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Mirrors > Home > ILE Home > Th. List > 1stexg | GIF version |
Description: Existence of the first member of a set. (Contributed by Jim Kingdon, 26-Jan-2019.) |
Ref | Expression |
---|---|
1stexg | ⊢ (𝐴 ∈ 𝑉 → (1st ‘𝐴) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2697 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
2 | fo1st 6055 | . . . 4 ⊢ 1st :V–onto→V | |
3 | fofn 5347 | . . . 4 ⊢ (1st :V–onto→V → 1st Fn V) | |
4 | 2, 3 | ax-mp 5 | . . 3 ⊢ 1st Fn V |
5 | funfvex 5438 | . . . 4 ⊢ ((Fun 1st ∧ 𝐴 ∈ dom 1st ) → (1st ‘𝐴) ∈ V) | |
6 | 5 | funfni 5223 | . . 3 ⊢ ((1st Fn V ∧ 𝐴 ∈ V) → (1st ‘𝐴) ∈ V) |
7 | 4, 6 | mpan 420 | . 2 ⊢ (𝐴 ∈ V → (1st ‘𝐴) ∈ V) |
8 | 1, 7 | syl 14 | 1 ⊢ (𝐴 ∈ 𝑉 → (1st ‘𝐴) ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 1480 Vcvv 2686 Fn wfn 5118 –onto→wfo 5121 ‘cfv 5123 1st c1st 6036 |
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 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-13 1491 ax-14 1492 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2121 ax-sep 4046 ax-pow 4098 ax-pr 4131 ax-un 4355 |
This theorem depends on definitions: df-bi 116 df-3an 964 df-tru 1334 df-nf 1437 df-sb 1736 df-eu 2002 df-mo 2003 df-clab 2126 df-cleq 2132 df-clel 2135 df-nfc 2270 df-ral 2421 df-rex 2422 df-v 2688 df-sbc 2910 df-un 3075 df-in 3077 df-ss 3084 df-pw 3512 df-sn 3533 df-pr 3534 df-op 3536 df-uni 3737 df-br 3930 df-opab 3990 df-mpt 3991 df-id 4215 df-xp 4545 df-rel 4546 df-cnv 4547 df-co 4548 df-dm 4549 df-rn 4550 df-iota 5088 df-fun 5125 df-fn 5126 df-f 5127 df-fo 5129 df-fv 5131 df-1st 6038 |
This theorem is referenced by: elxp7 6068 xpopth 6074 eqop 6075 2nd1st 6078 2ndrn 6081 releldm2 6083 reldm 6084 dfoprab3 6089 elopabi 6093 mpofvex 6101 dfmpo 6120 cnvf1olem 6121 cnvoprab 6131 f1od2 6132 disjxp1 6133 xpmapenlem 6743 cnref1o 9440 fsumcnv 11206 qredeu 11778 qnumval 11863 txbas 12427 txdis 12446 |
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