![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > MPE Home > Th. List > mptexd | Structured version Visualization version GIF version |
Description: If the domain of a function given by maps-to notation is a set, the function is a set. Deduction version of mptexg 6740. (Contributed by Glauco Siliprandi, 24-Dec-2020.) |
Ref | Expression |
---|---|
mptexd.1 | ⊢ (𝜑 → 𝐴 ∈ 𝑉) |
Ref | Expression |
---|---|
mptexd | ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mptexd.1 | . 2 ⊢ (𝜑 → 𝐴 ∈ 𝑉) | |
2 | mptexg 6740 | . 2 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2166 Vcvv 3414 ↦ cmpt 4952 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-rep 4994 ax-sep 5005 ax-nul 5013 ax-pr 5127 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ne 3000 df-ral 3122 df-rex 3123 df-reu 3124 df-rab 3126 df-v 3416 df-sbc 3663 df-csb 3758 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-iun 4742 df-br 4874 df-opab 4936 df-mpt 4953 df-id 5250 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-iota 6086 df-fun 6125 df-fn 6126 df-f 6127 df-f1 6128 df-fo 6129 df-f1o 6130 df-fv 6131 |
This theorem is referenced by: rrx0 23565 choicefi 40198 axccdom 40222 climeldmeqmpt 40695 climfveqmpt 40698 climfveqmpt3 40709 climeldmeqmpt3 40716 climfveqmpt2 40720 climeldmeqmpt2 40722 climeqmpt 40724 limsupresicompt 40783 liminfresicompt 40807 liminfvalxr 40810 iccvonmbllem 41686 vonioolem1 41688 vonioolem2 41689 vonicclem1 41691 vonicclem2 41692 smflimmpt 41810 smflimsuplem6 41825 uspgrbispr 42606 |
Copyright terms: Public domain | W3C validator |