfuccl 17649 catcfucclOLD 17650 estrcbasbas 17662 catcxpccl 17738 catcxpcclOLD 17739 ringcbasbas 45293 ringcbasbasALTV 45317 |
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