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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 6173 | . 2 ⊢ Fun (𝑥 ∈ 𝐴 ↦ 𝐵) | |
2 | eqid 2778 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
3 | 2 | dmmpt 5884 | . . . 4 ⊢ dom (𝑥 ∈ 𝐴 ↦ 𝐵) = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} |
4 | trud 1612 | . . . . . . 7 ⊢ (𝐵 ∈ V → ⊤) | |
5 | 4 | rgenw 3106 | . . . . . 6 ⊢ ∀𝑥 ∈ 𝐴 (𝐵 ∈ V → ⊤) |
6 | ss2rab 3899 | . . . . . 6 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} ⊆ {𝑥 ∈ 𝐴 ∣ ⊤} ↔ ∀𝑥 ∈ 𝐴 (𝐵 ∈ V → ⊤)) | |
7 | 5, 6 | mpbir 223 | . . . . 5 ⊢ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} ⊆ {𝑥 ∈ 𝐴 ∣ ⊤} |
8 | mptexgf.a | . . . . . 6 ⊢ Ⅎ𝑥𝐴 | |
9 | 8 | rabtru 3569 | . . . . 5 ⊢ {𝑥 ∈ 𝐴 ∣ ⊤} = 𝐴 |
10 | 7, 9 | sseqtri 3856 | . . . 4 ⊢ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} ⊆ 𝐴 |
11 | 3, 10 | eqsstri 3854 | . . 3 ⊢ dom (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ 𝐴 |
12 | ssexg 5041 | . . 3 ⊢ ((dom (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ 𝐴 ∧ 𝐴 ∈ 𝑉) → dom (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V) | |
13 | 11, 12 | mpan 680 | . 2 ⊢ (𝐴 ∈ 𝑉 → dom (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V) |
14 | funex 6754 | . 2 ⊢ ((Fun (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ dom (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V) | |
15 | 1, 13, 14 | sylancr 581 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ⊤wtru 1602 ∈ wcel 2107 Ⅎwnfc 2919 ∀wral 3090 {crab 3094 Vcvv 3398 ⊆ wss 3792 ↦ cmpt 4965 dom cdm 5355 Fun wfun 6129 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1839 ax-4 1853 ax-5 1953 ax-6 2021 ax-7 2055 ax-9 2116 ax-10 2135 ax-11 2150 ax-12 2163 ax-13 2334 ax-ext 2754 ax-rep 5006 ax-sep 5017 ax-nul 5025 ax-pr 5138 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 837 df-3an 1073 df-tru 1605 df-ex 1824 df-nf 1828 df-sb 2012 df-mo 2551 df-eu 2587 df-clab 2764 df-cleq 2770 df-clel 2774 df-nfc 2921 df-ne 2970 df-ral 3095 df-rex 3096 df-reu 3097 df-rab 3099 df-v 3400 df-sbc 3653 df-csb 3752 df-dif 3795 df-un 3797 df-in 3799 df-ss 3806 df-nul 4142 df-if 4308 df-sn 4399 df-pr 4401 df-op 4405 df-uni 4672 df-iun 4755 df-br 4887 df-opab 4949 df-mpt 4966 df-id 5261 df-xp 5361 df-rel 5362 df-cnv 5363 df-co 5364 df-dm 5365 df-rn 5366 df-res 5367 df-ima 5368 df-iota 6099 df-fun 6137 df-fn 6138 df-f 6139 df-f1 6140 df-fo 6141 df-f1o 6142 df-fv 6143 |
This theorem is referenced by: numclwwlk1lem2OLDOLD 27783 esumrnmpt2 30728 mptexf 40360 |
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