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Mirrors > Home > MPE Home > Th. List > fvelima | Structured version Visualization version GIF version |
Description: Function value in an image. Part of Theorem 4.4(iii) of [Monk1] p. 42. (Contributed by NM, 29-Apr-2004.) (Proof shortened by Andrew Salmon, 22-Oct-2011.) |
Ref | Expression |
---|---|
fvelima | ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ (𝐹 “ 𝐵)) → ∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | funbrfv 6947 | . . 3 ⊢ (Fun 𝐹 → (𝑥𝐹𝐴 → (𝐹‘𝑥) = 𝐴)) | |
2 | 1 | reximdv 3159 | . 2 ⊢ (Fun 𝐹 → (∃𝑥 ∈ 𝐵 𝑥𝐹𝐴 → ∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝐴)) |
3 | elimag 6068 | . . 3 ⊢ (𝐴 ∈ (𝐹 “ 𝐵) → (𝐴 ∈ (𝐹 “ 𝐵) ↔ ∃𝑥 ∈ 𝐵 𝑥𝐹𝐴)) | |
4 | 3 | ibi 266 | . 2 ⊢ (𝐴 ∈ (𝐹 “ 𝐵) → ∃𝑥 ∈ 𝐵 𝑥𝐹𝐴) |
5 | 2, 4 | impel 504 | 1 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ (𝐹 “ 𝐵)) → ∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 394 = wceq 1533 ∈ wcel 2098 ∃wrex 3059 class class class wbr 5149 “ cima 5681 Fun wfun 6543 ‘cfv 6549 |
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 5300 ax-nul 5307 ax-pr 5429 |
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-ral 3051 df-rex 3060 df-rab 3419 df-v 3463 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4323 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4910 df-br 5150 df-opab 5212 df-id 5576 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-iota 6501 df-fun 6551 df-fv 6557 |
This theorem is referenced by: funimassd 6964 ssimaex 6982 isofrlem 7347 fimaproj 8140 tz7.49 8466 rankwflemb 9818 tcrank 9909 zorn2lem5 10525 zorn2lem6 10526 uniimadom 10569 wunr1om 10744 tskr1om 10792 tskr1om2 10793 grur1 10845 imadrhmcl 20697 iscldtop 23043 kqfvima 23678 fmfnfmlem4 23905 fmfnfm 23906 qustgpopn 24068 cphsscph 25223 c1liplem1 25973 plypf1 26191 lrrecfr 27906 ltgseg 28472 axcontlem9 28855 uhgrspan1 29188 pthdlem2lem 29653 htthlem 30799 xrofsup 32619 tocyccntz 32957 rhmimaidl 33244 dimval 33429 dimvalfi 33430 txomap 33566 qtophaus 33568 erdszelem7 34938 erdszelem8 34939 mrsub0 35257 mrsubccat 35259 mrsubcn 35260 msubrn 35270 mthmblem 35321 ivthALT 35950 ftc2nc 37306 heibor1lem 37413 aks6d1c4 41727 imacrhmcl 41889 ismrc 42263 icccncfext 45413 dirkercncflem2 45630 smfpimbor1lem1 46324 |
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