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Mirrors > Home > MPE Home > Th. List > fvmptelcdm | Structured version Visualization version GIF version |
Description: The value of a function at a point of its domain belongs to its codomain. (Contributed by Glauco Siliprandi, 26-Jun-2021.) |
Ref | Expression |
---|---|
fvmptelcdm.1 | ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶𝐶) |
Ref | Expression |
---|---|
fvmptelcdm | ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fvmptelcdm.1 | . . 3 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶𝐶) | |
2 | eqid 2731 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
3 | 2 | fmpt 7094 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ↔ (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶𝐶) |
4 | 1, 3 | sylibr 233 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶) |
5 | 4 | r19.21bi 3247 | 1 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2106 ∀wral 3060 ↦ cmpt 5224 ⟶wf 6528 |
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 2702 ax-sep 5292 ax-nul 5299 ax-pr 5420 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2533 df-eu 2562 df-clab 2709 df-cleq 2723 df-clel 2809 df-nfc 2884 df-ral 3061 df-rex 3070 df-rab 3432 df-v 3475 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4523 df-sn 4623 df-pr 4625 df-op 4629 df-br 5142 df-opab 5204 df-mpt 5225 df-id 5567 df-xp 5675 df-rel 5676 df-cnv 5677 df-co 5678 df-dm 5679 df-rn 5680 df-res 5681 df-ima 5682 df-fun 6534 df-fn 6535 df-f 6536 |
This theorem is referenced by: rlimmptrcl 15534 lo1mptrcl 15548 o1mptrcl 15549 frlmgsum 21260 uvcresum 21281 psrass1lemOLD 21424 psrass1lem 21427 txcnp 23053 ptcnp 23055 ptcn 23060 cnmpt11 23096 cnmpt1t 23098 cnmpt12 23100 cnmptkp 23113 cnmptk1 23114 cnmptkk 23116 cnmptk1p 23118 cnmptk2 23119 cnmpt1plusg 23520 cnmpt1vsca 23627 cnmpt1ds 24287 cncfcompt2 24353 cncfmpt2ss 24361 cnmpt1ip 24693 divcncf 24893 mbfmptcl 25082 i1fposd 25154 itgss3 25261 dvmptcl 25405 dvmptco 25418 dvle 25453 dvfsumle 25467 dvfsumge 25468 dvmptrecl 25470 itgparts 25493 itgsubstlem 25494 itgsubst 25495 ulmss 25838 ulmdvlem2 25842 itgulm2 25850 logtayl 26097 intlewftc 40729 cncfcompt 44370 cncficcgt0 44375 itgsubsticclem 44462 sge0iunmptlemre 44902 hoicvrrex 45043 smfadd 45252 smfpimioompt 45273 smfinfmpt 45306 |
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