df1st2 - Intuitionistic Logic Explorer

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Theorem df1st2 5860

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 5804	. . . . 5 ⊢ 1^st :V–onto→V
2		fofn 5128	. . . . 5 ⊢ (1^st :V–onto→V → 1^st Fn V)
3		dffn5im 5240	. . . . 5 ⊢ (1^st Fn V → 1^st = (𝑤 ∈ V ↦ (1^st ‘𝑤)))
4	1, 2, 3	mp2b 8	. . . 4 ⊢ 1^st = (𝑤 ∈ V ↦ (1^st ‘𝑤))
5		mptv 3874	. . . 4 ⊢ (𝑤 ∈ V ↦ (1^st ‘𝑤)) = {⟨𝑤, 𝑧⟩ ∣ 𝑧 = (1^st ‘𝑤)}
6	4, 5	eqtri 2101	. . 3 ⊢ 1^st = {⟨𝑤, 𝑧⟩ ∣ 𝑧 = (1^st ‘𝑤)}
7	6	reseq1i 4626	. 2 ⊢ (1^st ↾ (V × V)) = ({⟨𝑤, 𝑧⟩ ∣ 𝑧 = (1^st ‘𝑤)} ↾ (V × V))
8		resopab 4672	. 2 ⊢ ({⟨𝑤, 𝑧⟩ ∣ 𝑧 = (1^st ‘𝑤)} ↾ (V × V)) = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ 𝑧 = (1^st ‘𝑤))}
9		vex 2604	. . . . 5 ⊢ 𝑥 ∈ V
10		vex 2604	. . . . 5 ⊢ 𝑦 ∈ V
11	9, 10	op1std 5795	. . . 4 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (1^st ‘𝑤) = 𝑥)
12	11	eqeq2d 2092	. . 3 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑧 = (1^st ‘𝑤) ↔ 𝑧 = 𝑥))
13	12	dfoprab3 5837	. 2 ⊢ {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ 𝑧 = (1^st ‘𝑤))} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}
14	7, 8, 13	3eqtrri 2106	1 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥} = (1^st ↾ (V × V))

Colors of variables: wff set class

Syntax hints: ∧ wa 102 = wceq 1284 ∈ wcel 1433 Vcvv 2601 ⟨cop 3401 {copab 3838 ↦ cmpt 3839 × cxp 4361 ↾ cres 4365 Fn wfn 4917 –onto→wfo 4920 ‘cfv 4922 {coprab 5533 1^st c1st 5785

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 104 ax-ia2 105 ax-ia3 106 ax-io 662 ax-5 1376 ax-7 1377 ax-gen 1378 ax-ie1 1422 ax-ie2 1423 ax-8 1435 ax-10 1436 ax-11 1437 ax-i12 1438 ax-bndl 1439 ax-4 1440 ax-13 1444 ax-14 1445 ax-17 1459 ax-i9 1463 ax-ial 1467 ax-i5r 1468 ax-ext 2063 ax-sep 3896 ax-pow 3948 ax-pr 3964 ax-un 4188

This theorem depends on definitions: df-bi 115 df-3an 921 df-tru 1287 df-nf 1390 df-sb 1686 df-eu 1944 df-mo 1945 df-clab 2068 df-cleq 2074 df-clel 2077 df-nfc 2208 df-ral 2353 df-rex 2354 df-v 2603 df-sbc 2816 df-un 2977 df-in 2979 df-ss 2986 df-pw 3384 df-sn 3404 df-pr 3405 df-op 3407 df-uni 3602 df-br 3786 df-opab 3840 df-mpt 3841 df-id 4048 df-xp 4369 df-rel 4370 df-cnv 4371 df-co 4372 df-dm 4373 df-rn 4374 df-res 4375 df-iota 4887 df-fun 4924 df-fn 4925 df-f 4926 df-fo 4928 df-fv 4930 df-oprab 5536 df-1st 5787 df-2nd 5788

This theorem is referenced by: (None)

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