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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 5067 | . 2 ⊢ (𝐹‘𝐴) = (℩𝑦𝐴𝐹𝑦) | |
2 | funfveu 5366 | . . 3 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → ∃!𝑦 𝐴𝐹𝑦) | |
3 | euiotaex 5040 | . . 3 ⊢ (∃!𝑦 𝐴𝐹𝑦 → (℩𝑦𝐴𝐹𝑦) ∈ V) | |
4 | 2, 3 | syl 14 | . 2 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (℩𝑦𝐴𝐹𝑦) ∈ V) |
5 | 1, 4 | syl5eqel 2186 | 1 ⊢ ((Fun 𝐹 ∧ 𝐴 ∈ dom 𝐹) → (𝐹‘𝐴) ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 103 ∈ wcel 1448 ∃!weu 1960 Vcvv 2641 class class class wbr 3875 dom cdm 4477 ℩cio 5022 Fun wfun 5053 ‘cfv 5059 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 671 ax-5 1391 ax-7 1392 ax-gen 1393 ax-ie1 1437 ax-ie2 1438 ax-8 1450 ax-10 1451 ax-11 1452 ax-i12 1453 ax-bndl 1454 ax-4 1455 ax-14 1460 ax-17 1474 ax-i9 1478 ax-ial 1482 ax-i5r 1483 ax-ext 2082 ax-sep 3986 ax-pow 4038 ax-pr 4069 |
This theorem depends on definitions: df-bi 116 df-3an 932 df-tru 1302 df-nf 1405 df-sb 1704 df-eu 1963 df-mo 1964 df-clab 2087 df-cleq 2093 df-clel 2096 df-nfc 2229 df-ral 2380 df-rex 2381 df-v 2643 df-sbc 2863 df-un 3025 df-in 3027 df-ss 3034 df-pw 3459 df-sn 3480 df-pr 3481 df-op 3483 df-uni 3684 df-br 3876 df-opab 3930 df-id 4153 df-cnv 4485 df-co 4486 df-dm 4487 df-iota 5024 df-fun 5061 df-fv 5067 |
This theorem is referenced by: fnbrfvb 5394 fvelrnb 5401 funimass4 5404 fvelimab 5409 fniinfv 5411 funfvdm 5416 dmfco 5421 fvco2 5422 eqfnfv 5450 fndmdif 5457 fndmin 5459 fvimacnvi 5466 fvimacnv 5467 funconstss 5470 fniniseg 5472 fniniseg2 5474 fnniniseg2 5475 rexsupp 5476 fvelrn 5483 rexrn 5489 ralrn 5490 dff3im 5497 fmptco 5518 fsn2 5526 fnressn 5538 resfunexg 5573 eufnfv 5580 funfvima3 5583 rexima 5588 ralima 5589 fniunfv 5595 elunirn 5599 dff13 5601 foeqcnvco 5623 f1eqcocnv 5624 isocnv2 5645 isoini 5651 f1oiso 5659 fnovex 5736 suppssof1 5930 offveqb 5932 1stexg 5996 2ndexg 5997 smoiso 6129 rdgtfr 6201 rdgruledefgg 6202 rdgivallem 6208 frectfr 6227 frecrdg 6235 en1 6623 fundmen 6630 fnfi 6753 ordiso2 6835 climshft2 10914 slotex 11768 strsetsid 11774 |
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