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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 2560 | . 2 ⊢ (A ∈ 𝑉 → A ∈ V) | |
2 | fo1st 5726 | . . . 4 ⊢ 1st :V–onto→V | |
3 | fofn 5051 | . . . 4 ⊢ (1st :V–onto→V → 1st Fn V) | |
4 | 2, 3 | ax-mp 7 | . . 3 ⊢ 1st Fn V |
5 | funfvex 5135 | . . . 4 ⊢ ((Fun 1st ∧ A ∈ dom 1st ) → (1st ‘A) ∈ V) | |
6 | 5 | funfni 4942 | . . 3 ⊢ ((1st Fn V ∧ A ∈ V) → (1st ‘A) ∈ V) |
7 | 4, 6 | mpan 400 | . 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 1390 Vcvv 2551 Fn wfn 4840 –onto→wfo 4843 ‘cfv 4845 1st c1st 5707 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-13 1401 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-sep 3866 ax-pow 3918 ax-pr 3935 ax-un 4136 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-v 2553 df-sbc 2759 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-br 3756 df-opab 3810 df-mpt 3811 df-id 4021 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-rn 4299 df-iota 4810 df-fun 4847 df-fn 4848 df-f 4849 df-fo 4851 df-fv 4853 df-1st 5709 |
This theorem is referenced by: elxp7 5739 xpopth 5744 eqop 5745 2nd1st 5748 2ndrn 5751 releldm2 5753 reldm 5754 dfoprab3 5759 elopabi 5763 mpt2fvex 5771 dfmpt2 5786 cnvf1olem 5787 cnref1o 8357 |
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