|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > fndmexd | Structured version Visualization version GIF version | ||
| Description: If a function is a set, its domain is a set. (Contributed by Rohan Ridenour, 13-May-2024.) | 
| Ref | Expression | 
|---|---|
| fndmexd.1 | ⊢ (𝜑 → 𝐹 ∈ 𝑉) | 
| fndmexd.2 | ⊢ (𝜑 → 𝐹 Fn 𝐷) | 
| Ref | Expression | 
|---|---|
| fndmexd | ⊢ (𝜑 → 𝐷 ∈ V) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | fndmexd.2 | . . 3 ⊢ (𝜑 → 𝐹 Fn 𝐷) | |
| 2 | 1 | fndmd 6672 | . 2 ⊢ (𝜑 → dom 𝐹 = 𝐷) | 
| 3 | fndmexd.1 | . . 3 ⊢ (𝜑 → 𝐹 ∈ 𝑉) | |
| 4 | 3 | dmexd 7926 | . 2 ⊢ (𝜑 → dom 𝐹 ∈ V) | 
| 5 | 2, 4 | eqeltrrd 2841 | 1 ⊢ (𝜑 → 𝐷 ∈ V) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∈ wcel 2107 Vcvv 3479 dom cdm 5684 Fn wfn 6555 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 ax-un 7756 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4907 df-br 5143 df-opab 5205 df-cnv 5692 df-dm 5694 df-rn 5695 df-fn 6563 | 
| This theorem is referenced by: fndmexb 7929 fsetdmprc0 8896 finnzfsuppd 9414 psrbagfsupp 21940 psrbaglecl 21944 psrbagaddcl 21945 psrbagcon 21946 psrbagleadd1 21949 psrbagconf1o 21950 gsumbagdiaglem 21951 psrass1lem 21953 psrbagev1 22102 psrbagev2 22103 tdeglem1 26098 tdeglem3 26099 tdeglem4 26100 gsumhashmul 33065 mhphf 42612 | 
| Copyright terms: Public domain | W3C validator |