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Theorem f2ndres 7723
 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 3413 . . . . . . . 8 𝑦 ∈ V
2 vex 3413 . . . . . . . 8 𝑧 ∈ V
31, 2op2nda 6061 . . . . . . 7 ran {⟨𝑦, 𝑧⟩} = 𝑧
43eleq1i 2842 . . . . . 6 ( ran {⟨𝑦, 𝑧⟩} ∈ 𝐵𝑧𝐵)
54biimpri 231 . . . . 5 (𝑧𝐵 ran {⟨𝑦, 𝑧⟩} ∈ 𝐵)
65adantl 485 . . . 4 ((𝑦𝐴𝑧𝐵) → ran {⟨𝑦, 𝑧⟩} ∈ 𝐵)
76rgen2 3132 . . 3 𝑦𝐴𝑧𝐵 ran {⟨𝑦, 𝑧⟩} ∈ 𝐵
8 sneq 4535 . . . . . . 7 (𝑥 = ⟨𝑦, 𝑧⟩ → {𝑥} = {⟨𝑦, 𝑧⟩})
98rneqd 5783 . . . . . 6 (𝑥 = ⟨𝑦, 𝑧⟩ → ran {𝑥} = ran {⟨𝑦, 𝑧⟩})
109unieqd 4815 . . . . 5 (𝑥 = ⟨𝑦, 𝑧⟩ → ran {𝑥} = ran {⟨𝑦, 𝑧⟩})
1110eleq1d 2836 . . . 4 (𝑥 = ⟨𝑦, 𝑧⟩ → ( ran {𝑥} ∈ 𝐵 ran {⟨𝑦, 𝑧⟩} ∈ 𝐵))
1211ralxp 5686 . . 3 (∀𝑥 ∈ (𝐴 × 𝐵) ran {𝑥} ∈ 𝐵 ↔ ∀𝑦𝐴𝑧𝐵 ran {⟨𝑦, 𝑧⟩} ∈ 𝐵)
137, 12mpbir 234 . 2 𝑥 ∈ (𝐴 × 𝐵) ran {𝑥} ∈ 𝐵
14 df-2nd 7699 . . . . 5 2nd = (𝑥 ∈ V ↦ ran {𝑥})
1514reseq1i 5823 . . . 4 (2nd ↾ (𝐴 × 𝐵)) = ((𝑥 ∈ V ↦ ran {𝑥}) ↾ (𝐴 × 𝐵))
16 ssv 3918 . . . . 5 (𝐴 × 𝐵) ⊆ V
17 resmpt 5881 . . . . 5 ((𝐴 × 𝐵) ⊆ V → ((𝑥 ∈ V ↦ ran {𝑥}) ↾ (𝐴 × 𝐵)) = (𝑥 ∈ (𝐴 × 𝐵) ↦ ran {𝑥}))
1816, 17ax-mp 5 . . . 4 ((𝑥 ∈ V ↦ ran {𝑥}) ↾ (𝐴 × 𝐵)) = (𝑥 ∈ (𝐴 × 𝐵) ↦ ran {𝑥})
1915, 18eqtri 2781 . . 3 (2nd ↾ (𝐴 × 𝐵)) = (𝑥 ∈ (𝐴 × 𝐵) ↦ ran {𝑥})
2019fmpt 6870 . 2 (∀𝑥 ∈ (𝐴 × 𝐵) ran {𝑥} ∈ 𝐵 ↔ (2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵)
2113, 20mpbi 233 1 (2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵
 Colors of variables: wff setvar class Syntax hints:   = wceq 1538   ∈ wcel 2111  ∀wral 3070  Vcvv 3409   ⊆ wss 3860  {csn 4525  ⟨cop 4531  ∪ cuni 4801   ↦ cmpt 5115   × cxp 5525  ran crn 5528   ↾ cres 5529  ⟶wf 6335  2nd c2nd 7697 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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5172  ax-nul 5179  ax-pr 5301 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3699  df-csb 3808  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-uni 4802  df-iun 4888  df-br 5036  df-opab 5098  df-mpt 5116  df-id 5433  df-xp 5533  df-rel 5534  df-cnv 5535  df-co 5536  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-iota 6298  df-fun 6341  df-fn 6342  df-f 6343  df-fv 6347  df-2nd 7699 This theorem is referenced by:  fo2ndres  7725  2ndcof  7729  fparlem2  7818  f2ndf  7826  eucalgcvga  15987  2ndfcl  17519  gaid  18501  tx2cn  22315  txkgen  22357  xpinpreima  31381  xpinpreima2  31382  2ndmbfm  31751  filnetlem4  34145  hausgraph  40557
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