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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 8302 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 8297 | . 2 ⊢ Undef = (𝑠 ∈ V ↦ 𝒫 ∪ 𝑠) | |
2 | unieq 4923 | . . 3 ⊢ (𝑠 = 𝑆 → ∪ 𝑠 = ∪ 𝑆) | |
3 | 2 | pweqd 4622 | . 2 ⊢ (𝑠 = 𝑆 → 𝒫 ∪ 𝑠 = 𝒫 ∪ 𝑆) |
4 | elex 3499 | . 2 ⊢ (𝑆 ∈ 𝑉 → 𝑆 ∈ V) | |
5 | uniexg 7759 | . . 3 ⊢ (𝑆 ∈ 𝑉 → ∪ 𝑆 ∈ V) | |
6 | 5 | pwexd 5385 | . 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 1537 ∈ wcel 2106 Vcvv 3478 𝒫 cpw 4605 ∪ cuni 4912 ‘cfv 6563 Undefcund 8296 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-ss 3980 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-iota 6516 df-fun 6565 df-fv 6571 df-undef 8297 |
This theorem is referenced by: undefnel2 8301 undefne0 8303 ndfatafv2undef 47162 |
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