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Mirrors > Home > ILE Home > Th. List > funfvex | GIF version |
Description: The value of a function exists. A special case of Corollary 6.13 of [TakeutiZaring] p. 27. (Contributed by Jim Kingdon, 29-Dec-2018.) |
Ref | Expression |
---|---|
funfvex | ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐹‘𝐴) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-fv 5101 | . 2 ⊢ (𝐹‘𝐴) = (℩𝑦𝐴𝐹𝑦) | |
2 | funfveu 5402 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ∃!𝑦 𝐴𝐹𝑦) | |
3 | euiotaex 5074 | . . 3 ⊢ (∃!𝑦 𝐴𝐹𝑦 → (℩𝑦𝐴𝐹𝑦) ∈ V) | |
4 | 2, 3 | syl 14 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (℩𝑦𝐴𝐹𝑦) ∈ V) |
5 | 1, 4 | eqeltrid 2204 | 1 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐹‘𝐴) ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∈ wcel 1465 ∃!weu 1977 Vcvv 2660 class class class wbr 3899 dom cdm 4509 ℩cio 5056 Fun wfun 5087 ‘cfv 5093 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 683 ax-5 1408 ax-7 1409 ax-gen 1410 ax-ie1 1454 ax-ie2 1455 ax-8 1467 ax-10 1468 ax-11 1469 ax-i12 1470 ax-bndl 1471 ax-4 1472 ax-14 1477 ax-17 1491 ax-i9 1495 ax-ial 1499 ax-i5r 1500 ax-ext 2099 ax-sep 4016 ax-pow 4068 ax-pr 4101 |
This theorem depends on definitions: df-bi 116 df-3an 949 df-tru 1319 df-nf 1422 df-sb 1721 df-eu 1980 df-mo 1981 df-clab 2104 df-cleq 2110 df-clel 2113 df-nfc 2247 df-ral 2398 df-rex 2399 df-v 2662 df-sbc 2883 df-un 3045 df-in 3047 df-ss 3054 df-pw 3482 df-sn 3503 df-pr 3504 df-op 3506 df-uni 3707 df-br 3900 df-opab 3960 df-id 4185 df-cnv 4517 df-co 4518 df-dm 4519 df-iota 5058 df-fun 5095 df-fv 5101 |
This theorem is referenced by: fnbrfvb 5430 fvelrnb 5437 funimass4 5440 fvelimab 5445 fniinfv 5447 funfvdm 5452 dmfco 5457 fvco2 5458 eqfnfv 5486 fndmdif 5493 fndmin 5495 fvimacnvi 5502 fvimacnv 5503 funconstss 5506 fniniseg 5508 fniniseg2 5510 fnniniseg2 5511 rexsupp 5512 fvelrn 5519 rexrn 5525 ralrn 5526 dff3im 5533 fmptco 5554 fsn2 5562 fnressn 5574 resfunexg 5609 eufnfv 5616 funfvima3 5619 rexima 5624 ralima 5625 fniunfv 5631 elunirn 5635 dff13 5637 foeqcnvco 5659 f1eqcocnv 5660 isocnv2 5681 isoini 5687 f1oiso 5695 fnovex 5772 suppssof1 5967 offveqb 5969 1stexg 6033 2ndexg 6034 smoiso 6167 rdgtfr 6239 rdgruledefgg 6240 rdgivallem 6246 frectfr 6265 frecrdg 6273 en1 6661 fundmen 6668 fnfi 6793 ordiso2 6888 climshft2 11030 slotex 11897 strsetsid 11903 |
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