df1st2 - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > df1st2		Structured version Visualization version GIF version

Theorem df1st2 8112

Description: An alternate possible definition of the 1^st function. (Contributed by NM, 14-Oct-2004.) (Revised by Mario Carneiro, 31-Aug-2015.)

**Assertion**
Ref	Expression
df1st2	⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥} = (1^st ↾ (V × V))

Distinct variable group: 𝑥,𝑦,𝑧

Proof of Theorem df1st2 Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fo1st 8022	. . . . . 6 ⊢ 1^st :V–onto→V
2		fofn 6799	. . . . . 6 ⊢ (1^st :V–onto→V → 1^st Fn V)
3	1, 2	ax-mp 5	. . . . 5 ⊢ 1^st Fn V
4		dffn5 6944	. . . . 5 ⊢ (1^st Fn V ↔ 1^st = (𝑤 ∈ V ↦ (1^st ‘𝑤)))
5	3, 4	mpbi 233	. . . 4 ⊢ 1^st = (𝑤 ∈ V ↦ (1^st ‘𝑤))
6		mptv 5210	. . . 4 ⊢ (𝑤 ∈ V ↦ (1^st ‘𝑤)) = {⟨𝑤, 𝑧⟩ ∣ 𝑧 = (1^st ‘𝑤)}
7	5, 6	eqtri 2783	. . 3 ⊢ 1^st = {⟨𝑤, 𝑧⟩ ∣ 𝑧 = (1^st ‘𝑤)}
8	7	reseq1i 5965	. 2 ⊢ (1^st ↾ (V × V)) = ({⟨𝑤, 𝑧⟩ ∣ 𝑧 = (1^st ‘𝑤)} ↾ (V × V))
9		resopab 6025	. 2 ⊢ ({⟨𝑤, 𝑧⟩ ∣ 𝑧 = (1^st ‘𝑤)} ↾ (V × V)) = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ 𝑧 = (1^st ‘𝑤))}
10		vex 3454	. . . . 5 ⊢ 𝑥 ∈ V
11		vex 3454	. . . . 5 ⊢ 𝑦 ∈ V
12	10, 11	op1std 8012	. . . 4 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (1^st ‘𝑤) = 𝑥)
13	12	eqeq2d 2771	. . 3 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑧 = (1^st ‘𝑤) ↔ 𝑧 = 𝑥))
14	13	dfoprab3 8066	. 2 ⊢ {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ 𝑧 = (1^st ‘𝑤))} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}
15	8, 9, 14	3eqtrri 2788	1 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥} = (1^st ↾ (V × V))

Colors of variables: wff setvar class

This proof depends on syntax axioms: ∧ wa 401 = wceq 1569 ∈ wcel 2144 Vcvv 3450 ⟨cop 4834 {copab 5166 ↦ cmpt 5185 × cxp 5648 ↾ cres 5652 Fn wfn 6534 –onto→wfo 6537 ‘cfv 6539 {coprab 7422 1^st c1st 8000

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-sep 5248 ax-nul 5259 ax-pr 5390 ax-un 7752

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-ral 3077 df-rex 3087 df-rab 3413 df-v 3452 df-sbc 3739 df-ss 3896 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-sn 4585 df-pr 4587 df-op 4835 df-uni 4867 df-br 5103 df-opab 5167 df-mpt 5186 df-id 5545 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-iota 6494 df-fun 6541 df-fn 6542 df-f 6543 df-fo 6545 df-fv 6547 df-oprab 7425 df-1st 8002 df-2nd 8003

This theorem is used by: df1stres 33345

W3C validator