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Theorem f1stres 5728
 Description: Mapping of a restriction of the 1st (first member of an ordered pair) function. (Contributed by NM, 11-Oct-2004.) (Revised by Mario Carneiro, 8-Sep-2013.)
Assertion
Ref Expression
f1stres (1st ↾ (A × B)):(A × B)⟶A

Proof of Theorem f1stres
Dummy variables x y z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 vex 2554 . . . . . . . 8 y V
2 vex 2554 . . . . . . . 8 z V
31, 2op1sta 4745 . . . . . . 7 dom {⟨y, z⟩} = y
43eleq1i 2100 . . . . . 6 ( dom {⟨y, z⟩} Ay A)
54biimpri 124 . . . . 5 (y A dom {⟨y, z⟩} A)
65adantr 261 . . . 4 ((y A z B) → dom {⟨y, z⟩} A)
76rgen2 2399 . . 3 y A z B dom {⟨y, z⟩} A
8 sneq 3378 . . . . . . 7 (x = ⟨y, z⟩ → {x} = {⟨y, z⟩})
98dmeqd 4480 . . . . . 6 (x = ⟨y, z⟩ → dom {x} = dom {⟨y, z⟩})
109unieqd 3582 . . . . 5 (x = ⟨y, z⟩ → dom {x} = dom {⟨y, z⟩})
1110eleq1d 2103 . . . 4 (x = ⟨y, z⟩ → ( dom {x} A dom {⟨y, z⟩} A))
1211ralxp 4422 . . 3 (x (A × B) dom {x} Ay A z B dom {⟨y, z⟩} A)
137, 12mpbir 134 . 2 x (A × B) dom {x} A
14 df-1st 5709 . . . . 5 1st = (x V ↦ dom {x})
1514reseq1i 4551 . . . 4 (1st ↾ (A × B)) = ((x V ↦ dom {x}) ↾ (A × B))
16 ssv 2959 . . . . 5 (A × B) ⊆ V
17 resmpt 4599 . . . . 5 ((A × B) ⊆ V → ((x V ↦ dom {x}) ↾ (A × B)) = (x (A × B) ↦ dom {x}))
1816, 17ax-mp 7 . . . 4 ((x V ↦ dom {x}) ↾ (A × B)) = (x (A × B) ↦ dom {x})
1915, 18eqtri 2057 . . 3 (1st ↾ (A × B)) = (x (A × B) ↦ dom {x})
2019fmpt 5262 . 2 (x (A × B) dom {x} A ↔ (1st ↾ (A × B)):(A × B)⟶A)
2113, 20mpbi 133 1 (1st ↾ (A × B)):(A × B)⟶A
 Colors of variables: wff set class Syntax hints:   = wceq 1242   ∈ wcel 1390  ∀wral 2300  Vcvv 2551   ⊆ wss 2911  {csn 3367  ⟨cop 3370  ∪ cuni 3571   ↦ cmpt 3809   × cxp 4286  dom cdm 4288   ↾ cres 4290  ⟶wf 4841  1st c1st 5707 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935 This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-fv 4853  df-1st 5709 This theorem is referenced by:  fo1stresm  5730  1stcof  5732
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