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Theorem fo1stres 8001
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 4347 . . . . . . 7 (𝐵 ≠ ∅ ↔ ∃𝑦 𝑦𝐵)
2 opelxp 5713 . . . . . . . . . 10 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ (𝑥𝐴𝑦𝐵))
3 fvres 6911 . . . . . . . . . . . 12 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → ((1st ↾ (𝐴 × 𝐵))‘⟨𝑥, 𝑦⟩) = (1st ‘⟨𝑥, 𝑦⟩))
4 vex 3479 . . . . . . . . . . . . 13 𝑥 ∈ V
5 vex 3479 . . . . . . . . . . . . 13 𝑦 ∈ V
64, 5op1st 7983 . . . . . . . . . . . 12 (1st ‘⟨𝑥, 𝑦⟩) = 𝑥
73, 6eqtr2di 2790 . . . . . . . . . . 11 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → 𝑥 = ((1st ↾ (𝐴 × 𝐵))‘⟨𝑥, 𝑦⟩))
8 f1stres 7999 . . . . . . . . . . . . 13 (1st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐴
9 ffn 6718 . . . . . . . . . . . . 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 7083 . . . . . . . . . . . 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 2834 . . . . . . . . . 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 3989 . . . . 5 (𝐵 ≠ ∅ → 𝐴 ⊆ ran (1st ↾ (𝐴 × 𝐵)))
19 frn 6725 . . . . . 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 3998 . . . 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 6810 . 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 2941  wss 3949  c0 4323  cop 4635   × cxp 5675  ran crn 5678  cres 5679   Fn wfn 6539  wf 6540  ontowfo 6542  cfv 6544  1st c1st 7973
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 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-fo 6550  df-fv 6552  df-1st 7975
This theorem is referenced by:  1stconst  8086  txcmpb  23148
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