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Theorem fo2ndres 7342
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 4078 . . . . . . 7 (𝐴 ≠ ∅ ↔ ∃𝑥 𝑥𝐴)
2 opelxp 5286 . . . . . . . . . 10 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ (𝑥𝐴𝑦𝐵))
3 fvres 6348 . . . . . . . . . . . 12 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → ((2nd ↾ (𝐴 × 𝐵))‘⟨𝑥, 𝑦⟩) = (2nd ‘⟨𝑥, 𝑦⟩))
4 vex 3354 . . . . . . . . . . . . 13 𝑥 ∈ V
5 vex 3354 . . . . . . . . . . . . 13 𝑦 ∈ V
64, 5op2nd 7324 . . . . . . . . . . . 12 (2nd ‘⟨𝑥, 𝑦⟩) = 𝑦
73, 6syl6req 2822 . . . . . . . . . . 11 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → 𝑦 = ((2nd ↾ (𝐴 × 𝐵))‘⟨𝑥, 𝑦⟩))
8 f2ndres 7340 . . . . . . . . . . . . 13 (2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵
9 ffn 6185 . . . . . . . . . . . . 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 6499 . . . . . . . . . . . 12 (((2nd ↾ (𝐴 × 𝐵)) Fn (𝐴 × 𝐵) ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)) → ((2nd ↾ (𝐴 × 𝐵))‘⟨𝑥, 𝑦⟩) ∈ ran (2nd ↾ (𝐴 × 𝐵)))
1210, 11mpan 670 . . . . . . . . . . 11 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → ((2nd ↾ (𝐴 × 𝐵))‘⟨𝑥, 𝑦⟩) ∈ ran (2nd ↾ (𝐴 × 𝐵)))
137, 12eqeltrd 2850 . . . . . . . . . 10 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → 𝑦 ∈ ran (2nd ↾ (𝐴 × 𝐵)))
142, 13sylbir 225 . . . . . . . . 9 ((𝑥𝐴𝑦𝐵) → 𝑦 ∈ ran (2nd ↾ (𝐴 × 𝐵)))
1514ex 397 . . . . . . . 8 (𝑥𝐴 → (𝑦𝐵𝑦 ∈ ran (2nd ↾ (𝐴 × 𝐵))))
1615exlimiv 2010 . . . . . . 7 (∃𝑥 𝑥𝐴 → (𝑦𝐵𝑦 ∈ ran (2nd ↾ (𝐴 × 𝐵))))
171, 16sylbi 207 . . . . . 6 (𝐴 ≠ ∅ → (𝑦𝐵𝑦 ∈ ran (2nd ↾ (𝐴 × 𝐵))))
1817ssrdv 3758 . . . . 5 (𝐴 ≠ ∅ → 𝐵 ⊆ ran (2nd ↾ (𝐴 × 𝐵)))
19 frn 6193 . . . . . 6 ((2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵 → ran (2nd ↾ (𝐴 × 𝐵)) ⊆ 𝐵)
208, 19ax-mp 5 . . . . 5 ran (2nd ↾ (𝐴 × 𝐵)) ⊆ 𝐵
2118, 20jctil 509 . . . 4 (𝐴 ≠ ∅ → (ran (2nd ↾ (𝐴 × 𝐵)) ⊆ 𝐵𝐵 ⊆ ran (2nd ↾ (𝐴 × 𝐵))))
22 eqss 3767 . . . 4 (ran (2nd ↾ (𝐴 × 𝐵)) = 𝐵 ↔ (ran (2nd ↾ (𝐴 × 𝐵)) ⊆ 𝐵𝐵 ⊆ ran (2nd ↾ (𝐴 × 𝐵))))
2321, 22sylibr 224 . . 3 (𝐴 ≠ ∅ → ran (2nd ↾ (𝐴 × 𝐵)) = 𝐵)
2423, 8jctil 509 . 2 (𝐴 ≠ ∅ → ((2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵 ∧ ran (2nd ↾ (𝐴 × 𝐵)) = 𝐵))
25 dffo2 6260 . 2 ((2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)–onto𝐵 ↔ ((2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵 ∧ ran (2nd ↾ (𝐴 × 𝐵)) = 𝐵))
2624, 25sylibr 224 1 (𝐴 ≠ ∅ → (2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)–onto𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wex 1852  wcel 2145  wne 2943  wss 3723  c0 4063  cop 4322   × cxp 5247  ran crn 5250  cres 5251   Fn wfn 6026  wf 6027  ontowfo 6029  cfv 6031  2nd c2nd 7314
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-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  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 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fo 6037  df-fv 6039  df-2nd 7316
This theorem is referenced by:  2ndconst  7417  txcmpb  21668
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