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Mirrors > Home > MPE Home > Th. List > wunfv | Structured version Visualization version GIF version |
Description: A weak universe is closed under the function value operator. (Contributed by Mario Carneiro, 3-Jan-2017.) |
Ref | Expression |
---|---|
wun0.1 | ⊢ (𝜑 → 𝑈 ∈ WUni) |
wunop.2 | ⊢ (𝜑 → 𝐴 ∈ 𝑈) |
Ref | Expression |
---|---|
wunfv | ⊢ (𝜑 → (𝐴‘𝐵) ∈ 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wun0.1 | . 2 ⊢ (𝜑 → 𝑈 ∈ WUni) | |
2 | wunop.2 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑈) | |
3 | 1, 2 | wunrn 10721 | . . 3 ⊢ (𝜑 → ran 𝐴 ∈ 𝑈) |
4 | 1, 3 | wununi 10698 | . 2 ⊢ (𝜑 → ∪ ran 𝐴 ∈ 𝑈) |
5 | fvssunirn 6915 | . . 3 ⊢ (𝐴‘𝐵) ⊆ ∪ ran 𝐴 | |
6 | 5 | a1i 11 | . 2 ⊢ (𝜑 → (𝐴‘𝐵) ⊆ ∪ ran 𝐴) |
7 | 1, 4, 6 | wunss 10704 | 1 ⊢ (𝜑 → (𝐴‘𝐵) ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2098 ⊆ wss 3941 ∪ cuni 4900 ran crn 5668 ‘cfv 6534 WUnicwun 10692 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pr 5418 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-ne 2933 df-ral 3054 df-rex 3063 df-rab 3425 df-v 3468 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-pw 4597 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-br 5140 df-opab 5202 df-tr 5257 df-cnv 5675 df-dm 5677 df-rn 5678 df-iota 6486 df-fv 6542 df-wun 10694 |
This theorem is referenced by: (None) |
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