df1st2 - Metamath Proof Explorer

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Theorem df1st2 7938

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 7851	. . . . . 6 ⊢ 1^st :V–onto→V
2		fofn 6690	. . . . . 6 ⊢ (1^st :V–onto→V → 1^st Fn V)
3	1, 2	ax-mp 5	. . . . 5 ⊢ 1^st Fn V
4		dffn5 6828	. . . . 5 ⊢ (1^st Fn V ↔ 1^st = (𝑤 ∈ V ↦ (1^st ‘𝑤)))
5	3, 4	mpbi 229	. . . 4 ⊢ 1^st = (𝑤 ∈ V ↦ (1^st ‘𝑤))
6		mptv 5190	. . . 4 ⊢ (𝑤 ∈ V ↦ (1^st ‘𝑤)) = {⟨𝑤, 𝑧⟩ ∣ 𝑧 = (1^st ‘𝑤)}
7	5, 6	eqtri 2766	. . 3 ⊢ 1^st = {⟨𝑤, 𝑧⟩ ∣ 𝑧 = (1^st ‘𝑤)}
8	7	reseq1i 5887	. 2 ⊢ (1^st ↾ (V × V)) = ({⟨𝑤, 𝑧⟩ ∣ 𝑧 = (1^st ‘𝑤)} ↾ (V × V))
9		resopab 5942	. 2 ⊢ ({⟨𝑤, 𝑧⟩ ∣ 𝑧 = (1^st ‘𝑤)} ↾ (V × V)) = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ 𝑧 = (1^st ‘𝑤))}
10		vex 3436	. . . . 5 ⊢ 𝑥 ∈ V
11		vex 3436	. . . . 5 ⊢ 𝑦 ∈ V
12	10, 11	op1std 7841	. . . 4 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (1^st ‘𝑤) = 𝑥)
13	12	eqeq2d 2749	. . 3 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑧 = (1^st ‘𝑤) ↔ 𝑧 = 𝑥))
14	13	dfoprab3 7894	. 2 ⊢ {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ 𝑧 = (1^st ‘𝑤))} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}
15	8, 9, 14	3eqtrri 2771	1 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥} = (1^st ↾ (V × V))

Colors of variables: wff setvar class

Syntax hints: ∧ wa 396 = wceq 1539 ∈ wcel 2106 Vcvv 3432 ⟨cop 4567 {copab 5136 ↦ cmpt 5157 × cxp 5587 ↾ cres 5591 Fn wfn 6428 –onto→wfo 6431 ‘cfv 6433 {coprab 7276 1^st c1st 7829

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-sep 5223 ax-nul 5230 ax-pr 5352 ax-un 7588

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ral 3069 df-rex 3070 df-rab 3073 df-v 3434 df-sbc 3717 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-fo 6439 df-fv 6441 df-oprab 7279 df-1st 7831 df-2nd 7832

This theorem is referenced by: df1stres 31036

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