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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 9551 | . . 3 ⊢ (𝜑 → ran 𝑆 ∈ 𝑈) |
4 | 1, 3 | wununi 9528 | . 2 ⊢ (𝜑 → ∪ ran 𝑆 ∈ 𝑈) |
5 | ndxarg.1 | . . . 4 ⊢ 𝐸 = Slot 𝑁 | |
6 | 5 | strfvss 15880 | . . 3 ⊢ (𝐸‘𝑆) ⊆ ∪ ran 𝑆 |
7 | 6 | a1i 11 | . 2 ⊢ (𝜑 → (𝐸‘𝑆) ⊆ ∪ ran 𝑆) |
8 | 1, 4, 7 | wunss 9534 | 1 ⊢ (𝜑 → (𝐸‘𝑆) ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1483 ∈ wcel 1990 ⊆ wss 3574 ∪ cuni 4436 ran crn 5115 ‘cfv 5888 WUnicwun 9522 Slot cslot 15856 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-sbc 3436 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-pw 4160 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-br 4654 df-opab 4713 df-mpt 4730 df-tr 4753 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-iota 5851 df-fun 5890 df-fv 5896 df-wun 9524 df-slot 15861 |
This theorem is referenced by: wunress 15940 1strwun 15982 wunfunc 16559 wunnat 16616 catcoppccl 16758 catcfuccl 16759 estrcbasbas 16771 catcxpccl 16847 ringcbasbas 42034 ringcbasbasALTV 42058 |
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