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Theorem fo1stres 7948
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 4307 . . . . . . 7 (𝐵 ≠ ∅ ↔ ∃𝑦 𝑦𝐵)
2 opelxp 5670 . . . . . . . . . 10 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ (𝑥𝐴𝑦𝐵))
3 fvres 6862 . . . . . . . . . . . 12 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → ((1st ↾ (𝐴 × 𝐵))‘⟨𝑥, 𝑦⟩) = (1st ‘⟨𝑥, 𝑦⟩))
4 vex 3450 . . . . . . . . . . . . 13 𝑥 ∈ V
5 vex 3450 . . . . . . . . . . . . 13 𝑦 ∈ V
64, 5op1st 7930 . . . . . . . . . . . 12 (1st ‘⟨𝑥, 𝑦⟩) = 𝑥
73, 6eqtr2di 2794 . . . . . . . . . . 11 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → 𝑥 = ((1st ↾ (𝐴 × 𝐵))‘⟨𝑥, 𝑦⟩))
8 f1stres 7946 . . . . . . . . . . . . 13 (1st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐴
9 ffn 6669 . . . . . . . . . . . . 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 7032 . . . . . . . . . . . 12 (((1st ↾ (𝐴 × 𝐵)) Fn (𝐴 × 𝐵) ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)) → ((1st ↾ (𝐴 × 𝐵))‘⟨𝑥, 𝑦⟩) ∈ ran (1st ↾ (𝐴 × 𝐵)))
1210, 11mpan 689 . . . . . . . . . . 11 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → ((1st ↾ (𝐴 × 𝐵))‘⟨𝑥, 𝑦⟩) ∈ ran (1st ↾ (𝐴 × 𝐵)))
137, 12eqeltrd 2838 . . . . . . . . . 10 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → 𝑥 ∈ ran (1st ↾ (𝐴 × 𝐵)))
142, 13sylbir 234 . . . . . . . . 9 ((𝑥𝐴𝑦𝐵) → 𝑥 ∈ ran (1st ↾ (𝐴 × 𝐵)))
1514expcom 415 . . . . . . . 8 (𝑦𝐵 → (𝑥𝐴𝑥 ∈ ran (1st ↾ (𝐴 × 𝐵))))
1615exlimiv 1934 . . . . . . 7 (∃𝑦 𝑦𝐵 → (𝑥𝐴𝑥 ∈ ran (1st ↾ (𝐴 × 𝐵))))
171, 16sylbi 216 . . . . . 6 (𝐵 ≠ ∅ → (𝑥𝐴𝑥 ∈ ran (1st ↾ (𝐴 × 𝐵))))
1817ssrdv 3951 . . . . 5 (𝐵 ≠ ∅ → 𝐴 ⊆ ran (1st ↾ (𝐴 × 𝐵)))
19 frn 6676 . . . . . 6 ((1st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐴 → ran (1st ↾ (𝐴 × 𝐵)) ⊆ 𝐴)
208, 19ax-mp 5 . . . . 5 ran (1st ↾ (𝐴 × 𝐵)) ⊆ 𝐴
2118, 20jctil 521 . . . 4 (𝐵 ≠ ∅ → (ran (1st ↾ (𝐴 × 𝐵)) ⊆ 𝐴𝐴 ⊆ ran (1st ↾ (𝐴 × 𝐵))))
22 eqss 3960 . . . 4 (ran (1st ↾ (𝐴 × 𝐵)) = 𝐴 ↔ (ran (1st ↾ (𝐴 × 𝐵)) ⊆ 𝐴𝐴 ⊆ ran (1st ↾ (𝐴 × 𝐵))))
2321, 22sylibr 233 . . 3 (𝐵 ≠ ∅ → ran (1st ↾ (𝐴 × 𝐵)) = 𝐴)
2423, 8jctil 521 . 2 (𝐵 ≠ ∅ → ((1st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐴 ∧ ran (1st ↾ (𝐴 × 𝐵)) = 𝐴))
25 dffo2 6761 . 2 ((1st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)–onto𝐴 ↔ ((1st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐴 ∧ ran (1st ↾ (𝐴 × 𝐵)) = 𝐴))
2624, 25sylibr 233 1 (𝐵 ≠ ∅ → (1st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)–onto𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wex 1782  wcel 2107  wne 2944  wss 3911  c0 4283  cop 4593   × cxp 5632  ran crn 5635  cres 5636   Fn wfn 6492  wf 6493  ontowfo 6495  cfv 6497  1st c1st 7920
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-sbc 3741  df-csb 3857  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-iun 4957  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fo 6503  df-fv 6505  df-1st 7922
This theorem is referenced by:  1stconst  8033  txcmpb  22998
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