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Mirrors > Home > ILE Home > Th. List > fmpttd | GIF version |
Description: Version of fmptd 5574 with inlined definition. Domain and codomain of the mapping operation; deduction form. (Contributed by Glauco Siliprandi, 23-Oct-2021.) (Proof shortened by BJ, 16-Aug-2022.) |
Ref | Expression |
---|---|
fmpttd.1 | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶) |
Ref | Expression |
---|---|
fmpttd | ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fmpttd.1 | . 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶) | |
2 | eqid 2139 | . 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
3 | 1, 2 | fmptd 5574 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∈ wcel 1480 ↦ cmpt 3989 ⟶wf 5119 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-rab 2425 df-v 2688 df-sbc 2910 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-res 4551 df-ima 4552 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fv 5131 |
This theorem is referenced by: fmpt3d 5576 ctmlemr 6993 ctssdclemn0 6995 ctssdc 6998 ismkvnex 7029 fsumf1o 11164 isumss 11165 fisumss 11166 fsumcl2lem 11172 fsumadd 11180 isumclim3 11197 isummulc2 11200 fsummulc2 11222 isumshft 11264 prodfdivap 11321 tgrest 12343 resttopon 12345 rest0 12353 cnpfval 12369 txcnp 12445 uptx 12448 cnmpt11 12457 bdxmet 12675 cncfmptc 12756 cncfmptid 12757 cdivcncfap 12761 mulcncf 12765 limcmpted 12806 dvfgg 12831 dvcnp2cntop 12837 dvmulxxbr 12840 dvcjbr 12846 dvexp 12849 dvrecap 12851 dvmptclx 12854 dvmptaddx 12855 dvmptmulx 12856 dvef 12861 subctctexmid 13201 nninffeq 13221 |
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