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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 |
---|---|
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 9873 | . . 3 ⊢ (𝜑 → ran 𝑆 ∈ 𝑈) |
4 | 1, 3 | wununi 9850 | . 2 ⊢ (𝜑 → ∪ ran 𝑆 ∈ 𝑈) |
5 | ndxarg.1 | . . . 4 ⊢ 𝐸 = Slot 𝑁 | |
6 | 5 | strfvss 16252 | . . 3 ⊢ (𝐸‘𝑆) ⊆ ∪ ran 𝑆 |
7 | 6 | a1i 11 | . 2 ⊢ (𝜑 → (𝐸‘𝑆) ⊆ ∪ ran 𝑆) |
8 | 1, 4, 7 | wunss 9856 | 1 ⊢ (𝜑 → (𝐸‘𝑆) ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1656 ∈ wcel 2164 ⊆ wss 3798 ∪ cuni 4660 ran crn 5347 ‘cfv 6127 WUnicwun 9844 Slot cslot 16228 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1894 ax-4 1908 ax-5 2009 ax-6 2075 ax-7 2112 ax-8 2166 ax-9 2173 ax-10 2192 ax-11 2207 ax-12 2220 ax-13 2389 ax-ext 2803 ax-sep 5007 ax-nul 5015 ax-pow 5067 ax-pr 5129 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 879 df-3an 1113 df-tru 1660 df-ex 1879 df-nf 1883 df-sb 2068 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4147 df-if 4309 df-pw 4382 df-sn 4400 df-pr 4402 df-op 4406 df-uni 4661 df-br 4876 df-opab 4938 df-mpt 4955 df-tr 4978 df-id 5252 df-xp 5352 df-rel 5353 df-cnv 5354 df-co 5355 df-dm 5356 df-rn 5357 df-iota 6090 df-fun 6129 df-fv 6135 df-wun 9846 df-slot 16233 |
This theorem is referenced by: wunress 16311 1strwun 16348 wunfunc 16918 wunnat 16975 catcoppccl 17117 catcfuccl 17118 estrcbasbas 17130 catcxpccl 17207 ringcbasbas 42899 ringcbasbasALTV 42923 |
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