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Theorem f2ndres 5729
Description: Mapping of a restriction of the 2nd (second member of an ordered pair) function. (Contributed by NM, 7-Aug-2006.) (Revised by Mario Carneiro, 8-Sep-2013.)
Assertion
Ref Expression
f2ndres (2nd ↾ (A × B)):(A × B)⟶B

Proof of Theorem f2ndres
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, 2op2nda 4748 . . . . . . 7 ran {⟨y, z⟩} = z
43eleq1i 2100 . . . . . 6 ( ran {⟨y, z⟩} Bz B)
54biimpri 124 . . . . 5 (z B ran {⟨y, z⟩} B)
65adantl 262 . . . 4 ((y A z B) → ran {⟨y, z⟩} B)
76rgen2 2399 . . 3 y A z B ran {⟨y, z⟩} B
8 sneq 3378 . . . . . . 7 (x = ⟨y, z⟩ → {x} = {⟨y, z⟩})
98rneqd 4506 . . . . . 6 (x = ⟨y, z⟩ → ran {x} = ran {⟨y, z⟩})
109unieqd 3582 . . . . 5 (x = ⟨y, z⟩ → ran {x} = ran {⟨y, z⟩})
1110eleq1d 2103 . . . 4 (x = ⟨y, z⟩ → ( ran {x} B ran {⟨y, z⟩} B))
1211ralxp 4422 . . 3 (x (A × B) ran {x} By A z B ran {⟨y, z⟩} B)
137, 12mpbir 134 . 2 x (A × B) ran {x} B
14 df-2nd 5710 . . . . 5 2nd = (x V ↦ ran {x})
1514reseq1i 4551 . . . 4 (2nd ↾ (A × B)) = ((x V ↦ ran {x}) ↾ (A × B))
16 ssv 2959 . . . . 5 (A × B) ⊆ V
17 resmpt 4599 . . . . 5 ((A × B) ⊆ V → ((x V ↦ ran {x}) ↾ (A × B)) = (x (A × B) ↦ ran {x}))
1816, 17ax-mp 7 . . . 4 ((x V ↦ ran {x}) ↾ (A × B)) = (x (A × B) ↦ ran {x})
1915, 18eqtri 2057 . . 3 (2nd ↾ (A × B)) = (x (A × B) ↦ ran {x})
2019fmpt 5262 . 2 (x (A × B) ran {x} B ↔ (2nd ↾ (A × B)):(A × B)⟶B)
2113, 20mpbi 133 1 (2nd ↾ (A × B)):(A × B)⟶B
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  ran crn 4289  cres 4290  wf 4841  2nd c2nd 5708
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-2nd 5710
This theorem is referenced by:  fo2ndresm  5731  2ndcof  5733  f2ndf  5789
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