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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 2610 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
2 | fo1st 5804 | . . . 4 ⊢ 1st :V–onto→V | |
3 | fofn 5128 | . . . 4 ⊢ (1st :V–onto→V → 1st Fn V) | |
4 | 2, 3 | ax-mp 7 | . . 3 ⊢ 1st Fn V |
5 | funfvex 5212 | . . . 4 ⊢ ((Fun 1st ∧ 𝐴 ∈ dom 1st ) → (1st ‘𝐴) ∈ V) | |
6 | 5 | funfni 5019 | . . 3 ⊢ ((1st Fn V ∧ 𝐴 ∈ V) → (1st ‘𝐴) ∈ V) |
7 | 4, 6 | mpan 414 | . 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 1433 Vcvv 2601 Fn wfn 4917 –onto→wfo 4920 ‘cfv 4922 1st c1st 5785 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188 |
This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-sbc 2816 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-mpt 3841 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-fo 4928 df-fv 4930 df-1st 5787 |
This theorem is referenced by: elxp7 5817 xpopth 5822 eqop 5823 2nd1st 5826 2ndrn 5829 releldm2 5831 reldm 5832 dfoprab3 5837 elopabi 5841 mpt2fvex 5849 dfmpt2 5864 cnvf1olem 5865 cnvoprab 5875 f1od2 5876 cnref1o 8733 qredeu 10479 |
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