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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 10767 | . . 3 ⊢ (𝜑 → ran 𝑆 ∈ 𝑈) |
4 | 1, 3 | wununi 10744 | . 2 ⊢ (𝜑 → ∪ ran 𝑆 ∈ 𝑈) |
5 | strfvss.e | . . . 4 ⊢ 𝐸 = Slot 𝑁 | |
6 | 5 | strfvss 17221 | . . 3 ⊢ (𝐸‘𝑆) ⊆ ∪ ran 𝑆 |
7 | 6 | a1i 11 | . 2 ⊢ (𝜑 → (𝐸‘𝑆) ⊆ ∪ ran 𝑆) |
8 | 1, 4, 7 | wunss 10750 | 1 ⊢ (𝜑 → (𝐸‘𝑆) ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2106 ⊆ wss 3963 ∪ cuni 4912 ran crn 5690 ‘cfv 6563 WUnicwun 10738 Slot cslot 17215 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-iota 6516 df-fun 6565 df-fv 6571 df-wun 10740 df-slot 17216 |
This theorem is referenced by: basndxelwund 17257 1strwunOLD 17266 wunress 17296 wunressOLD 17297 wunfunc 17952 wunfuncOLD 17953 wunnat 18011 wunnatOLD 18012 catcslotelcl 18167 catcoppccl 18171 catcoppcclOLD 18172 catcfuccl 18173 catcfucclOLD 18174 estrcbasbas 18186 catcxpccl 18263 catcxpcclOLD 18264 ringcbasbas 20690 ringcbasbasALTV 48156 |
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