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Theorem fo2ndres 7705
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 4307 . . . . . . 7 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
2 opelxp 5584 . . . . . . . . . 10 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ (𝑥𝐴𝑦𝐵))
3 fvres 6682 . . . . . . . . . . . 12 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → ((2nd ↾ (𝐴 × 𝐵))‘⟨𝑥, 𝑦⟩) = (2nd ‘⟨𝑥, 𝑦⟩))
4 vex 3495 . . . . . . . . . . . . 13 𝑥 ∈ V
5 vex 3495 . . . . . . . . . . . . 13 𝑦 ∈ V
64, 5op2nd 7687 . . . . . . . . . . . 12 (2nd ‘⟨𝑥, 𝑦⟩) = 𝑦
73, 6syl6req 2870 . . . . . . . . . . 11 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → 𝑦 = ((2nd ↾ (𝐴 × 𝐵))‘⟨𝑥, 𝑦⟩))
8 f2ndres 7703 . . . . . . . . . . . . 13 (2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵
9 ffn 6507 . . . . . . . . . . . . 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 6840 . . . . . . . . . . . 12 (((2nd ↾ (𝐴 × 𝐵)) Fn (𝐴 × 𝐵) ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)) → ((2nd ↾ (𝐴 × 𝐵))‘⟨𝑥, 𝑦⟩) ∈ ran (2nd ↾ (𝐴 × 𝐵)))
1210, 11mpan 686 . . . . . . . . . . 11 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → ((2nd ↾ (𝐴 × 𝐵))‘⟨𝑥, 𝑦⟩) ∈ ran (2nd ↾ (𝐴 × 𝐵)))
137, 12eqeltrd 2910 . . . . . . . . . 10 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → 𝑦 ∈ ran (2nd ↾ (𝐴 × 𝐵)))
142, 13sylbir 236 . . . . . . . . 9 ((𝑥𝐴𝑦𝐵) → 𝑦 ∈ ran (2nd ↾ (𝐴 × 𝐵)))
1514ex 413 . . . . . . . 8 (𝑥𝐴 → (𝑦𝐵𝑦 ∈ ran (2nd ↾ (𝐴 × 𝐵))))
1615exlimiv 1922 . . . . . . 7 (∃𝑥 𝑥𝐴 → (𝑦𝐵𝑦 ∈ ran (2nd ↾ (𝐴 × 𝐵))))
171, 16sylbi 218 . . . . . 6 (𝐴 ≠ ∅ → (𝑦𝐵𝑦 ∈ ran (2nd ↾ (𝐴 × 𝐵))))
1817ssrdv 3970 . . . . 5 (𝐴 ≠ ∅ → 𝐵 ⊆ ran (2nd ↾ (𝐴 × 𝐵)))
19 frn 6513 . . . . . 6 ((2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵 → ran (2nd ↾ (𝐴 × 𝐵)) ⊆ 𝐵)
208, 19ax-mp 5 . . . . 5 ran (2nd ↾ (𝐴 × 𝐵)) ⊆ 𝐵
2118, 20jctil 520 . . . 4 (𝐴 ≠ ∅ → (ran (2nd ↾ (𝐴 × 𝐵)) ⊆ 𝐵𝐵 ⊆ ran (2nd ↾ (𝐴 × 𝐵))))
22 eqss 3979 . . . 4 (ran (2nd ↾ (𝐴 × 𝐵)) = 𝐵 ↔ (ran (2nd ↾ (𝐴 × 𝐵)) ⊆ 𝐵𝐵 ⊆ ran (2nd ↾ (𝐴 × 𝐵))))
2321, 22sylibr 235 . . 3 (𝐴 ≠ ∅ → ran (2nd ↾ (𝐴 × 𝐵)) = 𝐵)
2423, 8jctil 520 . 2 (𝐴 ≠ ∅ → ((2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵 ∧ ran (2nd ↾ (𝐴 × 𝐵)) = 𝐵))
25 dffo2 6587 . 2 ((2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)–onto𝐵 ↔ ((2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵 ∧ ran (2nd ↾ (𝐴 × 𝐵)) = 𝐵))
2624, 25sylibr 235 1 (𝐴 ≠ ∅ → (2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)–onto𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396   = wceq 1528  wex 1771  wcel 2105  wne 3013  wss 3933  c0 4288  cop 4563   × cxp 5546  ran crn 5549  cres 5550   Fn wfn 6343  wf 6344  ontowfo 6346  cfv 6348  2nd c2nd 7677
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fo 6354  df-fv 6356  df-2nd 7679
This theorem is referenced by:  2ndconst  7785  txcmpb  22180
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