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Theorem fo1stresm 5819
Description: Onto mapping of a restriction of the 1st (first member of an ordered pair) function. (Contributed by Jim Kingdon, 24-Jan-2019.)
Assertion
Ref Expression
fo1stresm (∃𝑦 𝑦𝐵 → (1st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)–onto𝐴)
Distinct variable group:   𝑦,𝐵
Allowed substitution hint:   𝐴(𝑦)

Proof of Theorem fo1stresm
Dummy variables 𝑣 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq1 2142 . . 3 (𝑣 = 𝑦 → (𝑣𝐵𝑦𝐵))
21cbvexv 1837 . 2 (∃𝑣 𝑣𝐵 ↔ ∃𝑦 𝑦𝐵)
3 opelxp 4400 . . . . . . . . . 10 (⟨𝑢, 𝑣⟩ ∈ (𝐴 × 𝐵) ↔ (𝑢𝐴𝑣𝐵))
4 fvres 5230 . . . . . . . . . . . 12 (⟨𝑢, 𝑣⟩ ∈ (𝐴 × 𝐵) → ((1st ↾ (𝐴 × 𝐵))‘⟨𝑢, 𝑣⟩) = (1st ‘⟨𝑢, 𝑣⟩))
5 vex 2605 . . . . . . . . . . . . 13 𝑢 ∈ V
6 vex 2605 . . . . . . . . . . . . 13 𝑣 ∈ V
75, 6op1st 5804 . . . . . . . . . . . 12 (1st ‘⟨𝑢, 𝑣⟩) = 𝑢
84, 7syl6req 2131 . . . . . . . . . . 11 (⟨𝑢, 𝑣⟩ ∈ (𝐴 × 𝐵) → 𝑢 = ((1st ↾ (𝐴 × 𝐵))‘⟨𝑢, 𝑣⟩))
9 f1stres 5817 . . . . . . . . . . . . 13 (1st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐴
10 ffn 5077 . . . . . . . . . . . . 13 ((1st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐴 → (1st ↾ (𝐴 × 𝐵)) Fn (𝐴 × 𝐵))
119, 10ax-mp 7 . . . . . . . . . . . 12 (1st ↾ (𝐴 × 𝐵)) Fn (𝐴 × 𝐵)
12 fnfvelrn 5331 . . . . . . . . . . . 12 (((1st ↾ (𝐴 × 𝐵)) Fn (𝐴 × 𝐵) ∧ ⟨𝑢, 𝑣⟩ ∈ (𝐴 × 𝐵)) → ((1st ↾ (𝐴 × 𝐵))‘⟨𝑢, 𝑣⟩) ∈ ran (1st ↾ (𝐴 × 𝐵)))
1311, 12mpan 415 . . . . . . . . . . 11 (⟨𝑢, 𝑣⟩ ∈ (𝐴 × 𝐵) → ((1st ↾ (𝐴 × 𝐵))‘⟨𝑢, 𝑣⟩) ∈ ran (1st ↾ (𝐴 × 𝐵)))
148, 13eqeltrd 2156 . . . . . . . . . 10 (⟨𝑢, 𝑣⟩ ∈ (𝐴 × 𝐵) → 𝑢 ∈ ran (1st ↾ (𝐴 × 𝐵)))
153, 14sylbir 133 . . . . . . . . 9 ((𝑢𝐴𝑣𝐵) → 𝑢 ∈ ran (1st ↾ (𝐴 × 𝐵)))
1615expcom 114 . . . . . . . 8 (𝑣𝐵 → (𝑢𝐴𝑢 ∈ ran (1st ↾ (𝐴 × 𝐵))))
1716exlimiv 1530 . . . . . . 7 (∃𝑣 𝑣𝐵 → (𝑢𝐴𝑢 ∈ ran (1st ↾ (𝐴 × 𝐵))))
1817ssrdv 3006 . . . . . 6 (∃𝑣 𝑣𝐵𝐴 ⊆ ran (1st ↾ (𝐴 × 𝐵)))
19 frn 5083 . . . . . . 7 ((1st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐴 → ran (1st ↾ (𝐴 × 𝐵)) ⊆ 𝐴)
209, 19ax-mp 7 . . . . . 6 ran (1st ↾ (𝐴 × 𝐵)) ⊆ 𝐴
2118, 20jctil 305 . . . . 5 (∃𝑣 𝑣𝐵 → (ran (1st ↾ (𝐴 × 𝐵)) ⊆ 𝐴𝐴 ⊆ ran (1st ↾ (𝐴 × 𝐵))))
22 eqss 3015 . . . . 5 (ran (1st ↾ (𝐴 × 𝐵)) = 𝐴 ↔ (ran (1st ↾ (𝐴 × 𝐵)) ⊆ 𝐴𝐴 ⊆ ran (1st ↾ (𝐴 × 𝐵))))
2321, 22sylibr 132 . . . 4 (∃𝑣 𝑣𝐵 → ran (1st ↾ (𝐴 × 𝐵)) = 𝐴)
2423, 9jctil 305 . . 3 (∃𝑣 𝑣𝐵 → ((1st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐴 ∧ ran (1st ↾ (𝐴 × 𝐵)) = 𝐴))
25 dffo2 5141 . . 3 ((1st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)–onto𝐴 ↔ ((1st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐴 ∧ ran (1st ↾ (𝐴 × 𝐵)) = 𝐴))
2624, 25sylibr 132 . 2 (∃𝑣 𝑣𝐵 → (1st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)–onto𝐴)
272, 26sylbir 133 1 (∃𝑦 𝑦𝐵 → (1st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)–onto𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1285  wex 1422  wcel 1434  wss 2974  cop 3409   × cxp 4369  ran crn 4372  cres 4373   Fn wfn 4927  wf 4928  ontowfo 4930  cfv 4932  1st c1st 5796
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 2064  ax-sep 3904  ax-pow 3956  ax-pr 3972  ax-un 4196
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-un 2978  df-in 2980  df-ss 2987  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-id 4056  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-fo 4938  df-fv 4940  df-1st 5798
This theorem is referenced by:  1stconst  5873
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