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Mirrors > Home > ILE Home > Th. List > fmpttd | GIF version |
Description: Version of fmptd 5673 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 2177 | . 2 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
3 | 1, 2 | fmptd 5673 | 1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶𝐶) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 104 ∈ wcel 2148 ↦ cmpt 4066 ⟶wf 5214 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 709 ax-5 1447 ax-7 1448 ax-gen 1449 ax-ie1 1493 ax-ie2 1494 ax-8 1504 ax-10 1505 ax-11 1506 ax-i12 1507 ax-bndl 1509 ax-4 1510 ax-17 1526 ax-i9 1530 ax-ial 1534 ax-i5r 1535 ax-14 2151 ax-ext 2159 ax-sep 4123 ax-pow 4176 ax-pr 4211 |
This theorem depends on definitions: df-bi 117 df-3an 980 df-tru 1356 df-nf 1461 df-sb 1763 df-eu 2029 df-mo 2030 df-clab 2164 df-cleq 2170 df-clel 2173 df-nfc 2308 df-ral 2460 df-rex 2461 df-rab 2464 df-v 2741 df-sbc 2965 df-un 3135 df-in 3137 df-ss 3144 df-pw 3579 df-sn 3600 df-pr 3601 df-op 3603 df-uni 3812 df-br 4006 df-opab 4067 df-mpt 4068 df-id 4295 df-xp 4634 df-rel 4635 df-cnv 4636 df-co 4637 df-dm 4638 df-rn 4639 df-res 4640 df-ima 4641 df-iota 5180 df-fun 5220 df-fn 5221 df-f 5222 df-fv 5226 |
This theorem is referenced by: fmpt3d 5675 ctmlemr 7110 ctssdclemn0 7112 ctssdc 7115 infnninf 7125 nnnninf 7127 ismkvnex 7156 fsumf1o 11401 isumss 11402 fisumss 11403 fsumcl2lem 11409 fsumadd 11417 isumclim3 11434 isummulc2 11437 fsummulc2 11459 isumshft 11501 prodfdivap 11558 fprodf1o 11599 prodssdc 11600 fprodssdc 11601 fprodmul 11602 srglmhm 13187 srgrmhm 13188 tgrest 13830 resttopon 13832 rest0 13840 cnpfval 13856 txcnp 13932 uptx 13935 cnmpt11 13944 bdxmet 14162 cncfmptc 14243 cncfmptid 14244 cdivcncfap 14248 mulcncf 14252 limcmpted 14293 dvfgg 14318 dvcnp2cntop 14324 dvmulxxbr 14327 dvcjbr 14333 dvexp 14336 dvrecap 14338 dvmptclx 14341 dvmptaddx 14342 dvmptmulx 14343 dvmptcjx 14347 dvef 14349 subctctexmid 14912 nninffeq 14931 iswomni0 14961 dceqnconst 14970 dcapnconst 14971 |
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