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Theorem fo2ndresm 5816
 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 2116 . . 3 (𝑢 = 𝑥 → (𝑢𝐴𝑥𝐴))
21cbvexv 1811 . 2 (∃𝑢 𝑢𝐴 ↔ ∃𝑥 𝑥𝐴)
3 opelxp 4401 . . . . . . . . . 10 (⟨𝑢, 𝑣⟩ ∈ (𝐴 × 𝐵) ↔ (𝑢𝐴𝑣𝐵))
4 fvres 5225 . . . . . . . . . . . 12 (⟨𝑢, 𝑣⟩ ∈ (𝐴 × 𝐵) → ((2nd ↾ (𝐴 × 𝐵))‘⟨𝑢, 𝑣⟩) = (2nd ‘⟨𝑢, 𝑣⟩))
5 vex 2577 . . . . . . . . . . . . 13 𝑢 ∈ V
6 vex 2577 . . . . . . . . . . . . 13 𝑣 ∈ V
75, 6op2nd 5801 . . . . . . . . . . . 12 (2nd ‘⟨𝑢, 𝑣⟩) = 𝑣
84, 7syl6req 2105 . . . . . . . . . . 11 (⟨𝑢, 𝑣⟩ ∈ (𝐴 × 𝐵) → 𝑣 = ((2nd ↾ (𝐴 × 𝐵))‘⟨𝑢, 𝑣⟩))
9 f2ndres 5814 . . . . . . . . . . . . 13 (2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵
10 ffn 5073 . . . . . . . . . . . . 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 5326 . . . . . . . . . . . 12 (((2nd ↾ (𝐴 × 𝐵)) Fn (𝐴 × 𝐵) ∧ ⟨𝑢, 𝑣⟩ ∈ (𝐴 × 𝐵)) → ((2nd ↾ (𝐴 × 𝐵))‘⟨𝑢, 𝑣⟩) ∈ ran (2nd ↾ (𝐴 × 𝐵)))
1311, 12mpan 408 . . . . . . . . . . 11 (⟨𝑢, 𝑣⟩ ∈ (𝐴 × 𝐵) → ((2nd ↾ (𝐴 × 𝐵))‘⟨𝑢, 𝑣⟩) ∈ ran (2nd ↾ (𝐴 × 𝐵)))
148, 13eqeltrd 2130 . . . . . . . . . 10 (⟨𝑢, 𝑣⟩ ∈ (𝐴 × 𝐵) → 𝑣 ∈ ran (2nd ↾ (𝐴 × 𝐵)))
153, 14sylbir 129 . . . . . . . . 9 ((𝑢𝐴𝑣𝐵) → 𝑣 ∈ ran (2nd ↾ (𝐴 × 𝐵)))
1615ex 112 . . . . . . . 8 (𝑢𝐴 → (𝑣𝐵𝑣 ∈ ran (2nd ↾ (𝐴 × 𝐵))))
1716exlimiv 1505 . . . . . . 7 (∃𝑢 𝑢𝐴 → (𝑣𝐵𝑣 ∈ ran (2nd ↾ (𝐴 × 𝐵))))
1817ssrdv 2978 . . . . . 6 (∃𝑢 𝑢𝐴𝐵 ⊆ ran (2nd ↾ (𝐴 × 𝐵)))
19 frn 5079 . . . . . . 7 ((2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵 → ran (2nd ↾ (𝐴 × 𝐵)) ⊆ 𝐵)
209, 19ax-mp 7 . . . . . 6 ran (2nd ↾ (𝐴 × 𝐵)) ⊆ 𝐵
2118, 20jctil 299 . . . . 5 (∃𝑢 𝑢𝐴 → (ran (2nd ↾ (𝐴 × 𝐵)) ⊆ 𝐵𝐵 ⊆ ran (2nd ↾ (𝐴 × 𝐵))))
22 eqss 2987 . . . . 5 (ran (2nd ↾ (𝐴 × 𝐵)) = 𝐵 ↔ (ran (2nd ↾ (𝐴 × 𝐵)) ⊆ 𝐵𝐵 ⊆ ran (2nd ↾ (𝐴 × 𝐵))))
2321, 22sylibr 141 . . . 4 (∃𝑢 𝑢𝐴 → ran (2nd ↾ (𝐴 × 𝐵)) = 𝐵)
2423, 9jctil 299 . . 3 (∃𝑢 𝑢𝐴 → ((2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵 ∧ ran (2nd ↾ (𝐴 × 𝐵)) = 𝐵))
25 dffo2 5137 . . 3 ((2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)–onto𝐵 ↔ ((2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵 ∧ ran (2nd ↾ (𝐴 × 𝐵)) = 𝐵))
2624, 25sylibr 141 . 2 (∃𝑢 𝑢𝐴 → (2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)–onto𝐵)
272, 26sylbir 129 1 (∃𝑥 𝑥𝐴 → (2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)–onto𝐵)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 101   = wceq 1259  ∃wex 1397   ∈ wcel 1409   ⊆ wss 2944  ⟨cop 3405   × cxp 4370  ran crn 4373   ↾ cres 4374   Fn wfn 4924  ⟶wf 4925  –onto→wfo 4927  ‘cfv 4929  2nd c2nd 5793 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971  ax-un 4197 This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-rab 2332  df-v 2576  df-sbc 2787  df-csb 2880  df-un 2949  df-in 2951  df-ss 2958  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-uni 3608  df-iun 3686  df-br 3792  df-opab 3846  df-mpt 3847  df-id 4057  df-xp 4378  df-rel 4379  df-cnv 4380  df-co 4381  df-dm 4382  df-rn 4383  df-res 4384  df-ima 4385  df-iota 4894  df-fun 4931  df-fn 4932  df-f 4933  df-fo 4935  df-fv 4937  df-2nd 5795 This theorem is referenced by:  2ndconst  5870
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