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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 6983 | . . . . 5 ⊢ ((Fun 𝐹 ∧ 𝐵 ∈ dom 𝐹) → (𝐵 ∈ 𝐴 → (𝐹‘𝐵) ∈ (𝐹 “ 𝐴))) | |
2 | 1 | ex 413 | . . . 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 3958 | . 2 ⊢ (𝐴 ⊆ dom 𝐹 → (𝐵 ∈ 𝐴 → 𝐵 ∈ dom 𝐹)) | |
6 | 4, 5 | impel 506 | 1 ⊢ ((Fun 𝐹 ∧ 𝐴 ⊆ dom 𝐹) → (𝐵 ∈ 𝐴 → (𝐹‘𝐵) ∈ (𝐹 “ 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∈ wcel 2105 ⊆ wss 3933 dom cdm 5548 “ cima 5551 Fun wfun 6342 ‘cfv 6348 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1787 ax-4 1801 ax-5 1902 ax-6 1961 ax-7 2006 ax-8 2107 ax-9 2115 ax-10 2136 ax-11 2151 ax-12 2167 ax-ext 2790 ax-sep 5194 ax-nul 5201 ax-pr 5320 |
This theorem depends on definitions: df-bi 208 df-an 397 df-or 842 df-3an 1081 df-tru 1531 df-ex 1772 df-nf 1776 df-sb 2061 df-mo 2615 df-eu 2647 df-clab 2797 df-cleq 2811 df-clel 2890 df-nfc 2960 df-ral 3140 df-rex 3141 df-rab 3144 df-v 3494 df-sbc 3770 df-dif 3936 df-un 3938 df-in 3940 df-ss 3949 df-nul 4289 df-if 4464 df-sn 4558 df-pr 4560 df-op 4564 df-uni 4831 df-br 5058 df-opab 5120 df-id 5453 df-xp 5554 df-rel 5555 df-cnv 5556 df-co 5557 df-dm 5558 df-rn 5559 df-res 5560 df-ima 5561 df-iota 6307 df-fun 6350 df-fn 6351 df-fv 6356 |
This theorem is referenced by: funfvima2d 6985 fnfvima 6986 resfvresima 6988 f1oweALT 7662 tz7.49 8070 phimullem 16104 mrcuni 16880 frlmsslsp 20868 lindfrn 20893 iscldtop 21631 1stcfb 21981 2ndcomap 21994 rnelfm 22489 fmfnfmlem2 22491 fmfnfmlem4 22493 qtopbaslem 23294 tgqioo 23335 bndth 23489 volsup 24084 dyadmbllem 24127 opnmbllem 24129 itg1addlem4 24227 c1liplem1 24520 dvcnvrelem1 24541 dvcnvrelem2 24542 plyco0 24709 plyaddlem1 24730 plymullem1 24731 dvloglem 25158 logf1o2 25160 efopn 25168 axcontlem10 26686 imaelshi 29762 funimass4f 30310 sitgclg 31499 cvmliftlem3 32431 nocvxminlem 33144 nocvxmin 33145 ivthALT 33580 opnmbllem0 34809 ismtyres 34967 heibor1lem 34968 ismrc 39176 aomclem4 39535 |
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