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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 6587 | . 2 ⊢ Fun (𝑥 ∈ 𝐴 ↦ 𝐵) | |
2 | eqid 2733 | . . . . 5 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
3 | 2 | dmmpt 6240 | . . . 4 ⊢ dom (𝑥 ∈ 𝐴 ↦ 𝐵) = {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} |
4 | tru 1546 | . . . . . . 7 ⊢ ⊤ | |
5 | 4 | 2a1i 12 | . . . . . 6 ⊢ (𝑥 ∈ 𝐴 → (𝐵 ∈ V → ⊤)) |
6 | 5 | ss2rabi 4075 | . . . . 5 ⊢ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} ⊆ {𝑥 ∈ 𝐴 ∣ ⊤} |
7 | mptexgf.a | . . . . . 6 ⊢ Ⅎ𝑥𝐴 | |
8 | 7 | rabtru 3681 | . . . . 5 ⊢ {𝑥 ∈ 𝐴 ∣ ⊤} = 𝐴 |
9 | 6, 8 | sseqtri 4019 | . . . 4 ⊢ {𝑥 ∈ 𝐴 ∣ 𝐵 ∈ V} ⊆ 𝐴 |
10 | 3, 9 | eqsstri 4017 | . . 3 ⊢ dom (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ 𝐴 |
11 | ssexg 5324 | . . 3 ⊢ ((dom (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ 𝐴 ∧ 𝐴 ∈ 𝑉) → dom (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V) | |
12 | 10, 11 | mpan 689 | . 2 ⊢ (𝐴 ∈ 𝑉 → dom (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V) |
13 | funex 7221 | . 2 ⊢ ((Fun (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ dom (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V) | |
14 | 1, 12, 13 | sylancr 588 | 1 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ⊤wtru 1543 ∈ wcel 2107 Ⅎwnfc 2884 {crab 3433 Vcvv 3475 ⊆ wss 3949 ↦ cmpt 5232 dom cdm 5677 Fun wfun 6538 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-11 2155 ax-12 2172 ax-ext 2704 ax-rep 5286 ax-sep 5300 ax-nul 5307 ax-pr 5428 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-nfc 2886 df-ne 2942 df-ral 3063 df-rex 3072 df-reu 3378 df-rab 3434 df-v 3477 df-sbc 3779 df-csb 3895 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4910 df-iun 5000 df-br 5150 df-opab 5212 df-mpt 5233 df-id 5575 df-xp 5683 df-rel 5684 df-cnv 5685 df-co 5686 df-dm 5687 df-rn 5688 df-res 5689 df-ima 5690 df-iota 6496 df-fun 6546 df-fn 6547 df-f 6548 df-f1 6549 df-fo 6550 df-f1o 6551 df-fv 6552 |
This theorem is referenced by: esumrnmpt2 33066 exrecfnlem 36260 mptexf 43940 |
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