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Mirrors > Home > MPE Home > Th. List > funfvima2 | Structured version Visualization version GIF version |
Description: A function's value in an included preimage belongs to the image. (Contributed by NM, 3-Feb-1997.) |
Ref | Expression |
---|---|
funfvima2 | ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐵 ∈ 𝐴 → (𝐹‘𝐵) ∈ (𝐹 “ 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funfvima 7238 | . . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐵 ∈ dom 𝐹) → (𝐵 ∈ 𝐴 → (𝐹‘𝐵) ∈ (𝐹 “ 𝐴))) | |
2 | 1 | ex 411 | . . . 4 ⊢ (Fun 𝐹 → (𝐵 ∈ dom 𝐹 → (𝐵 ∈ 𝐴 → (𝐹‘𝐵) ∈ (𝐹 “ 𝐴)))) |
3 | 2 | com23 86 | . . 3 ⊢ (Fun 𝐹 → (𝐵 ∈ 𝐴 → (𝐵 ∈ dom 𝐹 → (𝐹‘𝐵) ∈ (𝐹 “ 𝐴)))) |
4 | 3 | a2d 29 | . 2 ⊢ (Fun 𝐹 → ((𝐵 ∈ 𝐴 → 𝐵 ∈ dom 𝐹) → (𝐵 ∈ 𝐴 → (𝐹‘𝐵) ∈ (𝐹 “ 𝐴)))) |
5 | ssel 3965 | . 2 ⊢ (𝐴 ⊆ dom 𝐹 → (𝐵 ∈ 𝐴 → 𝐵 ∈ dom 𝐹)) | |
6 | 4, 5 | impel 504 | 1 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐵 ∈ 𝐴 → (𝐹‘𝐵) ∈ (𝐹 “ 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 ∈ wcel 2098 ⊆ wss 3939 dom cdm 5672 “ cima 5675 Fun wfun 6537 ‘cfv 6543 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-12 2166 ax-ext 2696 ax-sep 5294 ax-nul 5301 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2528 df-eu 2557 df-clab 2703 df-cleq 2717 df-clel 2802 df-ne 2931 df-ral 3052 df-rex 3061 df-rab 3420 df-v 3465 df-dif 3942 df-un 3944 df-in 3946 df-ss 3956 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5144 df-opab 5206 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6495 df-fun 6545 df-fn 6546 df-fv 6551 |
This theorem is referenced by: funfvima2d 7240 fnfvima 7241 resfvresima 7243 f1oweALT 7974 tz7.49 8464 phimullem 16747 mrcuni 17600 frlmsslsp 21734 lindfrn 21759 iscldtop 23017 1stcfb 23367 2ndcomap 23380 rnelfm 23875 fmfnfmlem2 23877 fmfnfmlem4 23879 qtopbaslem 24693 tgqioo 24734 bndth 24902 volsup 25503 dyadmbllem 25546 opnmbllem 25548 itg1addlem4 25646 itg1addlem4OLD 25647 c1liplem1 25947 dvcnvrelem1 25968 dvcnvrelem2 25969 plyco0 26144 plyaddlem1 26165 plymullem1 26166 dvloglem 26600 logf1o2 26602 efopn 26610 nocvxminlem 27728 nocvxmin 27729 axcontlem10 28828 imaelshi 31912 funimass4f 32467 sitgclg 34019 cvmliftlem3 34954 ivthALT 35876 opnmbllem0 37186 ismtyres 37338 heibor1lem 37339 ismrc 42186 aomclem4 42546 |
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