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Theorem fo1stres 7152
 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 3913 . . . . . . 7 (𝐵 ≠ ∅ ↔ ∃𝑦 𝑦𝐵)
2 opelxp 5116 . . . . . . . . . 10 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) ↔ (𝑥𝐴𝑦𝐵))
3 fvres 6174 . . . . . . . . . . . 12 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → ((1st ↾ (𝐴 × 𝐵))‘⟨𝑥, 𝑦⟩) = (1st ‘⟨𝑥, 𝑦⟩))
4 vex 3193 . . . . . . . . . . . . 13 𝑥 ∈ V
5 vex 3193 . . . . . . . . . . . . 13 𝑦 ∈ V
64, 5op1st 7136 . . . . . . . . . . . 12 (1st ‘⟨𝑥, 𝑦⟩) = 𝑥
73, 6syl6req 2672 . . . . . . . . . . 11 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → 𝑥 = ((1st ↾ (𝐴 × 𝐵))‘⟨𝑥, 𝑦⟩))
8 f1stres 7150 . . . . . . . . . . . . 13 (1st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐴
9 ffn 6012 . . . . . . . . . . . . 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 6322 . . . . . . . . . . . 12 (((1st ↾ (𝐴 × 𝐵)) Fn (𝐴 × 𝐵) ∧ ⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵)) → ((1st ↾ (𝐴 × 𝐵))‘⟨𝑥, 𝑦⟩) ∈ ran (1st ↾ (𝐴 × 𝐵)))
1210, 11mpan 705 . . . . . . . . . . 11 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → ((1st ↾ (𝐴 × 𝐵))‘⟨𝑥, 𝑦⟩) ∈ ran (1st ↾ (𝐴 × 𝐵)))
137, 12eqeltrd 2698 . . . . . . . . . 10 (⟨𝑥, 𝑦⟩ ∈ (𝐴 × 𝐵) → 𝑥 ∈ ran (1st ↾ (𝐴 × 𝐵)))
142, 13sylbir 225 . . . . . . . . 9 ((𝑥𝐴𝑦𝐵) → 𝑥 ∈ ran (1st ↾ (𝐴 × 𝐵)))
1514expcom 451 . . . . . . . 8 (𝑦𝐵 → (𝑥𝐴𝑥 ∈ ran (1st ↾ (𝐴 × 𝐵))))
1615exlimiv 1855 . . . . . . 7 (∃𝑦 𝑦𝐵 → (𝑥𝐴𝑥 ∈ ran (1st ↾ (𝐴 × 𝐵))))
171, 16sylbi 207 . . . . . 6 (𝐵 ≠ ∅ → (𝑥𝐴𝑥 ∈ ran (1st ↾ (𝐴 × 𝐵))))
1817ssrdv 3594 . . . . 5 (𝐵 ≠ ∅ → 𝐴 ⊆ ran (1st ↾ (𝐴 × 𝐵)))
19 frn 6020 . . . . . 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 3603 . . . 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 6086 . 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 384   = wceq 1480  ∃wex 1701   ∈ wcel 1987   ≠ wne 2790   ⊆ wss 3560  ∅c0 3897  ⟨cop 4161   × cxp 5082  ran crn 5085   ↾ cres 5086   Fn wfn 5852  ⟶wf 5853  –onto→wfo 5855  ‘cfv 5857  1st c1st 7126 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4751  ax-nul 4759  ax-pow 4813  ax-pr 4877  ax-un 6914 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-iun 4494  df-br 4624  df-opab 4684  df-mpt 4685  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-res 5096  df-ima 5097  df-iota 5820  df-fun 5859  df-fn 5860  df-f 5861  df-fo 5863  df-fv 5865  df-1st 7128 This theorem is referenced by:  1stconst  7225  txcmpb  21387
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