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Theorem fo2ndresm 6067
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 2203 . . 3 (𝑢 = 𝑥 → (𝑢𝐴𝑥𝐴))
21cbvexv 1891 . 2 (∃𝑢 𝑢𝐴 ↔ ∃𝑥 𝑥𝐴)
3 opelxp 4576 . . . . . . . . . 10 (⟨𝑢, 𝑣⟩ ∈ (𝐴 × 𝐵) ↔ (𝑢𝐴𝑣𝐵))
4 fvres 5452 . . . . . . . . . . . 12 (⟨𝑢, 𝑣⟩ ∈ (𝐴 × 𝐵) → ((2nd ↾ (𝐴 × 𝐵))‘⟨𝑢, 𝑣⟩) = (2nd ‘⟨𝑢, 𝑣⟩))
5 vex 2692 . . . . . . . . . . . . 13 𝑢 ∈ V
6 vex 2692 . . . . . . . . . . . . 13 𝑣 ∈ V
75, 6op2nd 6052 . . . . . . . . . . . 12 (2nd ‘⟨𝑢, 𝑣⟩) = 𝑣
84, 7eqtr2di 2190 . . . . . . . . . . 11 (⟨𝑢, 𝑣⟩ ∈ (𝐴 × 𝐵) → 𝑣 = ((2nd ↾ (𝐴 × 𝐵))‘⟨𝑢, 𝑣⟩))
9 f2ndres 6065 . . . . . . . . . . . . 13 (2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵
10 ffn 5279 . . . . . . . . . . . . 13 ((2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵 → (2nd ↾ (𝐴 × 𝐵)) Fn (𝐴 × 𝐵))
119, 10ax-mp 5 . . . . . . . . . . . 12 (2nd ↾ (𝐴 × 𝐵)) Fn (𝐴 × 𝐵)
12 fnfvelrn 5559 . . . . . . . . . . . 12 (((2nd ↾ (𝐴 × 𝐵)) Fn (𝐴 × 𝐵) ∧ ⟨𝑢, 𝑣⟩ ∈ (𝐴 × 𝐵)) → ((2nd ↾ (𝐴 × 𝐵))‘⟨𝑢, 𝑣⟩) ∈ ran (2nd ↾ (𝐴 × 𝐵)))
1311, 12mpan 421 . . . . . . . . . . 11 (⟨𝑢, 𝑣⟩ ∈ (𝐴 × 𝐵) → ((2nd ↾ (𝐴 × 𝐵))‘⟨𝑢, 𝑣⟩) ∈ ran (2nd ↾ (𝐴 × 𝐵)))
148, 13eqeltrd 2217 . . . . . . . . . 10 (⟨𝑢, 𝑣⟩ ∈ (𝐴 × 𝐵) → 𝑣 ∈ ran (2nd ↾ (𝐴 × 𝐵)))
153, 14sylbir 134 . . . . . . . . 9 ((𝑢𝐴𝑣𝐵) → 𝑣 ∈ ran (2nd ↾ (𝐴 × 𝐵)))
1615ex 114 . . . . . . . 8 (𝑢𝐴 → (𝑣𝐵𝑣 ∈ ran (2nd ↾ (𝐴 × 𝐵))))
1716exlimiv 1578 . . . . . . 7 (∃𝑢 𝑢𝐴 → (𝑣𝐵𝑣 ∈ ran (2nd ↾ (𝐴 × 𝐵))))
1817ssrdv 3107 . . . . . 6 (∃𝑢 𝑢𝐴𝐵 ⊆ ran (2nd ↾ (𝐴 × 𝐵)))
19 frn 5288 . . . . . . 7 ((2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵 → ran (2nd ↾ (𝐴 × 𝐵)) ⊆ 𝐵)
209, 19ax-mp 5 . . . . . 6 ran (2nd ↾ (𝐴 × 𝐵)) ⊆ 𝐵
2118, 20jctil 310 . . . . 5 (∃𝑢 𝑢𝐴 → (ran (2nd ↾ (𝐴 × 𝐵)) ⊆ 𝐵𝐵 ⊆ ran (2nd ↾ (𝐴 × 𝐵))))
22 eqss 3116 . . . . 5 (ran (2nd ↾ (𝐴 × 𝐵)) = 𝐵 ↔ (ran (2nd ↾ (𝐴 × 𝐵)) ⊆ 𝐵𝐵 ⊆ ran (2nd ↾ (𝐴 × 𝐵))))
2321, 22sylibr 133 . . . 4 (∃𝑢 𝑢𝐴 → ran (2nd ↾ (𝐴 × 𝐵)) = 𝐵)
2423, 9jctil 310 . . 3 (∃𝑢 𝑢𝐴 → ((2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵 ∧ ran (2nd ↾ (𝐴 × 𝐵)) = 𝐵))
25 dffo2 5356 . . 3 ((2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)–onto𝐵 ↔ ((2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐵 ∧ ran (2nd ↾ (𝐴 × 𝐵)) = 𝐵))
2624, 25sylibr 133 . 2 (∃𝑢 𝑢𝐴 → (2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)–onto𝐵)
272, 26sylbir 134 1 (∃𝑥 𝑥𝐴 → (2nd ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)–onto𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103   = wceq 1332  wex 1469  wcel 1481  wss 3075  cop 3534   × cxp 4544  ran crn 4547  cres 4548   Fn wfn 5125  wf 5126  ontowfo 5128  cfv 5130  2nd c2nd 6044
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138  ax-un 4362
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-sbc 2913  df-csb 3007  df-un 3079  df-in 3081  df-ss 3088  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-iun 3822  df-br 3937  df-opab 3997  df-mpt 3998  df-id 4222  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-fo 5136  df-fv 5138  df-2nd 6046
This theorem is referenced by:  2ndconst  6126
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