df1st2 - Metamath Proof Explorer

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Theorem df1st2 8080

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 7991	. . . . . 6 ⊢ 1^st :V–onto→V
2		fofn 6777	. . . . . 6 ⊢ (1^st :V–onto→V → 1^st Fn V)
3	1, 2	ax-mp 5	. . . . 5 ⊢ 1^st Fn V
4		dffn5 6922	. . . . 5 ⊢ (1^st Fn V ↔ 1^st = (𝑤 ∈ V ↦ (1^st ‘𝑤)))
5	3, 4	mpbi 230	. . . 4 ⊢ 1^st = (𝑤 ∈ V ↦ (1^st ‘𝑤))
6		mptv 5216	. . . 4 ⊢ (𝑤 ∈ V ↦ (1^st ‘𝑤)) = {⟨𝑤, 𝑧⟩ ∣ 𝑧 = (1^st ‘𝑤)}
7	5, 6	eqtri 2753	. . 3 ⊢ 1^st = {⟨𝑤, 𝑧⟩ ∣ 𝑧 = (1^st ‘𝑤)}
8	7	reseq1i 5949	. 2 ⊢ (1^st ↾ (V × V)) = ({⟨𝑤, 𝑧⟩ ∣ 𝑧 = (1^st ‘𝑤)} ↾ (V × V))
9		resopab 6008	. 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 7981	. . . 4 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (1^st ‘𝑤) = 𝑥)
13	12	eqeq2d 2741	. . 3 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑧 = (1^st ‘𝑤) ↔ 𝑧 = 𝑥))
14	13	dfoprab3 8036	. 2 ⊢ {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ 𝑧 = (1^st ‘𝑤))} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}
15	8, 9, 14	3eqtrri 2758	1 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥} = (1^st ↾ (V × V))

Colors of variables: wff setvar class

Syntax hints: ∧ wa 395 = wceq 1540 ∈ wcel 2109 Vcvv 3450 ⟨cop 4598 {copab 5172 ↦ cmpt 5191 × cxp 5639 ↾ cres 5643 Fn wfn 6509 –onto→wfo 6512 ‘cfv 6514 {coprab 7391 1^st c1st 7969

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2008 ax-8 2111 ax-9 2119 ax-10 2142 ax-11 2158 ax-12 2178 ax-ext 2702 ax-sep 5254 ax-nul 5264 ax-pr 5390 ax-un 7714

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2066 df-mo 2534 df-eu 2563 df-clab 2709 df-cleq 2722 df-clel 2804 df-nfc 2879 df-ne 2927 df-ral 3046 df-rex 3055 df-rab 3409 df-v 3452 df-sbc 3757 df-dif 3920 df-un 3922 df-in 3924 df-ss 3934 df-nul 4300 df-if 4492 df-sn 4593 df-pr 4595 df-op 4599 df-uni 4875 df-br 5111 df-opab 5173 df-mpt 5192 df-id 5536 df-xp 5647 df-rel 5648 df-cnv 5649 df-co 5650 df-dm 5651 df-rn 5652 df-res 5653 df-iota 6467 df-fun 6516 df-fn 6517 df-f 6518 df-fo 6520 df-fv 6522 df-oprab 7394 df-1st 7971 df-2nd 7972

This theorem is referenced by: df1stres 32634

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