ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  fo2ndresm GIF version

Theorem fo2ndresm 5868
Description: Onto mapping of a restriction of the 2nd (second member of an ordered pair) function. (Contributed by Jim Kingdon, 24-Jan-2019.)
Assertion
Ref Expression
fo2ndresm (∃𝑥 𝑥𝐴 → (2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)–onto𝐵)
Distinct variable group:   𝑥,𝐴
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem fo2ndresm
Dummy variables 𝑣 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq1 2145 . . 3 (𝑢 = 𝑥 → (𝑢𝐴𝑥𝐴))
21cbvexv 1838 . 2 (∃𝑢 𝑢𝐴 ↔ ∃𝑥 𝑥𝐴)
3 opelxp 4430 . . . . . . . . . 10 (⟨𝑢, 𝑣⟩ ∈ (𝐴 × 𝐵) ↔ (𝑢𝐴𝑣𝐵))
4 fvres 5274 . . . . . . . . . . . 12 (⟨𝑢, 𝑣⟩ ∈ (𝐴 × 𝐵) → ((2nd ↾ (𝐴 × 𝐵))‘⟨𝑢, 𝑣⟩) = (2nd ‘⟨𝑢, 𝑣⟩))
5 vex 2615 . . . . . . . . . . . . 13 𝑢 ∈ V
6 vex 2615 . . . . . . . . . . . . 13 𝑣 ∈ V
75, 6op2nd 5853 . . . . . . . . . . . 12 (2nd ‘⟨𝑢, 𝑣⟩) = 𝑣
84, 7syl6req 2132 . . . . . . . . . . 11 (⟨𝑢, 𝑣⟩ ∈ (𝐴 × 𝐵) → 𝑣 = ((2nd ↾ (𝐴 × 𝐵))‘⟨𝑢, 𝑣⟩))
9 f2ndres 5866 . . . . . . . . . . . . 13 (2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵
10 ffn 5114 . . . . . . . . . . . . 13 ((2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵 → (2nd ↾ (𝐴 × 𝐵)) Fn (𝐴 × 𝐵))
119, 10ax-mp 7 . . . . . . . . . . . 12 (2nd ↾ (𝐴 × 𝐵)) Fn (𝐴 × 𝐵)
12 fnfvelrn 5376 . . . . . . . . . . . 12 (((2nd ↾ (𝐴 × 𝐵)) Fn (𝐴 × 𝐵) ∧ ⟨𝑢, 𝑣⟩ ∈ (𝐴 × 𝐵)) → ((2nd ↾ (𝐴 × 𝐵))‘⟨𝑢, 𝑣⟩) ∈ ran (2nd ↾ (𝐴 × 𝐵)))
1311, 12mpan 415 . . . . . . . . . . 11 (⟨𝑢, 𝑣⟩ ∈ (𝐴 × 𝐵) → ((2nd ↾ (𝐴 × 𝐵))‘⟨𝑢, 𝑣⟩) ∈ ran (2nd ↾ (𝐴 × 𝐵)))
148, 13eqeltrd 2159 . . . . . . . . . 10 (⟨𝑢, 𝑣⟩ ∈ (𝐴 × 𝐵) → 𝑣 ∈ ran (2nd ↾ (𝐴 × 𝐵)))
153, 14sylbir 133 . . . . . . . . 9 ((𝑢𝐴𝑣𝐵) → 𝑣 ∈ ran (2nd ↾ (𝐴 × 𝐵)))
1615ex 113 . . . . . . . 8 (𝑢𝐴 → (𝑣𝐵𝑣 ∈ ran (2nd ↾ (𝐴 × 𝐵))))
1716exlimiv 1530 . . . . . . 7 (∃𝑢 𝑢𝐴 → (𝑣𝐵𝑣 ∈ ran (2nd ↾ (𝐴 × 𝐵))))
1817ssrdv 3016 . . . . . 6 (∃𝑢 𝑢𝐴𝐵 ⊆ ran (2nd ↾ (𝐴 × 𝐵)))
19 frn 5121 . . . . . . 7 ((2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵 → ran (2nd ↾ (𝐴 × 𝐵)) ⊆ 𝐵)
209, 19ax-mp 7 . . . . . 6 ran (2nd ↾ (𝐴 × 𝐵)) ⊆ 𝐵
2118, 20jctil 305 . . . . 5 (∃𝑢 𝑢𝐴 → (ran (2nd ↾ (𝐴 × 𝐵)) ⊆ 𝐵𝐵 ⊆ ran (2nd ↾ (𝐴 × 𝐵))))
22 eqss 3025 . . . . 5 (ran (2nd ↾ (𝐴 × 𝐵)) = 𝐵 ↔ (ran (2nd ↾ (𝐴 × 𝐵)) ⊆ 𝐵𝐵 ⊆ ran (2nd ↾ (𝐴 × 𝐵))))
2321, 22sylibr 132 . . . 4 (∃𝑢 𝑢𝐴 → ran (2nd ↾ (𝐴 × 𝐵)) = 𝐵)
2423, 9jctil 305 . . 3 (∃𝑢 𝑢𝐴 → ((2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵 ∧ ran (2nd ↾ (𝐴 × 𝐵)) = 𝐵))
25 dffo2 5185 . . 3 ((2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)–onto𝐵 ↔ ((2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵 ∧ ran (2nd ↾ (𝐴 × 𝐵)) = 𝐵))
2624, 25sylibr 132 . 2 (∃𝑢 𝑢𝐴 → (2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)–onto𝐵)
272, 26sylbir 133 1 (∃𝑥 𝑥𝐴 → (2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)–onto𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1285  wex 1422  wcel 1434  wss 2984  cop 3425   × cxp 4399  ran crn 4402  cres 4403   Fn wfn 4964  wf 4965  ontowfo 4967  cfv 4969  2nd c2nd 5845
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3922  ax-pow 3974  ax-pr 4000  ax-un 4224
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-rab 2362  df-v 2614  df-sbc 2827  df-csb 2920  df-un 2988  df-in 2990  df-ss 2997  df-pw 3408  df-sn 3428  df-pr 3429  df-op 3431  df-uni 3628  df-iun 3706  df-br 3812  df-opab 3866  df-mpt 3867  df-id 4084  df-xp 4407  df-rel 4408  df-cnv 4409  df-co 4410  df-dm 4411  df-rn 4412  df-res 4413  df-ima 4414  df-iota 4934  df-fun 4971  df-fn 4972  df-f 4973  df-fo 4975  df-fv 4977  df-2nd 5847
This theorem is referenced by:  2ndconst  5922
  Copyright terms: Public domain W3C validator