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Theorem fo1stres 8003
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 4346 . . . . . . 7 (𝐵 ≠ ∅ ↔ ∃𝑦 𝑦𝐵)
2 opelxp 5712 . . . . . . . . . 10 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ (𝑥𝐴𝑦𝐵))
3 fvres 6910 . . . . . . . . . . . 12 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → ((1st ↾ (𝐴 × 𝐵))‘⟨𝑥, 𝑦⟩) = (1st ‘⟨𝑥, 𝑦⟩))
4 vex 3478 . . . . . . . . . . . . 13 𝑥 ∈ V
5 vex 3478 . . . . . . . . . . . . 13 𝑦 ∈ V
64, 5op1st 7985 . . . . . . . . . . . 12 (1st ‘⟨𝑥, 𝑦⟩) = 𝑥
73, 6eqtr2di 2789 . . . . . . . . . . 11 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → 𝑥 = ((1st ↾ (𝐴 × 𝐵))‘⟨𝑥, 𝑦⟩))
8 f1stres 8001 . . . . . . . . . . . . 13 (1st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐴
9 ffn 6717 . . . . . . . . . . . . 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 7082 . . . . . . . . . . . 12 (((1st ↾ (𝐴 × 𝐵)) Fn (𝐴 × 𝐵) ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)) → ((1st ↾ (𝐴 × 𝐵))‘⟨𝑥, 𝑦⟩) ∈ ran (1st ↾ (𝐴 × 𝐵)))
1210, 11mpan 688 . . . . . . . . . . 11 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → ((1st ↾ (𝐴 × 𝐵))‘⟨𝑥, 𝑦⟩) ∈ ran (1st ↾ (𝐴 × 𝐵)))
137, 12eqeltrd 2833 . . . . . . . . . 10 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → 𝑥 ∈ ran (1st ↾ (𝐴 × 𝐵)))
142, 13sylbir 234 . . . . . . . . 9 ((𝑥𝐴𝑦𝐵) → 𝑥 ∈ ran (1st ↾ (𝐴 × 𝐵)))
1514expcom 414 . . . . . . . 8 (𝑦𝐵 → (𝑥𝐴𝑥 ∈ ran (1st ↾ (𝐴 × 𝐵))))
1615exlimiv 1933 . . . . . . 7 (∃𝑦 𝑦𝐵 → (𝑥𝐴𝑥 ∈ ran (1st ↾ (𝐴 × 𝐵))))
171, 16sylbi 216 . . . . . 6 (𝐵 ≠ ∅ → (𝑥𝐴𝑥 ∈ ran (1st ↾ (𝐴 × 𝐵))))
1817ssrdv 3988 . . . . 5 (𝐵 ≠ ∅ → 𝐴 ⊆ ran (1st ↾ (𝐴 × 𝐵)))
19 frn 6724 . . . . . 6 ((1st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐴 → ran (1st ↾ (𝐴 × 𝐵)) ⊆ 𝐴)
208, 19ax-mp 5 . . . . 5 ran (1st ↾ (𝐴 × 𝐵)) ⊆ 𝐴
2118, 20jctil 520 . . . 4 (𝐵 ≠ ∅ → (ran (1st ↾ (𝐴 × 𝐵)) ⊆ 𝐴𝐴 ⊆ ran (1st ↾ (𝐴 × 𝐵))))
22 eqss 3997 . . . 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 520 . 2 (𝐵 ≠ ∅ → ((1st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐴 ∧ ran (1st ↾ (𝐴 × 𝐵)) = 𝐴))
25 dffo2 6809 . 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 396   = wceq 1541  wex 1781  wcel 2106  wne 2940  wss 3948  c0 4322  cop 4634   × cxp 5674  ran crn 5677  cres 5678   Fn wfn 6538  wf 6539  ontowfo 6541  cfv 6543  1st c1st 7975
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7727
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fo 6549  df-fv 6551  df-1st 7977
This theorem is referenced by:  1stconst  8088  txcmpb  23155
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