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Mirrors > Home > MPE Home > Th. List > mptfvmpt | Structured version Visualization version GIF version |
Description: A function in maps-to notation as the value of another function in maps-to notation. (Contributed by AV, 20-Aug-2022.) |
Ref | Expression |
---|---|
mptfvmpt.y | ⊢ (𝑦 = 𝑌 → 𝑀 = (𝑥 ∈ 𝑉 ↦ 𝐴)) |
mptfvmpt.g | ⊢ 𝐺 = (𝑦 ∈ 𝑊 ↦ 𝑀) |
mptfvmpt.v | ⊢ 𝑉 = (𝐹‘𝑋) |
Ref | Expression |
---|---|
mptfvmpt | ⊢ (𝑌 ∈ 𝑊 → (𝐺‘𝑌) = (𝑥 ∈ 𝑉 ↦ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mptfvmpt.y | . 2 ⊢ (𝑦 = 𝑌 → 𝑀 = (𝑥 ∈ 𝑉 ↦ 𝐴)) | |
2 | mptfvmpt.g | . 2 ⊢ 𝐺 = (𝑦 ∈ 𝑊 ↦ 𝑀) | |
3 | mptfvmpt.v | . . . 4 ⊢ 𝑉 = (𝐹‘𝑋) | |
4 | 3 | fvexi 6920 | . . 3 ⊢ 𝑉 ∈ V |
5 | 4 | mptex 7242 | . 2 ⊢ (𝑥 ∈ 𝑉 ↦ 𝐴) ∈ V |
6 | 1, 2, 5 | fvmpt 7015 | 1 ⊢ (𝑌 ∈ 𝑊 → (𝐺‘𝑌) = (𝑥 ∈ 𝑉 ↦ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1536 ∈ wcel 2105 ↦ cmpt 5230 ‘cfv 6562 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-rep 5284 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-reu 3378 df-rab 3433 df-v 3479 df-sbc 3791 df-csb 3908 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-iun 4997 df-br 5148 df-opab 5210 df-mpt 5231 df-id 5582 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-iota 6515 df-fun 6564 df-fn 6565 df-f 6566 df-f1 6567 df-fo 6568 df-f1o 6569 df-fv 6570 |
This theorem is referenced by: cidfval 17720 idafval 18110 grpinvfvalALT 19009 grplactfval 19071 odfvalALT 19565 asclfval 21916 ig1pval 26229 ishlg 28624 htthlem 30945 sgnsv 33162 mvrsval 35489 mvhfval 35517 msrfval 35521 lkrfval 39068 pmapfval 39738 watfvalN 39974 ldilfset 40090 ltrnfset 40099 dilfsetN 40134 trnfsetN 40137 trlfset 40142 tgrpfset 40726 tendofset 40740 tendoi 40776 erngfset 40781 erngfset-rN 40789 dvafset 40986 diaffval 41012 dvhfset 41062 docaffvalN 41103 djaffvalN 41115 dibffval 41122 dicffval 41156 dihffval 41212 dihfval 41213 dochffval 41331 djhffval 41378 lcfrlem8 41531 lcdfval 41570 mapdffval 41608 mapdfval 41609 hvmapffval 41740 hdmap1ffval 41777 hdmapffval 41808 hdmapfval 41809 hgmapffval 41867 hgmapfval 41868 hbtlem1 43111 hbtlem7 43113 |
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