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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 8303 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 8298 | . 2 ⊢ Undef = (𝑠 ∈ V ↦ 𝒫 ∪ 𝑠) | |
| 2 | unieq 4918 | . . 3 ⊢ (𝑠 = 𝑆 → ∪ 𝑠 = ∪ 𝑆) | |
| 3 | 2 | pweqd 4617 | . 2 ⊢ (𝑠 = 𝑆 → 𝒫 ∪ 𝑠 = 𝒫 ∪ 𝑆) |
| 4 | elex 3501 | . 2 ⊢ (𝑆 ∈ 𝑉 → 𝑆 ∈ V) | |
| 5 | uniexg 7760 | . . 3 ⊢ (𝑆 ∈ 𝑉 → ∪ 𝑆 ∈ V) | |
| 6 | 5 | pwexd 5379 | . 2 ⊢ (𝑆 ∈ 𝑉 → 𝒫 ∪ 𝑆 ∈ V) |
| 7 | 1, 3, 4, 6 | fvmptd3 7039 | 1 ⊢ (𝑆 ∈ 𝑉 → (Undef‘𝑆) = 𝒫 ∪ 𝑆) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 = wceq 1540 ∈ wcel 2108 Vcvv 3480 𝒫 cpw 4600 ∪ cuni 4907 ‘cfv 6561 Undefcund 8297 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pow 5365 ax-pr 5432 ax-un 7755 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-ss 3968 df-nul 4334 df-if 4526 df-pw 4602 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-iota 6514 df-fun 6563 df-fv 6569 df-undef 8298 |
| This theorem is referenced by: undefnel2 8302 undefne0 8304 ndfatafv2undef 47224 |
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