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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 2737 | . . . 4 ⊢ (𝑥 ∈ 𝐴 ↦ 𝐵) = (𝑥 ∈ 𝐴 ↦ 𝐵) | |
| 3 | 2 | fmpt 7130 | . . 3 ⊢ (∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶 ↔ (𝑥 ∈ 𝐴 ↦ 𝐵):𝐴⟶𝐶) |
| 4 | 1, 3 | sylibr 234 | . 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 𝐵 ∈ 𝐶) |
| 5 | 4 | r19.21bi 3251 | 1 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → 𝐵 ∈ 𝐶) |
| Colors of variables: wff setvar class |
| Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2108 ∀wral 3061 ↦ cmpt 5225 ⟶wf 6557 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-mpt 5226 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-fun 6563 df-fn 6564 df-f 6565 |
| This theorem is referenced by: rlimmptrcl 15644 lo1mptrcl 15658 o1mptrcl 15659 frlmgsum 21792 uvcresum 21813 psrass1lem 21952 txcnp 23628 ptcnp 23630 ptcn 23635 cnmpt11 23671 cnmpt1t 23673 cnmpt12 23675 cnmptkp 23688 cnmptk1 23689 cnmptkk 23691 cnmptk1p 23693 cnmptk2 23694 cnmpt1plusg 24095 cnmpt1vsca 24202 cnmpt1ds 24864 cncfcompt2 24934 cncfmpt2ss 24942 cnmpt1ip 25281 divcncf 25482 mbfmptcl 25671 i1fposd 25742 itgss3 25850 dvmptcl 25997 dvmptco 26010 dvle 26046 dvfsumle 26060 dvfsumleOLD 26061 dvfsumge 26062 dvmptrecl 26064 itgparts 26088 itgsubstlem 26089 itgsubst 26090 ulmss 26440 ulmdvlem2 26444 itgulm2 26452 logtayl 26702 intlewftc 42062 cncfcompt 45898 cncficcgt0 45903 itgsubsticclem 45990 sge0iunmptlemre 46430 hoicvrrex 46571 smfadd 46780 smfpimioompt 46801 |
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