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Theorem f2ndres 7341
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 3354 . . . . . . . 8 𝑦 ∈ V
2 vex 3354 . . . . . . . 8 𝑧 ∈ V
31, 2op2nda 5765 . . . . . . 7 ran {⟨𝑦, 𝑧⟩} = 𝑧
43eleq1i 2841 . . . . . 6 ( ran {⟨𝑦, 𝑧⟩} ∈ 𝐵𝑧𝐵)
54biimpri 218 . . . . 5 (𝑧𝐵 ran {⟨𝑦, 𝑧⟩} ∈ 𝐵)
65adantl 467 . . . 4 ((𝑦𝐴𝑧𝐵) → ran {⟨𝑦, 𝑧⟩} ∈ 𝐵)
76rgen2 3124 . . 3 𝑦𝐴𝑧𝐵 ran {⟨𝑦, 𝑧⟩} ∈ 𝐵
8 sneq 4327 . . . . . . 7 (𝑥 = ⟨𝑦, 𝑧⟩ → {𝑥} = {⟨𝑦, 𝑧⟩})
98rneqd 5492 . . . . . 6 (𝑥 = ⟨𝑦, 𝑧⟩ → ran {𝑥} = ran {⟨𝑦, 𝑧⟩})
109unieqd 4585 . . . . 5 (𝑥 = ⟨𝑦, 𝑧⟩ → ran {𝑥} = ran {⟨𝑦, 𝑧⟩})
1110eleq1d 2835 . . . 4 (𝑥 = ⟨𝑦, 𝑧⟩ → ( ran {𝑥} ∈ 𝐵 ran {⟨𝑦, 𝑧⟩} ∈ 𝐵))
1211ralxp 5403 . . 3 (∀𝑥 ∈ (𝐴 × 𝐵) ran {𝑥} ∈ 𝐵 ↔ ∀𝑦𝐴𝑧𝐵 ran {⟨𝑦, 𝑧⟩} ∈ 𝐵)
137, 12mpbir 221 . 2 𝑥 ∈ (𝐴 × 𝐵) ran {𝑥} ∈ 𝐵
14 df-2nd 7317 . . . . 5 2nd = (𝑥 ∈ V ↦ ran {𝑥})
1514reseq1i 5531 . . . 4 (2nd ↾ (𝐴 × 𝐵)) = ((𝑥 ∈ V ↦ ran {𝑥}) ↾ (𝐴 × 𝐵))
16 ssv 3775 . . . . 5 (𝐴 × 𝐵) ⊆ V
17 resmpt 5591 . . . . 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 2793 . . 3 (2nd ↾ (𝐴 × 𝐵)) = (𝑥 ∈ (𝐴 × 𝐵) ↦ ran {𝑥})
2019fmpt 6524 . 2 (∀𝑥 ∈ (𝐴 × 𝐵) ran {𝑥} ∈ 𝐵 ↔ (2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵)
2113, 20mpbi 220 1 (2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵
Colors of variables: wff setvar class
Syntax hints:   = wceq 1631  wcel 2145  wral 3061  Vcvv 3351  wss 3724  {csn 4317  cop 4323   cuni 4575  cmpt 4864   × cxp 5248  ran crn 5251  cres 5252  wf 6028  2nd c2nd 7315
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4916  ax-nul 4924  ax-pr 5035
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 829  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3589  df-csb 3684  df-dif 3727  df-un 3729  df-in 3731  df-ss 3738  df-nul 4065  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-iun 4657  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5995  df-fun 6034  df-fn 6035  df-f 6036  df-fv 6040  df-2nd 7317
This theorem is referenced by:  fo2ndres  7343  2ndcof  7347  fparlem2  7430  f2ndf  7435  eucalgcvga  15508  2ndfcl  17047  gaid  17940  tx2cn  21635  txkgen  21677  xpinpreima  30293  xpinpreima2  30294  2ndmbfm  30664  filnetlem4  32714  hausgraph  38317
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