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Theorem fo2ndres 7527
Description: Onto mapping of a restriction of the 2nd (second member of an ordered pair) function. (Contributed by NM, 14-Dec-2008.)
Assertion
Ref Expression
fo2ndres (𝐴 ≠ ∅ → (2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)–onto𝐵)

Proof of Theorem fo2ndres
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 n0 4191 . . . . . . 7 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
2 opelxp 5440 . . . . . . . . . 10 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ (𝑥𝐴𝑦𝐵))
3 fvres 6516 . . . . . . . . . . . 12 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → ((2nd ↾ (𝐴 × 𝐵))‘⟨𝑥, 𝑦⟩) = (2nd ‘⟨𝑥, 𝑦⟩))
4 vex 3413 . . . . . . . . . . . . 13 𝑥 ∈ V
5 vex 3413 . . . . . . . . . . . . 13 𝑦 ∈ V
64, 5op2nd 7509 . . . . . . . . . . . 12 (2nd ‘⟨𝑥, 𝑦⟩) = 𝑦
73, 6syl6req 2826 . . . . . . . . . . 11 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → 𝑦 = ((2nd ↾ (𝐴 × 𝐵))‘⟨𝑥, 𝑦⟩))
8 f2ndres 7525 . . . . . . . . . . . . 13 (2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵
9 ffn 6342 . . . . . . . . . . . . 13 ((2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵 → (2nd ↾ (𝐴 × 𝐵)) Fn (𝐴 × 𝐵))
108, 9ax-mp 5 . . . . . . . . . . . 12 (2nd ↾ (𝐴 × 𝐵)) Fn (𝐴 × 𝐵)
11 fnfvelrn 6672 . . . . . . . . . . . 12 (((2nd ↾ (𝐴 × 𝐵)) Fn (𝐴 × 𝐵) ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)) → ((2nd ↾ (𝐴 × 𝐵))‘⟨𝑥, 𝑦⟩) ∈ ran (2nd ↾ (𝐴 × 𝐵)))
1210, 11mpan 678 . . . . . . . . . . 11 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → ((2nd ↾ (𝐴 × 𝐵))‘⟨𝑥, 𝑦⟩) ∈ ran (2nd ↾ (𝐴 × 𝐵)))
137, 12eqeltrd 2861 . . . . . . . . . 10 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → 𝑦 ∈ ran (2nd ↾ (𝐴 × 𝐵)))
142, 13sylbir 227 . . . . . . . . 9 ((𝑥𝐴𝑦𝐵) → 𝑦 ∈ ran (2nd ↾ (𝐴 × 𝐵)))
1514ex 405 . . . . . . . 8 (𝑥𝐴 → (𝑦𝐵𝑦 ∈ ran (2nd ↾ (𝐴 × 𝐵))))
1615exlimiv 1890 . . . . . . 7 (∃𝑥 𝑥𝐴 → (𝑦𝐵𝑦 ∈ ran (2nd ↾ (𝐴 × 𝐵))))
171, 16sylbi 209 . . . . . 6 (𝐴 ≠ ∅ → (𝑦𝐵𝑦 ∈ ran (2nd ↾ (𝐴 × 𝐵))))
1817ssrdv 3859 . . . . 5 (𝐴 ≠ ∅ → 𝐵 ⊆ ran (2nd ↾ (𝐴 × 𝐵)))
19 frn 6348 . . . . . 6 ((2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵 → ran (2nd ↾ (𝐴 × 𝐵)) ⊆ 𝐵)
208, 19ax-mp 5 . . . . 5 ran (2nd ↾ (𝐴 × 𝐵)) ⊆ 𝐵
2118, 20jctil 512 . . . 4 (𝐴 ≠ ∅ → (ran (2nd ↾ (𝐴 × 𝐵)) ⊆ 𝐵𝐵 ⊆ ran (2nd ↾ (𝐴 × 𝐵))))
22 eqss 3868 . . . 4 (ran (2nd ↾ (𝐴 × 𝐵)) = 𝐵 ↔ (ran (2nd ↾ (𝐴 × 𝐵)) ⊆ 𝐵𝐵 ⊆ ran (2nd ↾ (𝐴 × 𝐵))))
2321, 22sylibr 226 . . 3 (𝐴 ≠ ∅ → ran (2nd ↾ (𝐴 × 𝐵)) = 𝐵)
2423, 8jctil 512 . 2 (𝐴 ≠ ∅ → ((2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵 ∧ ran (2nd ↾ (𝐴 × 𝐵)) = 𝐵))
25 dffo2 6421 . 2 ((2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)–onto𝐵 ↔ ((2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵 ∧ ran (2nd ↾ (𝐴 × 𝐵)) = 𝐵))
2624, 25sylibr 226 1 (𝐴 ≠ ∅ → (2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)–onto𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 387   = wceq 1508  wex 1743  wcel 2051  wne 2962  wss 3824  c0 4173  cop 4442   × cxp 5402  ran crn 5405  cres 5406   Fn wfn 6181  wf 6182  ontowfo 6184  cfv 6186  2nd c2nd 7499
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2745  ax-sep 5057  ax-nul 5064  ax-pow 5116  ax-pr 5183  ax-un 7278
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2754  df-cleq 2766  df-clel 2841  df-nfc 2913  df-ne 2963  df-ral 3088  df-rex 3089  df-rab 3092  df-v 3412  df-sbc 3677  df-csb 3782  df-dif 3827  df-un 3829  df-in 3831  df-ss 3838  df-nul 4174  df-if 4346  df-sn 4437  df-pr 4439  df-op 4443  df-uni 4710  df-iun 4791  df-br 4927  df-opab 4989  df-mpt 5006  df-id 5309  df-xp 5410  df-rel 5411  df-cnv 5412  df-co 5413  df-dm 5414  df-rn 5415  df-res 5416  df-ima 5417  df-iota 6150  df-fun 6188  df-fn 6189  df-f 6190  df-fo 6192  df-fv 6194  df-2nd 7501
This theorem is referenced by:  2ndconst  7603  txcmpb  21972
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