Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > fsetfcdm | Structured version Visualization version GIF version |
Description: The class of functions with a given domain and a given codomain is mapped, through evaluation at a point of the domain, into the codomain. (Contributed by AV, 15-Sep-2024.) |
Ref | Expression |
---|---|
fsetfocdm.f | ⊢ 𝐹 = {𝑓 ∣ 𝑓:𝐴⟶𝐵} |
fsetfocdm.s | ⊢ 𝑆 = (𝑔 ∈ 𝐹 ↦ (𝑔‘𝑋)) |
Ref | Expression |
---|---|
fsetfcdm | ⊢ (𝑋 ∈ 𝐴 → 𝑆:𝐹⟶𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 3413 | . . . . 5 ⊢ 𝑔 ∈ V | |
2 | feq1 6484 | . . . . 5 ⊢ (𝑓 = 𝑔 → (𝑓:𝐴⟶𝐵 ↔ 𝑔:𝐴⟶𝐵)) | |
3 | fsetfocdm.f | . . . . 5 ⊢ 𝐹 = {𝑓 ∣ 𝑓:𝐴⟶𝐵} | |
4 | 1, 2, 3 | elab2 3593 | . . . 4 ⊢ (𝑔 ∈ 𝐹 ↔ 𝑔:𝐴⟶𝐵) |
5 | ffvelrn 6846 | . . . . 5 ⊢ ((𝑔:𝐴⟶𝐵 ∧ 𝑋 ∈ 𝐴) → (𝑔‘𝑋) ∈ 𝐵) | |
6 | 5 | expcom 417 | . . . 4 ⊢ (𝑋 ∈ 𝐴 → (𝑔:𝐴⟶𝐵 → (𝑔‘𝑋) ∈ 𝐵)) |
7 | 4, 6 | syl5bi 245 | . . 3 ⊢ (𝑋 ∈ 𝐴 → (𝑔 ∈ 𝐹 → (𝑔‘𝑋) ∈ 𝐵)) |
8 | 7 | imp 410 | . 2 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑔 ∈ 𝐹) → (𝑔‘𝑋) ∈ 𝐵) |
9 | fsetfocdm.s | . 2 ⊢ 𝑆 = (𝑔 ∈ 𝐹 ↦ (𝑔‘𝑋)) | |
10 | 8, 9 | fmptd 6875 | 1 ⊢ (𝑋 ∈ 𝐴 → 𝑆:𝐹⟶𝐵) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1538 ∈ wcel 2111 {cab 2735 ↦ cmpt 5116 ⟶wf 6336 ‘cfv 6340 |
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 2729 ax-sep 5173 ax-nul 5180 ax-pr 5302 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2557 df-eu 2588 df-clab 2736 df-cleq 2750 df-clel 2830 df-nfc 2901 df-ral 3075 df-rex 3076 df-rab 3079 df-v 3411 df-sbc 3699 df-dif 3863 df-un 3865 df-in 3867 df-ss 3877 df-nul 4228 df-if 4424 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4802 df-br 5037 df-opab 5099 df-mpt 5117 df-id 5434 df-xp 5534 df-rel 5535 df-cnv 5536 df-co 5537 df-dm 5538 df-rn 5539 df-res 5540 df-ima 5541 df-iota 6299 df-fun 6342 df-fn 6343 df-f 6344 df-fv 6348 |
This theorem is referenced by: fsetfocdm 8456 |
Copyright terms: Public domain | W3C validator |