![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 | elimag 5711 | . . . 4 ⊢ (𝐴 ∈ (𝐹 “ 𝐵) → (𝐴 ∈ (𝐹 “ 𝐵) ↔ ∃𝑥 ∈ 𝐵 𝑥𝐹𝐴)) | |
2 | 1 | ibi 259 | . . 3 ⊢ (𝐴 ∈ (𝐹 “ 𝐵) → ∃𝑥 ∈ 𝐵 𝑥𝐹𝐴) |
3 | funbrfv 6480 | . . . 4 ⊢ (Fun 𝐹 → (𝑥𝐹𝐴 → (𝐹‘𝑥) = 𝐴)) | |
4 | 3 | reximdv 3224 | . . 3 ⊢ (Fun 𝐹 → (∃𝑥 ∈ 𝐵 𝑥𝐹𝐴 → ∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝐴)) |
5 | 2, 4 | syl5 34 | . 2 ⊢ (Fun 𝐹 → (𝐴 ∈ (𝐹 “ 𝐵) → ∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝐴)) |
6 | 5 | imp 397 | 1 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ (𝐹 “ 𝐵)) → ∃𝑥 ∈ 𝐵 (𝐹‘𝑥) = 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 386 = wceq 1658 ∈ wcel 2166 ∃wrex 3118 class class class wbr 4873 “ cima 5345 Fun wfun 6117 ‘cfv 6123 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1896 ax-4 1910 ax-5 2011 ax-6 2077 ax-7 2114 ax-9 2175 ax-10 2194 ax-11 2209 ax-12 2222 ax-13 2391 ax-ext 2803 ax-sep 5005 ax-nul 5013 ax-pr 5127 |
This theorem depends on definitions: df-bi 199 df-an 387 df-or 881 df-3an 1115 df-tru 1662 df-ex 1881 df-nf 1885 df-sb 2070 df-mo 2605 df-eu 2640 df-clab 2812 df-cleq 2818 df-clel 2821 df-nfc 2958 df-ral 3122 df-rex 3123 df-rab 3126 df-v 3416 df-sbc 3663 df-dif 3801 df-un 3803 df-in 3805 df-ss 3812 df-nul 4145 df-if 4307 df-sn 4398 df-pr 4400 df-op 4404 df-uni 4659 df-br 4874 df-opab 4936 df-id 5250 df-xp 5348 df-rel 5349 df-cnv 5350 df-co 5351 df-dm 5352 df-rn 5353 df-res 5354 df-ima 5355 df-iota 6086 df-fun 6125 df-fv 6131 |
This theorem is referenced by: ssimaex 6510 isofrlem 6845 tz7.49 7806 rankwflemb 8933 tcrank 9024 zorn2lem5 9637 zorn2lem6 9638 uniimadom 9681 wunr1om 9856 tskr1om 9904 tskr1om2 9905 grur1 9957 iscldtop 21270 kqfvima 21904 fmfnfmlem4 22131 fmfnfm 22132 qustgpopn 22293 cphsscph 23419 c1liplem1 24158 plypf1 24367 ltgseg 25908 axcontlem9 26271 uhgrspan1 26600 pthdlem2lem 27069 htthlem 28329 xrofsup 30080 fimaproj 30445 txomap 30446 qtophaus 30448 erdszelem7 31725 erdszelem8 31726 mrsub0 31959 mrsubccat 31961 mrsubcn 31962 msubrn 31972 mthmblem 32023 ivthALT 32868 ftc2nc 34037 heibor1lem 34150 ismrc 38108 funimassd 40233 icccncfext 40895 dirkercncflem2 41115 smfpimbor1lem1 41799 |
Copyright terms: Public domain | W3C validator |