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Theorem fo1stres 7236
Description: Onto mapping of a restriction of the 1st (first member of an ordered pair) function. (Contributed by NM, 14-Dec-2008.)
Assertion
Ref Expression
fo1stres (𝐵 ≠ ∅ → (1st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)–onto𝐴)

Proof of Theorem fo1stres
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 n0 3964 . . . . . . 7 (𝐵 ≠ ∅ ↔ ∃𝑦 𝑦𝐵)
2 opelxp 5180 . . . . . . . . . 10 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ (𝑥𝐴𝑦𝐵))
3 fvres 6245 . . . . . . . . . . . 12 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → ((1st ↾ (𝐴 × 𝐵))‘⟨𝑥, 𝑦⟩) = (1st ‘⟨𝑥, 𝑦⟩))
4 vex 3234 . . . . . . . . . . . . 13 𝑥 ∈ V
5 vex 3234 . . . . . . . . . . . . 13 𝑦 ∈ V
64, 5op1st 7218 . . . . . . . . . . . 12 (1st ‘⟨𝑥, 𝑦⟩) = 𝑥
73, 6syl6req 2702 . . . . . . . . . . 11 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → 𝑥 = ((1st ↾ (𝐴 × 𝐵))‘⟨𝑥, 𝑦⟩))
8 f1stres 7234 . . . . . . . . . . . . 13 (1st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐴
9 ffn 6083 . . . . . . . . . . . . 13 ((1st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐴 → (1st ↾ (𝐴 × 𝐵)) Fn (𝐴 × 𝐵))
108, 9ax-mp 5 . . . . . . . . . . . 12 (1st ↾ (𝐴 × 𝐵)) Fn (𝐴 × 𝐵)
11 fnfvelrn 6396 . . . . . . . . . . . 12 (((1st ↾ (𝐴 × 𝐵)) Fn (𝐴 × 𝐵) ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)) → ((1st ↾ (𝐴 × 𝐵))‘⟨𝑥, 𝑦⟩) ∈ ran (1st ↾ (𝐴 × 𝐵)))
1210, 11mpan 706 . . . . . . . . . . 11 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → ((1st ↾ (𝐴 × 𝐵))‘⟨𝑥, 𝑦⟩) ∈ ran (1st ↾ (𝐴 × 𝐵)))
137, 12eqeltrd 2730 . . . . . . . . . 10 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → 𝑥 ∈ ran (1st ↾ (𝐴 × 𝐵)))
142, 13sylbir 225 . . . . . . . . 9 ((𝑥𝐴𝑦𝐵) → 𝑥 ∈ ran (1st ↾ (𝐴 × 𝐵)))
1514expcom 450 . . . . . . . 8 (𝑦𝐵 → (𝑥𝐴𝑥 ∈ ran (1st ↾ (𝐴 × 𝐵))))
1615exlimiv 1898 . . . . . . 7 (∃𝑦 𝑦𝐵 → (𝑥𝐴𝑥 ∈ ran (1st ↾ (𝐴 × 𝐵))))
171, 16sylbi 207 . . . . . 6 (𝐵 ≠ ∅ → (𝑥𝐴𝑥 ∈ ran (1st ↾ (𝐴 × 𝐵))))
1817ssrdv 3642 . . . . 5 (𝐵 ≠ ∅ → 𝐴 ⊆ ran (1st ↾ (𝐴 × 𝐵)))
19 frn 6091 . . . . . 6 ((1st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐴 → ran (1st ↾ (𝐴 × 𝐵)) ⊆ 𝐴)
208, 19ax-mp 5 . . . . 5 ran (1st ↾ (𝐴 × 𝐵)) ⊆ 𝐴
2118, 20jctil 559 . . . 4 (𝐵 ≠ ∅ → (ran (1st ↾ (𝐴 × 𝐵)) ⊆ 𝐴𝐴 ⊆ ran (1st ↾ (𝐴 × 𝐵))))
22 eqss 3651 . . . 4 (ran (1st ↾ (𝐴 × 𝐵)) = 𝐴 ↔ (ran (1st ↾ (𝐴 × 𝐵)) ⊆ 𝐴𝐴 ⊆ ran (1st ↾ (𝐴 × 𝐵))))
2321, 22sylibr 224 . . 3 (𝐵 ≠ ∅ → ran (1st ↾ (𝐴 × 𝐵)) = 𝐴)
2423, 8jctil 559 . 2 (𝐵 ≠ ∅ → ((1st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐴 ∧ ran (1st ↾ (𝐴 × 𝐵)) = 𝐴))
25 dffo2 6157 . 2 ((1st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)–onto𝐴 ↔ ((1st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐴 ∧ ran (1st ↾ (𝐴 × 𝐵)) = 𝐴))
2624, 25sylibr 224 1 (𝐵 ≠ ∅ → (1st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)–onto𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383   = wceq 1523  wex 1744  wcel 2030  wne 2823  wss 3607  c0 3948  cop 4216   × cxp 5141  ran crn 5144  cres 5145   Fn wfn 5921  wf 5922  ontowfo 5924  cfv 5926  1st c1st 7208
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fo 5932  df-fv 5934  df-1st 7210
This theorem is referenced by:  1stconst  7310  txcmpb  21495
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