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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 6684 | . . 3 ⊢ 𝑉 ∈ V |
5 | 4 | mptex 6986 | . 2 ⊢ (𝑥 ∈ 𝑉 ↦ 𝐴) ∈ V |
6 | 1, 2, 5 | fvmpt 6768 | 1 ⊢ (𝑌 ∈ 𝑊 → (𝐺‘𝑌) = (𝑥 ∈ 𝑉 ↦ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1537 ∈ wcel 2114 ↦ cmpt 5146 ‘cfv 6355 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-rep 5190 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 |
This theorem is referenced by: cidfval 16947 idafval 17317 grpinvfvalALT 18143 grplactfval 18200 odfvalALT 18661 asclfval 20108 ig1pval 24766 ishlg 26388 htthlem 28694 sgnsv 30802 mvrsval 32752 mvhfval 32780 msrfval 32784 lkrfval 36238 pmapfval 36907 watfvalN 37143 ldilfset 37259 ltrnfset 37268 dilfsetN 37303 trnfsetN 37306 trlfset 37311 tgrpfset 37895 tendofset 37909 tendoi 37945 erngfset 37950 erngfset-rN 37958 dvafset 38155 diaffval 38181 dvhfset 38231 docaffvalN 38272 djaffvalN 38284 dibffval 38291 dicffval 38325 dihffval 38381 dihfval 38382 dochffval 38500 djhffval 38547 lcfrlem8 38700 lcdfval 38739 mapdffval 38777 mapdfval 38778 hvmapffval 38909 hdmap1ffval 38946 hdmapffval 38977 hdmapfval 38978 hgmapffval 39036 hgmapfval 39037 hbtlem1 39743 hbtlem7 39745 |
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