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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 5210 | . 2 ⊢ {𝑦 ∈ 𝐴 ∣ 𝜑} ∈ V |
3 | 2 | mptex 7008 | 1 ⊢ (𝑥 ∈ {𝑦 ∈ 𝐴 ∣ 𝜑} ↦ 𝐵) ∈ V |
Colors of variables: wff setvar class |
Syntax hints: ∈ wcel 2114 {crab 3058 Vcvv 3400 ↦ cmpt 5120 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1975 ax-7 2020 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2162 ax-12 2179 ax-ext 2711 ax-rep 5164 ax-sep 5177 ax-nul 5184 ax-pr 5306 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1787 df-nf 1791 df-sb 2075 df-mo 2541 df-eu 2571 df-clab 2718 df-cleq 2731 df-clel 2812 df-nfc 2882 df-ne 2936 df-ral 3059 df-rex 3060 df-reu 3061 df-rab 3063 df-v 3402 df-sbc 3686 df-csb 3801 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-nul 4222 df-if 4425 df-sn 4527 df-pr 4529 df-op 4533 df-uni 4807 df-iun 4893 df-br 5041 df-opab 5103 df-mpt 5121 df-id 5439 df-xp 5541 df-rel 5542 df-cnv 5543 df-co 5544 df-dm 5545 df-rn 5546 df-res 5547 df-ima 5548 df-iota 6307 df-fun 6351 df-fn 6352 df-f 6353 df-f1 6354 df-fo 6355 df-f1o 6356 df-fv 6357 |
This theorem is referenced by: odzval 16240 pmtrfval 18708 dmdprd 19251 dprdval 19256 psrlidm 20794 psrass23l 20799 psrass23 20801 mplsubrg 20833 mplmonmul 20859 mplbas2 20865 fusgrfis 27284 wlksnwwlknvbij 27858 clwwlkvbij 28062 sitgval 31881 fwddifnval 34120 diafval 38700 dicfval 38844 |
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