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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 5199 | . 2 ⊢ {𝑦 ∈ 𝐴 ∣ 𝜑} ∈ V |
3 | 2 | mptex 6963 | 1 ⊢ (𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑} ↦ 𝐵) ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2111 {crab 3110 Vcvv 3441 ↦ cmpt 5110 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pr 5295 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-reu 3113 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-iun 4883 df-br 5031 df-opab 5093 df-mpt 5111 df-id 5425 df-xp 5525 df-rel 5526 df-cnv 5527 df-co 5528 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-iota 6283 df-fun 6326 df-fn 6327 df-f 6328 df-f1 6329 df-fo 6330 df-f1o 6331 df-fv 6332 |
This theorem is referenced by: odzval 16118 pmtrfval 18570 dmdprd 19113 dprdval 19118 psrlidm 20641 psrass23l 20646 psrass23 20648 mplsubrg 20678 mplmonmul 20704 mplbas2 20710 fusgrfis 27120 wlksnwwlknvbij 27694 clwwlkvbij 27898 sitgval 31700 fwddifnval 33737 diafval 38327 dicfval 38471 |
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