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Theorem f2ndres 7947
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 3450 . . . . . . . 8 𝑦 ∈ V
2 vex 3450 . . . . . . . 8 𝑧 ∈ V
31, 2op2nda 6181 . . . . . . 7 ran {⟨𝑦, 𝑧⟩} = 𝑧
43eleq1i 2829 . . . . . 6 ( ran {⟨𝑦, 𝑧⟩} ∈ 𝐵𝑧𝐵)
54biimpri 227 . . . . 5 (𝑧𝐵 ran {⟨𝑦, 𝑧⟩} ∈ 𝐵)
65adantl 483 . . . 4 ((𝑦𝐴𝑧𝐵) → ran {⟨𝑦, 𝑧⟩} ∈ 𝐵)
76rgen2 3195 . . 3 𝑦𝐴𝑧𝐵 ran {⟨𝑦, 𝑧⟩} ∈ 𝐵
8 sneq 4597 . . . . . . 7 (𝑥 = ⟨𝑦, 𝑧⟩ → {𝑥} = {⟨𝑦, 𝑧⟩})
98rneqd 5894 . . . . . 6 (𝑥 = ⟨𝑦, 𝑧⟩ → ran {𝑥} = ran {⟨𝑦, 𝑧⟩})
109unieqd 4880 . . . . 5 (𝑥 = ⟨𝑦, 𝑧⟩ → ran {𝑥} = ran {⟨𝑦, 𝑧⟩})
1110eleq1d 2823 . . . 4 (𝑥 = ⟨𝑦, 𝑧⟩ → ( ran {𝑥} ∈ 𝐵 ran {⟨𝑦, 𝑧⟩} ∈ 𝐵))
1211ralxp 5798 . . 3 (∀𝑥 ∈ (𝐴 × 𝐵) ran {𝑥} ∈ 𝐵 ↔ ∀𝑦𝐴𝑧𝐵 ran {⟨𝑦, 𝑧⟩} ∈ 𝐵)
137, 12mpbir 230 . 2 𝑥 ∈ (𝐴 × 𝐵) ran {𝑥} ∈ 𝐵
14 df-2nd 7923 . . . . 5 2nd = (𝑥 ∈ V ↦ ran {𝑥})
1514reseq1i 5934 . . . 4 (2nd ↾ (𝐴 × 𝐵)) = ((𝑥 ∈ V ↦ ran {𝑥}) ↾ (𝐴 × 𝐵))
16 ssv 3969 . . . . 5 (𝐴 × 𝐵) ⊆ V
17 resmpt 5992 . . . . 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 2765 . . 3 (2nd ↾ (𝐴 × 𝐵)) = (𝑥 ∈ (𝐴 × 𝐵) ↦ ran {𝑥})
2019fmpt 7059 . 2 (∀𝑥 ∈ (𝐴 × 𝐵) ran {𝑥} ∈ 𝐵 ↔ (2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵)
2113, 20mpbi 229 1 (2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵
Colors of variables: wff setvar class
Syntax hints:   = wceq 1542  wcel 2107  wral 3065  Vcvv 3446  wss 3911  {csn 4587  cop 4593   cuni 4866  cmpt 5189   × cxp 5632  ran crn 5635  cres 5636  wf 6493  2nd c2nd 7921
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-fun 6499  df-fn 6500  df-f 6501  df-2nd 7923
This theorem is referenced by:  fo2ndres  7949  2ndcof  7953  fparlem2  8046  f2ndf  8053  eucalgcvga  16463  2ndfcl  18087  gaid  19080  tx2cn  22964  txkgen  23006  xpinpreima  32490  xpinpreima2  32491  2ndmbfm  32864  filnetlem4  34856  hausgraph  41542
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