1stexg - Intuitionistic Logic Explorer

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Theorem 1stexg 6361

Description: Existence of the first member of a set. (Contributed by Jim Kingdon, 26-Jan-2019.)

**Assertion**
Ref	Expression
1stexg	⊢ (𝐴 ∈ 𝑉 → (1^st ‘𝐴) ∈ V)

**Proof of Theorem 1stexg**
Step	Hyp	Ref	Expression
1		elex 2825	. 2 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ V)
2		fo1st 6351	. . . 4 ⊢ 1^st :V–onto→V
3		fofn 5592	. . . 4 ⊢ (1^st :V–onto→V → 1^st Fn V)
4	2, 3	ax-mp 5	. . 3 ⊢ 1^st Fn V
5		funfvex 5687	. . . 4 ⊢ ((Fun 1^st ∧ 𝐴 ∈ dom 1^st ) → (1^st ‘𝐴) ∈ V)
6	5	funfni 5458	. . 3 ⊢ ((1^st Fn V ∧ 𝐴 ∈ V) → (1^st ‘𝐴) ∈ V)
7	4, 6	mpan 424	. 2 ⊢ (𝐴 ∈ V → (1^st ‘𝐴) ∈ V)
8	1, 7	syl 14	1 ⊢ (𝐴 ∈ 𝑉 → (1^st ‘𝐴) ∈ V)

Colors of variables: wff set class

Syntax hints: → wi 4 ∈ wcel 2203 Vcvv 2813 Fn wfn 5347 –onto→wfo 5350 ‘cfv 5352 1^st c1st 6332

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 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2205 ax-14 2206 ax-ext 2214 ax-sep 4228 ax-pow 4287 ax-pr 4322 ax-un 4554

This theorem depends on definitions: df-bi 117 df-3an 1007 df-tru 1401 df-nf 1510 df-sb 1812 df-eu 2083 df-mo 2084 df-clab 2219 df-cleq 2225 df-clel 2228 df-nfc 2373 df-ral 2525 df-rex 2526 df-v 2815 df-sbc 3043 df-un 3215 df-in 3217 df-ss 3224 df-pw 3671 df-sn 3695 df-pr 3696 df-op 3698 df-uni 3915 df-br 4110 df-opab 4172 df-mpt 4173 df-id 4414 df-xp 4755 df-rel 4756 df-cnv 4757 df-co 4758 df-dm 4759 df-rn 4760 df-iota 5312 df-fun 5354 df-fn 5355 df-f 5356 df-fo 5358 df-fv 5360 df-1st 6334

This theorem is referenced by: elxp7 6364 xpopth 6370 eqop 6371 2nd1st 6374 2ndrn 6377 releldm2 6379 reldm 6380 dfoprab3 6385 elopabi 6391 mpofvex 6401 dfmpo 6419 cnvf1olem 6420 cnvoprab 6430 f1od2 6431 disjxp1 6432 elmpom 6434 xpmapenlem 7102 cnref1o 9983 fsumcnv 12123 fprodcnv 12311 qredeu 12794 qnumval 12882 xpsff1o 13562 txbas 15123 txdis 15142 vtxvalg 16011 vtxex 16013 wlkelvv 16344 wlk2f 16346