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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 6386 | . 2 ⊢ Fun (𝑥 ∈ 𝐴 ↦ 𝐵) | |
2 | eqid 2818 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
3 | 2 | dmmpt 6087 | . . . 4 ⊢ dom (𝑥 ∈ 𝐴 ↦ 𝐵) = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} |
4 | trud 1538 | . . . . . . 7 ⊢ (𝐵 ∈ V → ⊤) | |
5 | 4 | rgenw 3147 | . . . . . 6 ⊢ ∀𝑥 ∈ 𝐴 (𝐵 ∈ V → ⊤) |
6 | ss2rab 4044 | . . . . . 6 ⊢ ({𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} ⊆ {𝑥 ∈ 𝐴 ∣ ⊤} ↔ ∀𝑥 ∈ 𝐴 (𝐵 ∈ V → ⊤)) | |
7 | 5, 6 | mpbir 232 | . . . . 5 ⊢ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} ⊆ {𝑥 ∈ 𝐴 ∣ ⊤} |
8 | mptexgf.a | . . . . . 6 ⊢ Ⅎ𝑥𝐴 | |
9 | 8 | rabtru 3674 | . . . . 5 ⊢ {𝑥 ∈ 𝐴 ∣ ⊤} = 𝐴 |
10 | 7, 9 | sseqtri 4000 | . . . 4 ⊢ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} ⊆ 𝐴 |
11 | 3, 10 | eqsstri 3998 | . . 3 ⊢ dom (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ 𝐴 |
12 | ssexg 5218 | . . 3 ⊢ ((dom (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ 𝐴 ∧ 𝐴 ∈ 𝑉) → dom (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V) | |
13 | 11, 12 | mpan 686 | . 2 ⊢ (𝐴 ∈ 𝑉 → dom (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V) |
14 | funex 6973 | . 2 ⊢ ((Fun (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ dom (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V) | |
15 | 1, 13, 14 | sylancr 587 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ⊤wtru 1529 ∈ wcel 2105 Ⅎwnfc 2958 ∀wral 3135 {crab 3139 Vcvv 3492 ⊆ wss 3933 ↦ cmpt 5137 dom cdm 5548 Fun wfun 6342 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-rep 5181 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ne 3014 df-ral 3140 df-rex 3141 df-reu 3142 df-rab 3144 df-v 3494 df-sbc 3770 df-csb 3881 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-iun 4912 df-br 5058 df-opab 5120 df-mpt 5138 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-f 6352 df-f1 6353 df-fo 6354 df-f1o 6355 df-fv 6356 |
This theorem is referenced by: esumrnmpt2 31226 exrecfnlem 34542 mptexf 41383 |
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