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Mirrors > Home > ILE Home > Th. List > mptexg | GIF version |
Description: If the domain of a function given by maps-to notation is a set, the function is a set. (Contributed by FL, 6-Jun-2011.) (Revised by Mario Carneiro, 31-Aug-2015.) |
Ref | Expression |
---|---|
mptexg | ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funmpt 5003 | . 2 ⊢ Fun (𝑥 ∈ 𝐴 ↦ 𝐵) | |
2 | eqid 2083 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
3 | 2 | dmmptss 4879 | . . 3 ⊢ dom (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ 𝐴 |
4 | ssexg 3943 | . . 3 ⊢ ((dom (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ 𝐴 ∧ 𝐴 ∈ 𝑉) → dom (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V) | |
5 | 3, 4 | mpan 415 | . 2 ⊢ (𝐴 ∈ 𝑉 → dom (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V) |
6 | funex 5458 | . 2 ⊢ ((Fun (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ dom (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V) | |
7 | 1, 5, 6 | sylancr 405 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1434 Vcvv 2612 ⊆ wss 2984 ↦ cmpt 3865 dom cdm 4399 Fun wfun 4961 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 663 ax-5 1377 ax-7 1378 ax-gen 1379 ax-ie1 1423 ax-ie2 1424 ax-8 1436 ax-10 1437 ax-11 1438 ax-i12 1439 ax-bndl 1440 ax-4 1441 ax-14 1446 ax-17 1460 ax-i9 1464 ax-ial 1468 ax-i5r 1469 ax-ext 2065 ax-coll 3919 ax-sep 3922 ax-pow 3974 ax-pr 3999 |
This theorem depends on definitions: df-bi 115 df-3an 922 df-tru 1288 df-nf 1391 df-sb 1688 df-eu 1946 df-mo 1947 df-clab 2070 df-cleq 2076 df-clel 2079 df-nfc 2212 df-ral 2358 df-rex 2359 df-reu 2360 df-rab 2362 df-v 2614 df-sbc 2827 df-csb 2920 df-un 2988 df-in 2990 df-ss 2997 df-pw 3408 df-sn 3428 df-pr 3429 df-op 3431 df-uni 3628 df-iun 3706 df-br 3812 df-opab 3866 df-mpt 3867 df-id 4083 df-xp 4405 df-rel 4406 df-cnv 4407 df-co 4408 df-dm 4409 df-rn 4410 df-res 4411 df-ima 4412 df-iota 4932 df-fun 4969 df-fn 4970 df-f 4971 df-f1 4972 df-fo 4973 df-f1o 4974 df-fv 4975 |
This theorem is referenced by: mptex 5461 offval 5796 abrexexg 5822 xpexgALT 5837 offval3 5838 iunon 5979 updjud 6678 negfi 10482 climmpt 10511 |
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