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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 4740 | . 2 ⊢ {𝑦 ∈ 𝐴 ∣ 𝜑} ∈ V |
3 | 2 | mptex 6390 | 1 ⊢ (𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑} ↦ 𝐵) ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 1977 {crab 2900 Vcvv 3173 ↦ cmpt 4643 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 |
This theorem is referenced by: odzval 15334 pmtrfval 17693 dmdprd 18220 dprdval 18225 psrlidm 19224 psrass23l 19229 psrass23 19231 mplsubrg 19261 mplmonmul 19285 mplbas2 19291 wlknwwlknbij2 26242 wlkiswwlkbij2 26249 wlknwwlknvbij 26268 clwwlkbij 26327 clwwlkvbij 26329 sitgval 29721 fwddifnval 31440 diafval 35338 dicfval 35482 fusgrfis 40549 wlknwwlksnbij2 41089 wlkwwlkbij2 41096 wlksnwwlknvbij 41114 clwwlksbij 41227 clwwlksvbij 41229 |
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