Theorem df1st2 6428

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} = (1st ↾ (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 6364	. . . . 5 ⊢ 1st :V–onto→V
2		fofn 5597	. . . . 5 ⊢ (1st :V–onto→V → 1st Fn V)
3		dffn5im 5727	. . . . 5 ⊢ (1st Fn V → 1st = (𝑤 ∈ V ↦ (1st ‘𝑤)))
4	1, 2, 3	mp2b 8	. . . 4 ⊢ 1st = (𝑤 ∈ V ↦ (1st ‘𝑤))
5		mptv 4212	. . . 4 ⊢ (𝑤 ∈ V ↦ (1st ‘𝑤)) = {⟨𝑤, 𝑧⟩ ∣ 𝑧 = (1st ‘𝑤)}
6	4, 5	eqtri 2255	. . 3 ⊢ 1st = {⟨𝑤, 𝑧⟩ ∣ 𝑧 = (1st ‘𝑤)}
7	6	reseq1i 5039	. 2 ⊢ (1st ↾ (V × V)) = ({⟨𝑤, 𝑧⟩ ∣ 𝑧 = (1st ‘𝑤)} ↾ (V × V))
8		resopab 5087	. 2 ⊢ ({⟨𝑤, 𝑧⟩ ∣ 𝑧 = (1st ‘𝑤)} ↾ (V × V)) = {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ 𝑧 = (1st ‘𝑤))}
9		vex 2818	. . . . 5 ⊢ 𝑥 ∈ V
10		vex 2818	. . . . 5 ⊢ 𝑦 ∈ V
11	9, 10	op1std 6355	. . . 4 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (1st ‘𝑤) = 𝑥)
12	11	eqeq2d 2246	. . 3 ⊢ (𝑤 = ⟨𝑥, 𝑦⟩ → (𝑧 = (1st ‘𝑤) ↔ 𝑧 = 𝑥))
13	12	dfoprab3 6398	. 2 ⊢ {⟨𝑤, 𝑧⟩ ∣ (𝑤 ∈ (V × V) ∧ 𝑧 = (1st ‘𝑤))} = {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥}
14	7, 8, 13	3eqtrri 2260	1 ⊢ {⟨⟨𝑥, 𝑦⟩, 𝑧⟩ ∣ 𝑧 = 𝑥} = (1st ↾ (V × V))

Syntax hints:   ∧ wa 104   = wceq 1398   ∈ wcel 2205  Vcvv 2815  ⟨cop 3697  {copab 4175   ↦ cmpt 4176   × cxp 4752   ↾ cres 4756   Fn wfn 5352  –onto→wfo 5355  ‘cfv 5357  {coprab 6059  1^st c1st 6345

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-sbc 3046  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-mpt 4178  df-id 4419  df-xp 4760  df-rel 4761  df-cnv 4762  df-co 4763  df-dm 4764  df-rn 4765  df-res 4766  df-iota 5317  df-fun 5359  df-fn 5360  df-f 5361  df-fo 5363  df-fv 5365  df-oprab 6062  df-1st 6347  df-2nd 6348

This theorem is referenced by: (None)