![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > undefval | Structured version Visualization version GIF version |
Description: Value of the undefined value function. Normally we will not reference the explicit value but will use undefnel 7927 instead. (Contributed by NM, 15-Sep-2011.) (Revised by Mario Carneiro, 24-Dec-2016.) |
Ref | Expression |
---|---|
undefval | ⊢ (𝑆 ∈ 𝑉 → (Undef‘𝑆) = 𝒫 ∪ 𝑆) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-undef 7922 | . 2 ⊢ Undef = (𝑠 ∈ V ↦ 𝒫 ∪ 𝑠) | |
2 | unieq 4811 | . . 3 ⊢ (𝑠 = 𝑆 → ∪ 𝑠 = ∪ 𝑆) | |
3 | 2 | pweqd 4516 | . 2 ⊢ (𝑠 = 𝑆 → 𝒫 ∪ 𝑠 = 𝒫 ∪ 𝑆) |
4 | elex 3459 | . 2 ⊢ (𝑆 ∈ 𝑉 → 𝑆 ∈ V) | |
5 | uniexg 7446 | . . 3 ⊢ (𝑆 ∈ 𝑉 → ∪ 𝑆 ∈ V) | |
6 | 5 | pwexd 5245 | . 2 ⊢ (𝑆 ∈ 𝑉 → 𝒫 ∪ 𝑆 ∈ V) |
7 | 1, 3, 4, 6 | fvmptd3 6768 | 1 ⊢ (𝑆 ∈ 𝑉 → (Undef‘𝑆) = 𝒫 ∪ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 Vcvv 3441 𝒫 cpw 4497 ∪ cuni 4800 ‘cfv 6324 Undefcund 7921 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ral 3111 df-rex 3112 df-v 3443 df-sbc 3721 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-iota 6283 df-fun 6326 df-fv 6332 df-undef 7922 |
This theorem is referenced by: undefnel2 7926 undefne0 7928 ndfatafv2undef 43768 |
Copyright terms: Public domain | W3C validator |