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Mirrors > Home > MPE Home > Th. List > fvelrn | Structured version Visualization version GIF version |
Description: A function's value belongs to its range. (Contributed by NM, 14-Oct-1996.) |
Ref | Expression |
---|---|
fvelrn | ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐹‘𝐴) ∈ ran 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eleq1 2676 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ dom 𝐹 ↔ 𝐴 ∈ dom 𝐹)) | |
2 | 1 | anbi2d 736 | . . . 4 ⊢ (𝑥 = 𝐴 → ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) ↔ (Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹))) |
3 | fveq2 6103 | . . . . 5 ⊢ (𝑥 = 𝐴 → (𝐹‘𝑥) = (𝐹‘𝐴)) | |
4 | 3 | eleq1d 2672 | . . . 4 ⊢ (𝑥 = 𝐴 → ((𝐹‘𝑥) ∈ ran 𝐹 ↔ (𝐹‘𝐴) ∈ ran 𝐹)) |
5 | 2, 4 | imbi12d 333 | . . 3 ⊢ (𝑥 = 𝐴 → (((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) ∈ ran 𝐹) ↔ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐹‘𝐴) ∈ ran 𝐹))) |
6 | funfvop 6237 | . . . . 5 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → 〈𝑥, (𝐹‘𝑥)〉 ∈ 𝐹) | |
7 | vex 3176 | . . . . . 6 ⊢ 𝑥 ∈ V | |
8 | opeq1 4340 | . . . . . . 7 ⊢ (𝑦 = 𝑥 → 〈𝑦, (𝐹‘𝑥)〉 = 〈𝑥, (𝐹‘𝑥)〉) | |
9 | 8 | eleq1d 2672 | . . . . . 6 ⊢ (𝑦 = 𝑥 → (〈𝑦, (𝐹‘𝑥)〉 ∈ 𝐹 ↔ 〈𝑥, (𝐹‘𝑥)〉 ∈ 𝐹)) |
10 | 7, 9 | spcev 3273 | . . . . 5 ⊢ (〈𝑥, (𝐹‘𝑥)〉 ∈ 𝐹 → ∃𝑦〈𝑦, (𝐹‘𝑥)〉 ∈ 𝐹) |
11 | 6, 10 | syl 17 | . . . 4 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → ∃𝑦〈𝑦, (𝐹‘𝑥)〉 ∈ 𝐹) |
12 | fvex 6113 | . . . . 5 ⊢ (𝐹‘𝑥) ∈ V | |
13 | 12 | elrn2 5286 | . . . 4 ⊢ ((𝐹‘𝑥) ∈ ran 𝐹 ↔ ∃𝑦〈𝑦, (𝐹‘𝑥)〉 ∈ 𝐹) |
14 | 11, 13 | sylibr 223 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝑥 ∈ dom 𝐹) → (𝐹‘𝑥) ∈ ran 𝐹) |
15 | 5, 14 | vtoclg 3239 | . 2 ⊢ (𝐴 ∈ dom 𝐹 → ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐹‘𝐴) ∈ ran 𝐹)) |
16 | 15 | anabsi7 856 | 1 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐹‘𝐴) ∈ ran 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 383 = wceq 1475 ∃wex 1695 ∈ wcel 1977 〈cop 4131 dom cdm 5038 ran crn 5039 Fun wfun 5798 ‘cfv 5804 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-id 4953 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-iota 5768 df-fun 5806 df-fn 5807 df-fv 5812 |
This theorem is referenced by: nelrnfvne 6261 fnfvelrn 6264 eldmrexrn 6273 fvn0fvelrn 6335 funfvima 6396 elunirn 6413 rankwflemb 8539 dfac9 8841 fin1a2lem6 9110 gsumpropd2lem 17096 usgraedg3 25915 nbgraf1olem5 25974 usgrwlknloop 26093 usgra2wlkspthlem2 26148 nvnencycllem 26171 wlkiswwlk1 26218 usg2wlkonot 26410 usg2wotspth 26411 opfv 28828 nofv 31054 sltres 31061 bdayelon 31079 nodenselem3 31082 bj-elccinfty 32278 bj-minftyccb 32289 icoreunrn 32383 indexdom 32699 diaclN 35357 dia1elN 35361 docaclN 35431 dibclN 35469 dfac21 36654 gneispace 37452 cncmpmax 38214 icccncfext 38773 stoweidlem27 38920 stoweidlem29 38922 stoweidlem59 38952 fourierdlem20 39020 fourierdlem63 39062 fourierdlem76 39075 fourierdlem82 39081 fourierdlem93 39092 fourierdlem113 39112 fge0iccico 39263 sge0sn 39272 sge0tsms 39273 sge0cl 39274 sge0isum 39320 hoicvr 39438 afvelrn 39897 usgredg3 40443 ushgredgedga 40456 ushgredgedgaloop 40458 subgruhgredgd 40508 edginwlk 40839 iedginwlk 40841 suppdm 42094 |
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