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Mirrors > Home > MPE Home > Th. List > mptfi | Structured version Visualization version GIF version |
Description: A finite mapping set is finite. (Contributed by Mario Carneiro, 31-Aug-2015.) |
Ref | Expression |
---|---|
mptfi | ⊢ (𝐴 ∈ Fin → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ Fin) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funmpt 6576 | . . 3 ⊢ Fun (𝑥 ∈ 𝐴 ↦ 𝐵) | |
2 | funfn 6568 | . . 3 ⊢ (Fun (𝑥 ∈ 𝐴 ↦ 𝐵) ↔ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn dom (𝑥 ∈ 𝐴 ↦ 𝐵)) | |
3 | 1, 2 | mpbi 229 | . 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) Fn dom (𝑥 ∈ 𝐴 ↦ 𝐵) |
4 | eqid 2732 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
5 | 4 | dmmptss 6230 | . . 3 ⊢ dom (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ 𝐴 |
6 | ssfi 9158 | . . 3 ⊢ ((𝐴 ∈ Fin ∧ dom (𝑥 ∈ 𝐴 ↦ 𝐵) ⊆ 𝐴) → dom (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ Fin) | |
7 | 5, 6 | mpan2 689 | . 2 ⊢ (𝐴 ∈ Fin → dom (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ Fin) |
8 | fnfi 9166 | . 2 ⊢ (((𝑥 ∈ 𝐴 ↦ 𝐵) Fn dom (𝑥 ∈ 𝐴 ↦ 𝐵) ∧ dom (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ Fin) → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ Fin) | |
9 | 3, 7, 8 | sylancr 587 | 1 ⊢ (𝐴 ∈ Fin → (𝑥 ∈ 𝐴 ↦ 𝐵) ∈ Fin) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∈ wcel 2106 ⊆ wss 3945 ↦ cmpt 5225 dom cdm 5670 Fun wfun 6527 Fn wfn 6528 Fincfn 8924 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2703 ax-sep 5293 ax-nul 5300 ax-pr 5421 ax-un 7709 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-reu 3377 df-rab 3433 df-v 3476 df-sbc 3775 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-pss 3964 df-nul 4320 df-if 4524 df-pw 4599 df-sn 4624 df-pr 4626 df-op 4630 df-uni 4903 df-br 5143 df-opab 5205 df-mpt 5226 df-tr 5260 df-id 5568 df-eprel 5574 df-po 5582 df-so 5583 df-fr 5625 df-we 5627 df-xp 5676 df-rel 5677 df-cnv 5678 df-co 5679 df-dm 5680 df-rn 5681 df-res 5682 df-ima 5683 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-om 7840 df-1o 8450 df-en 8925 df-fin 8928 |
This theorem is referenced by: abrexfi 9337 ccatalpha 14527 prdsmet 23807 gsummpt2co 32135 carsgclctunlem2 33213 carsgclctunlem3 33214 breprexplema 33537 istotbnd3 36508 sstotbnd 36512 totbndbnd 36526 rnmptfi 43702 choicefi 43734 stoweidlem39 44592 fourierdlem31 44691 aacllem 47560 |
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