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Theorem f2ndres 6067
 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 2693 . . . . . . . 8 𝑦 ∈ V
2 vex 2693 . . . . . . . 8 𝑧 ∈ V
31, 2op2nda 5032 . . . . . . 7 ran {⟨𝑦, 𝑧⟩} = 𝑧
43eleq1i 2206 . . . . . 6 ( ran {⟨𝑦, 𝑧⟩} ∈ 𝐵𝑧𝐵)
54biimpri 132 . . . . 5 (𝑧𝐵 ran {⟨𝑦, 𝑧⟩} ∈ 𝐵)
65adantl 275 . . . 4 ((𝑦𝐴𝑧𝐵) → ran {⟨𝑦, 𝑧⟩} ∈ 𝐵)
76rgen2 2522 . . 3 𝑦𝐴𝑧𝐵 ran {⟨𝑦, 𝑧⟩} ∈ 𝐵
8 sneq 3544 . . . . . . 7 (𝑥 = ⟨𝑦, 𝑧⟩ → {𝑥} = {⟨𝑦, 𝑧⟩})
98rneqd 4777 . . . . . 6 (𝑥 = ⟨𝑦, 𝑧⟩ → ran {𝑥} = ran {⟨𝑦, 𝑧⟩})
109unieqd 3756 . . . . 5 (𝑥 = ⟨𝑦, 𝑧⟩ → ran {𝑥} = ran {⟨𝑦, 𝑧⟩})
1110eleq1d 2209 . . . 4 (𝑥 = ⟨𝑦, 𝑧⟩ → ( ran {𝑥} ∈ 𝐵 ran {⟨𝑦, 𝑧⟩} ∈ 𝐵))
1211ralxp 4691 . . 3 (∀𝑥 ∈ (𝐴 × 𝐵) ran {𝑥} ∈ 𝐵 ↔ ∀𝑦𝐴𝑧𝐵 ran {⟨𝑦, 𝑧⟩} ∈ 𝐵)
137, 12mpbir 145 . 2 𝑥 ∈ (𝐴 × 𝐵) ran {𝑥} ∈ 𝐵
14 df-2nd 6048 . . . . 5 2nd = (𝑥 ∈ V ↦ ran {𝑥})
1514reseq1i 4824 . . . 4 (2nd ↾ (𝐴 × 𝐵)) = ((𝑥 ∈ V ↦ ran {𝑥}) ↾ (𝐴 × 𝐵))
16 ssv 3125 . . . . 5 (𝐴 × 𝐵) ⊆ V
17 resmpt 4876 . . . . 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 2161 . . 3 (2nd ↾ (𝐴 × 𝐵)) = (𝑥 ∈ (𝐴 × 𝐵) ↦ ran {𝑥})
2019fmpt 5579 . 2 (∀𝑥 ∈ (𝐴 × 𝐵) ran {𝑥} ∈ 𝐵 ↔ (2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵)
2113, 20mpbi 144 1 (2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵
 Colors of variables: wff set class Syntax hints:   = wceq 1332   ∈ wcel 1481  ∀wral 2417  Vcvv 2690   ⊆ wss 3077  {csn 3533  ⟨cop 3536  ∪ cuni 3745   ↦ cmpt 3998   × cxp 4546  ran crn 4549   ↾ cres 4550  ⟶wf 5128  2nd c2nd 6046 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4055  ax-pow 4107  ax-pr 4140 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2692  df-sbc 2915  df-csb 3009  df-un 3081  df-in 3083  df-ss 3090  df-pw 3518  df-sn 3539  df-pr 3540  df-op 3542  df-uni 3746  df-iun 3824  df-br 3939  df-opab 3999  df-mpt 4000  df-id 4224  df-xp 4554  df-rel 4555  df-cnv 4556  df-co 4557  df-dm 4558  df-rn 4559  df-res 4560  df-ima 4561  df-iota 5097  df-fun 5134  df-fn 5135  df-f 5136  df-fv 5140  df-2nd 6048 This theorem is referenced by:  fo2ndresm  6069  2ndcof  6071  f2ndf  6132  eucalgcvga  11795  tx2cn  12498
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