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Theorem f2ndres 8038
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 3482 . . . . . . . 8 𝑦 ∈ V
2 vex 3482 . . . . . . . 8 𝑧 ∈ V
31, 2op2nda 6250 . . . . . . 7 ran {⟨𝑦, 𝑧⟩} = 𝑧
43eleq1i 2830 . . . . . 6 ( ran {⟨𝑦, 𝑧⟩} ∈ 𝐵𝑧𝐵)
54biimpri 228 . . . . 5 (𝑧𝐵 ran {⟨𝑦, 𝑧⟩} ∈ 𝐵)
65adantl 481 . . . 4 ((𝑦𝐴𝑧𝐵) → ran {⟨𝑦, 𝑧⟩} ∈ 𝐵)
76rgen2 3197 . . 3 𝑦𝐴𝑧𝐵 ran {⟨𝑦, 𝑧⟩} ∈ 𝐵
8 sneq 4641 . . . . . . 7 (𝑥 = ⟨𝑦, 𝑧⟩ → {𝑥} = {⟨𝑦, 𝑧⟩})
98rneqd 5952 . . . . . 6 (𝑥 = ⟨𝑦, 𝑧⟩ → ran {𝑥} = ran {⟨𝑦, 𝑧⟩})
109unieqd 4925 . . . . 5 (𝑥 = ⟨𝑦, 𝑧⟩ → ran {𝑥} = ran {⟨𝑦, 𝑧⟩})
1110eleq1d 2824 . . . 4 (𝑥 = ⟨𝑦, 𝑧⟩ → ( ran {𝑥} ∈ 𝐵 ran {⟨𝑦, 𝑧⟩} ∈ 𝐵))
1211ralxp 5855 . . 3 (∀𝑥 ∈ (𝐴 × 𝐵) ran {𝑥} ∈ 𝐵 ↔ ∀𝑦𝐴𝑧𝐵 ran {⟨𝑦, 𝑧⟩} ∈ 𝐵)
137, 12mpbir 231 . 2 𝑥 ∈ (𝐴 × 𝐵) ran {𝑥} ∈ 𝐵
14 df-2nd 8014 . . . . 5 2nd = (𝑥 ∈ V ↦ ran {𝑥})
1514reseq1i 5996 . . . 4 (2nd ↾ (𝐴 × 𝐵)) = ((𝑥 ∈ V ↦ ran {𝑥}) ↾ (𝐴 × 𝐵))
16 ssv 4020 . . . . 5 (𝐴 × 𝐵) ⊆ V
17 resmpt 6057 . . . . 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 2763 . . 3 (2nd ↾ (𝐴 × 𝐵)) = (𝑥 ∈ (𝐴 × 𝐵) ↦ ran {𝑥})
2019fmpt 7130 . 2 (∀𝑥 ∈ (𝐴 × 𝐵) ran {𝑥} ∈ 𝐵 ↔ (2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵)
2113, 20mpbi 230 1 (2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵
Colors of variables: wff setvar class
Syntax hints:   = wceq 1537  wcel 2106  wral 3059  Vcvv 3478  wss 3963  {csn 4631  cop 4637   cuni 4912  cmpt 5231   × cxp 5687  ran crn 5690  cres 5691  wf 6559  2nd c2nd 8012
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-fun 6565  df-fn 6566  df-f 6567  df-2nd 8014
This theorem is referenced by:  fo2ndres  8040  2ndcof  8044  fparlem2  8137  f2ndf  8144  eucalgcvga  16620  2ndfcl  18254  gaid  19330  tx2cn  23634  txkgen  23676  xpinpreima  33867  xpinpreima2  33868  2ndmbfm  34243  filnetlem4  36364  hausgraph  43194
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