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| Mirrors > Home > MPE Home > Th. List > mptexgf | 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. (Contributed by FL, 6-Jun-2011.) (Revised by Mario Carneiro, 31-Aug-2015.) (Revised by Thierry Arnoux, 17-May-2020.) | 
| Ref | Expression | 
|---|---|
| mptexgf.a | ⊢ Ⅎ𝑥𝐴 | 
| Ref | Expression | 
|---|---|
| mptexgf | ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | funmpt 6603 | . 2 ⊢ Fun (𝑥 ∈ 𝐴 ↦ 𝐵) | |
| 2 | eqid 2736 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
| 3 | 2 | dmmpt 6259 | . . . 4 ⊢ dom (𝑥 ∈ 𝐴 ↦ 𝐵) = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} | 
| 4 | tru 1543 | . . . . . . 7 ⊢ ⊤ | |
| 5 | 4 | 2a1i 12 | . . . . . 6 ⊢ (𝑥 ∈ 𝐴 → (𝐵 ∈ V → ⊤)) | 
| 6 | 5 | ss2rabi 4076 | . . . . 5 ⊢ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} ⊆ {𝑥 ∈ 𝐴 ∣ ⊤} | 
| 7 | mptexgf.a | . . . . . 6 ⊢ Ⅎ𝑥𝐴 | |
| 8 | 7 | rabtru 3688 | . . . . 5 ⊢ {𝑥 ∈ 𝐴 ∣ ⊤} = 𝐴 | 
| 9 | 6, 8 | sseqtri 4031 | . . . 4 ⊢ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} ⊆ 𝐴 | 
| 10 | 3, 9 | eqsstri 4029 | . . 3 ⊢ dom (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ 𝐴 | 
| 11 | ssexg 5322 | . . 3 ⊢ ((dom (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ 𝐴 ∧ 𝐴 ∈ 𝑉) → dom (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V) | |
| 12 | 10, 11 | mpan 690 | . 2 ⊢ (𝐴 ∈ 𝑉 → dom (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V) | 
| 13 | funex 7240 | . 2 ⊢ ((Fun (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ dom (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V) | |
| 14 | 1, 12, 13 | sylancr 587 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ⊤wtru 1540 ∈ wcel 2107 Ⅎwnfc 2889 {crab 3435 Vcvv 3479 ⊆ wss 3950 ↦ cmpt 5224 dom cdm 5684 Fun wfun 6554 | 
| 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-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-rep 5278 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| 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-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-reu 3380 df-rab 3436 df-v 3481 df-sbc 3788 df-csb 3899 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-iun 4992 df-br 5143 df-opab 5205 df-mpt 5225 df-id 5577 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-iota 6513 df-fun 6562 df-fn 6563 df-f 6564 df-f1 6565 df-fo 6566 df-f1o 6567 df-fv 6568 | 
| This theorem is referenced by: esumrnmpt2 34070 exrecfnlem 37381 mptexf 45248 | 
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