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Mirrors > Home > MPE Home > Th. List > wunpm | Structured version Visualization version GIF version |
Description: A weak universe is closed under partial mappings. (Contributed by Mario Carneiro, 2-Jan-2017.) |
Ref | Expression |
---|---|
wun0.1 | ⊢ (𝜑 → 𝑈 ∈ WUni) |
wunop.2 | ⊢ (𝜑 → 𝐴 ∈ 𝑈) |
wunop.3 | ⊢ (𝜑 → 𝐵 ∈ 𝑈) |
Ref | Expression |
---|---|
wunpm | ⊢ (𝜑 → (𝐴 ↑pm 𝐵) ∈ 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wun0.1 | . 2 ⊢ (𝜑 → 𝑈 ∈ WUni) | |
2 | wunop.3 | . . . 4 ⊢ (𝜑 → 𝐵 ∈ 𝑈) | |
3 | wunop.2 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑈) | |
4 | 1, 2, 3 | wunxp 10135 | . . 3 ⊢ (𝜑 → (𝐵 × 𝐴) ∈ 𝑈) |
5 | 1, 4 | wunpw 10118 | . 2 ⊢ (𝜑 → 𝒫 (𝐵 × 𝐴) ∈ 𝑈) |
6 | pmsspw 8431 | . . 3 ⊢ (𝐴 ↑pm 𝐵) ⊆ 𝒫 (𝐵 × 𝐴) | |
7 | 6 | a1i 11 | . 2 ⊢ (𝜑 → (𝐴 ↑pm 𝐵) ⊆ 𝒫 (𝐵 × 𝐴)) |
8 | 1, 5, 7 | wunss 10123 | 1 ⊢ (𝜑 → (𝐴 ↑pm 𝐵) ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2105 ⊆ wss 3935 𝒫 cpw 4537 × cxp 5547 (class class class)co 7145 ↑pm cpm 8397 WUnicwun 10111 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2793 ax-sep 5195 ax-nul 5202 ax-pow 5258 ax-pr 5321 ax-un 7450 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3497 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4466 df-pw 4539 df-sn 4560 df-pr 4562 df-op 4566 df-uni 4833 df-iun 4914 df-br 5059 df-opab 5121 df-mpt 5139 df-tr 5165 df-id 5454 df-xp 5555 df-rel 5556 df-cnv 5557 df-co 5558 df-dm 5559 df-rn 5560 df-res 5561 df-ima 5562 df-iota 6308 df-fun 6351 df-fn 6352 df-f 6353 df-fv 6357 df-ov 7148 df-oprab 7149 df-mpo 7150 df-1st 7680 df-2nd 7681 df-pm 8399 df-wun 10113 |
This theorem is referenced by: wunmap 10137 catcfuccl 17359 catcxpccl 17447 |
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