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Mirrors > Home > ILE Home > Th. List > 2ndexg | GIF version |
Description: Existence of the first member of a set. (Contributed by Jim Kingdon, 26-Jan-2019.) |
Ref | Expression |
---|---|
2ndexg | ⊢ (𝐴 ∈ 𝑉 → (2nd ‘𝐴) ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | elex 2760 | . 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V) | |
2 | fo2nd 6173 | . . . 4 ⊢ 2nd :V–onto→V | |
3 | fofn 5452 | . . . 4 ⊢ (2nd :V–onto→V → 2nd Fn V) | |
4 | 2, 3 | ax-mp 5 | . . 3 ⊢ 2nd Fn V |
5 | funfvex 5544 | . . . 4 ⊢ ((Fun 2nd ∧ 𝐴 ∈ dom 2nd ) → (2nd ‘𝐴) ∈ V) | |
6 | 5 | funfni 5328 | . . 3 ⊢ ((2nd Fn V ∧ 𝐴 ∈ V) → (2nd ‘𝐴) ∈ V) |
7 | 4, 6 | mpan 424 | . 2 ⊢ (𝐴 ∈ V → (2nd ‘𝐴) ∈ V) |
8 | 1, 7 | syl 14 | 1 ⊢ (𝐴 ∈ 𝑉 → (2nd ‘𝐴) ∈ V) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∈ wcel 2158 Vcvv 2749 Fn wfn 5223 –onto→wfo 5226 ‘cfv 5228 2nd c2nd 6154 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-io 710 ax-5 1457 ax-7 1458 ax-gen 1459 ax-ie1 1503 ax-ie2 1504 ax-8 1514 ax-10 1515 ax-11 1516 ax-i12 1517 ax-bndl 1519 ax-4 1520 ax-17 1536 ax-i9 1540 ax-ial 1544 ax-i5r 1545 ax-13 2160 ax-14 2161 ax-ext 2169 ax-sep 4133 ax-pow 4186 ax-pr 4221 ax-un 4445 |
This theorem depends on definitions: df-bi 117 df-3an 981 df-tru 1366 df-nf 1471 df-sb 1773 df-eu 2039 df-mo 2040 df-clab 2174 df-cleq 2180 df-clel 2183 df-nfc 2318 df-ral 2470 df-rex 2471 df-v 2751 df-sbc 2975 df-un 3145 df-in 3147 df-ss 3154 df-pw 3589 df-sn 3610 df-pr 3611 df-op 3613 df-uni 3822 df-br 4016 df-opab 4077 df-mpt 4078 df-id 4305 df-xp 4644 df-rel 4645 df-cnv 4646 df-co 4647 df-dm 4648 df-rn 4649 df-iota 5190 df-fun 5230 df-fn 5231 df-f 5232 df-fo 5234 df-fv 5236 df-2nd 6156 |
This theorem is referenced by: elxp7 6185 xpopth 6191 eqop 6192 op1steq 6194 2nd1st 6195 2ndrn 6198 dfoprab3 6206 elopabi 6210 mpofvex 6218 dfmpo 6238 cnvf1olem 6239 cnvoprab 6249 f1od2 6250 xpmapenlem 6863 cc2lem 7279 cnref1o 9664 fsumcnv 11459 fprodcnv 11647 qredeu 12111 qdenval 12200 xpsff1o 12787 txbas 14054 txdis 14073 |
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