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Mirrors > Home > MPE Home > Th. List > wunrn | Structured version Visualization version GIF version |
Description: A weak universe is closed under the range operator. (Contributed by Mario Carneiro, 2-Jan-2017.) |
Ref | Expression |
---|---|
wun0.1 | ⊢ (𝜑 → 𝑈 ∈ WUni) |
wunop.2 | ⊢ (𝜑 → 𝐴 ∈ 𝑈) |
Ref | Expression |
---|---|
wunrn | ⊢ (𝜑 → ran 𝐴 ∈ 𝑈) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wun0.1 | . 2 ⊢ (𝜑 → 𝑈 ∈ WUni) | |
2 | wunop.2 | . . . 4 ⊢ (𝜑 → 𝐴 ∈ 𝑈) | |
3 | 1, 2 | wununi 9528 | . . 3 ⊢ (𝜑 → ∪ 𝐴 ∈ 𝑈) |
4 | 1, 3 | wununi 9528 | . 2 ⊢ (𝜑 → ∪ ∪ 𝐴 ∈ 𝑈) |
5 | ssun2 3777 | . . . 4 ⊢ ran 𝐴 ⊆ (dom 𝐴 ∪ ran 𝐴) | |
6 | dmrnssfld 5384 | . . . 4 ⊢ (dom 𝐴 ∪ ran 𝐴) ⊆ ∪ ∪ 𝐴 | |
7 | 5, 6 | sstri 3612 | . . 3 ⊢ ran 𝐴 ⊆ ∪ ∪ 𝐴 |
8 | 7 | a1i 11 | . 2 ⊢ (𝜑 → ran 𝐴 ⊆ ∪ ∪ 𝐴) |
9 | 1, 4, 8 | wunss 9534 | 1 ⊢ (𝜑 → ran 𝐴 ∈ 𝑈) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 1990 ∪ cun 3572 ⊆ wss 3574 ∪ cuni 4436 dom cdm 5114 ran crn 5115 WUnicwun 9522 |
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-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 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-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-tr 4753 df-cnv 5122 df-dm 5124 df-rn 5125 df-wun 9524 |
This theorem is referenced by: wuncnv 9552 wunfv 9554 wunco 9555 wuntpos 9556 wunstr 15881 wunfunc 16559 wunnat 16616 catcoppccl 16758 catcfuccl 16759 catcxpccl 16847 |
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