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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 7686 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 7681 | . 2 ⊢ Undef = (𝑠 ∈ V ↦ 𝒫 ∪ 𝑠) | |
2 | unieq 4679 | . . 3 ⊢ (𝑠 = 𝑆 → ∪ 𝑠 = ∪ 𝑆) | |
3 | 2 | pweqd 4384 | . 2 ⊢ (𝑠 = 𝑆 → 𝒫 ∪ 𝑠 = 𝒫 ∪ 𝑆) |
4 | elex 3414 | . 2 ⊢ (𝑆 ∈ 𝑉 → 𝑆 ∈ V) | |
5 | uniexg 7232 | . . 3 ⊢ (𝑆 ∈ 𝑉 → ∪ 𝑆 ∈ V) | |
6 | 5 | pwexd 5091 | . 2 ⊢ (𝑆 ∈ 𝑉 → 𝒫 ∪ 𝑆 ∈ V) |
7 | 1, 3, 4, 6 | fvmptd3 6564 | 1 ⊢ (𝑆 ∈ 𝑉 → (Undef‘𝑆) = 𝒫 ∪ 𝑆) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1601 ∈ wcel 2107 Vcvv 3398 𝒫 cpw 4379 ∪ cuni 4671 ‘cfv 6135 Undefcund 7680 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-8 2109 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-sep 5017 ax-nul 5025 ax-pow 5077 ax-pr 5138 ax-un 7226 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ral 3095 df-rex 3096 df-rab 3099 df-v 3400 df-sbc 3653 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-pw 4381 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4672 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-iota 6099 df-fun 6137 df-fv 6143 df-undef 7681 |
This theorem is referenced by: undefnel2 7685 undefne0 7687 ndfatafv2undef 42253 |
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