df1st2 - Metamath Proof Explorer

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Theorem df1st2 8048

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 7962	. . . . . 6 ⊢ 1^st :V–onto→V
2		fofn 6754	. . . . . 6 ⊢ (1^st :V–onto→V → 1^st Fn V)
3	1, 2	ax-mp 5	. . . . 5 ⊢ 1^st Fn V
4		dffn5 6898	. . . . 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 5191	. . . 4 ⊢ (𝑤 ∈ V ↦ (1^st ‘𝑤)) = {⟨𝑤, 𝑧⟩ ∣ 𝑧 = (1^st ‘𝑤)}
7	5, 6	eqtri 2759	. . 3 ⊢ 1^st = {⟨𝑤, 𝑧⟩ ∣ 𝑧 = (1^st ‘𝑤)}
8	7	reseq1i 5940	. 2 ⊢ (1^st ↾ (V × V)) = ({⟨𝑤, 𝑧⟩ ∣ 𝑧 = (1^st ‘𝑤)} ↾ (V × V))
9		resopab 5999	. 2 ⊢ ({⟨𝑤, 𝑧⟩ ∣ 𝑧 = (1^st ‘𝑤)} ↾ (V × V)) = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ 𝑧 = (1^st ‘𝑤))}
10		vex 3433	. . . . 5 ⊢ 𝑥 ∈ V
11		vex 3433	. . . . 5 ⊢ 𝑦 ∈ V
12	10, 11	op1std 7952	. . . 4 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (1^st ‘𝑤) = 𝑥)
13	12	eqeq2d 2747	. . 3 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑧 = (1^st ‘𝑤) ↔ 𝑧 = 𝑥))
14	13	dfoprab3 8007	. 2 ⊢ {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ 𝑧 = (1^st ‘𝑤))} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}
15	8, 9, 14	3eqtrri 2764	1 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥} = (1^st ↾ (V × V))

Colors of variables: wff setvar class

Syntax hints: ∧ wa 395 = wceq 1542 ∈ wcel 2114 Vcvv 3429 ⟨cop 4573 {copab 5147 ↦ cmpt 5166 × cxp 5629 ↾ cres 5633 Fn wfn 6493 –onto→wfo 6496 ‘cfv 6498 {coprab 7368 1^st c1st 7940

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2708 ax-sep 5231 ax-nul 5241 ax-pr 5375 ax-un 7689

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2539 df-eu 2569 df-clab 2715 df-cleq 2728 df-clel 2811 df-nfc 2885 df-ne 2933 df-ral 3052 df-rex 3062 df-rab 3390 df-v 3431 df-sbc 3729 df-dif 3892 df-un 3894 df-in 3896 df-ss 3906 df-nul 4274 df-if 4467 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4851 df-br 5086 df-opab 5148 df-mpt 5167 df-id 5526 df-xp 5637 df-rel 5638 df-cnv 5639 df-co 5640 df-dm 5641 df-rn 5642 df-res 5643 df-iota 6454 df-fun 6500 df-fn 6501 df-f 6502 df-fo 6504 df-fv 6506 df-oprab 7371 df-1st 7942 df-2nd 7943

This theorem is referenced by: df1stres 32777

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