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Mirrors > Home > MPE Home > Th. List > mptrabex | Structured version Visualization version GIF version |
Description: If the domain of a function given by maps-to notation is a class abstraction based on a set, the function is a set. (Contributed by AV, 16-Jul-2019.) (Revised by AV, 26-Mar-2021.) |
Ref | Expression |
---|---|
mptrabex.1 | ⊢ 𝐴 ∈ V |
Ref | Expression |
---|---|
mptrabex | ⊢ (𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑} ↦ 𝐵) ∈ V |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mptrabex.1 | . . 3 ⊢ 𝐴 ∈ V | |
2 | 1 | rabex 5331 | . 2 ⊢ {𝑦 ∈ 𝐴 ∣ 𝜑} ∈ V |
3 | 2 | mptex 7226 | 1 ⊢ (𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑} ↦ 𝐵) ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2104 {crab 3430 Vcvv 3472 ↦ cmpt 5230 |
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 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-10 2135 ax-11 2152 ax-12 2169 ax-ext 2701 ax-rep 5284 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2532 df-eu 2561 df-clab 2708 df-cleq 2722 df-clel 2808 df-nfc 2883 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3375 df-rab 3431 df-v 3474 df-sbc 3777 df-csb 3893 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-iun 4998 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 |
This theorem is referenced by: odzval 16728 pmtrfval 19359 dmdprd 19909 dprdval 19914 psrlidm 21742 psrass23l 21747 psrass23 21749 mplsubrg 21783 mplmonmul 21810 mplbas2 21816 fusgrfis 28854 wlksnwwlknvbij 29429 clwwlkvbij 29633 sitgval 33629 fwddifnval 35439 diafval 40205 dicfval 40349 |
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