![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 7181 | . . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐵 ∈ dom 𝐹) → (𝐵 ∈ 𝐴 → (𝐹‘𝐵) ∈ (𝐹 “ 𝐴))) | |
2 | 1 | ex 414 | . . . 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 3938 | . 2 ⊢ (𝐴 ⊆ dom 𝐹 → (𝐵 ∈ 𝐴 → 𝐵 ∈ dom 𝐹)) | |
6 | 4, 5 | impel 507 | 1 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐵 ∈ 𝐴 → (𝐹‘𝐵) ∈ (𝐹 “ 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 397 ∈ wcel 2107 ⊆ wss 3911 dom cdm 5634 “ cima 5637 Fun wfun 6491 ‘cfv 6497 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-10 2138 ax-12 2172 ax-ext 2704 ax-sep 5257 ax-nul 5264 ax-pr 5385 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-3an 1090 df-tru 1545 df-fal 1555 df-ex 1783 df-nf 1787 df-sb 2069 df-mo 2535 df-eu 2564 df-clab 2711 df-cleq 2725 df-clel 2811 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3407 df-v 3446 df-dif 3914 df-un 3916 df-in 3918 df-ss 3928 df-nul 4284 df-if 4488 df-sn 4588 df-pr 4590 df-op 4594 df-uni 4867 df-br 5107 df-opab 5169 df-id 5532 df-xp 5640 df-rel 5641 df-cnv 5642 df-co 5643 df-dm 5644 df-rn 5645 df-res 5646 df-ima 5647 df-iota 6449 df-fun 6499 df-fn 6500 df-fv 6505 |
This theorem is referenced by: funfvima2d 7183 fnfvima 7184 resfvresima 7186 f1oweALT 7906 tz7.49 8392 phimullem 16656 mrcuni 17506 frlmsslsp 21218 lindfrn 21243 iscldtop 22462 1stcfb 22812 2ndcomap 22825 rnelfm 23320 fmfnfmlem2 23322 fmfnfmlem4 23324 qtopbaslem 24138 tgqioo 24179 bndth 24337 volsup 24936 dyadmbllem 24979 opnmbllem 24981 itg1addlem4 25079 itg1addlem4OLD 25080 c1liplem1 25376 dvcnvrelem1 25397 dvcnvrelem2 25398 plyco0 25569 plyaddlem1 25590 plymullem1 25591 dvloglem 26019 logf1o2 26021 efopn 26029 nocvxminlem 27139 nocvxmin 27140 axcontlem10 27964 imaelshi 31042 funimass4f 31597 sitgclg 32999 cvmliftlem3 33938 ivthALT 34853 opnmbllem0 36160 ismtyres 36313 heibor1lem 36314 ismrc 41067 aomclem4 41427 |
Copyright terms: Public domain | W3C validator |