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Theorem fo1stresm 6099
 Description: Onto mapping of a restriction of the 1st (first member of an ordered pair) function. (Contributed by Jim Kingdon, 24-Jan-2019.)
Assertion
Ref Expression
fo1stresm (∃𝑦 𝑦𝐵 → (1st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)–onto𝐴)
Distinct variable group:   𝑦,𝐵
Allowed substitution hint:   𝐴(𝑦)

Proof of Theorem fo1stresm
Dummy variables 𝑣 𝑢 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq1 2217 . . 3 (𝑣 = 𝑦 → (𝑣𝐵𝑦𝐵))
21cbvexv 1895 . 2 (∃𝑣 𝑣𝐵 ↔ ∃𝑦 𝑦𝐵)
3 opelxp 4609 . . . . . . . . . 10 (⟨𝑢, 𝑣⟩ ∈ (𝐴 × 𝐵) ↔ (𝑢𝐴𝑣𝐵))
4 fvres 5485 . . . . . . . . . . . 12 (⟨𝑢, 𝑣⟩ ∈ (𝐴 × 𝐵) → ((1st ↾ (𝐴 × 𝐵))‘⟨𝑢, 𝑣⟩) = (1st ‘⟨𝑢, 𝑣⟩))
5 vex 2712 . . . . . . . . . . . . 13 𝑢 ∈ V
6 vex 2712 . . . . . . . . . . . . 13 𝑣 ∈ V
75, 6op1st 6084 . . . . . . . . . . . 12 (1st ‘⟨𝑢, 𝑣⟩) = 𝑢
84, 7eqtr2di 2204 . . . . . . . . . . 11 (⟨𝑢, 𝑣⟩ ∈ (𝐴 × 𝐵) → 𝑢 = ((1st ↾ (𝐴 × 𝐵))‘⟨𝑢, 𝑣⟩))
9 f1stres 6097 . . . . . . . . . . . . 13 (1st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐴
10 ffn 5312 . . . . . . . . . . . . 13 ((1st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐴 → (1st ↾ (𝐴 × 𝐵)) Fn (𝐴 × 𝐵))
119, 10ax-mp 5 . . . . . . . . . . . 12 (1st ↾ (𝐴 × 𝐵)) Fn (𝐴 × 𝐵)
12 fnfvelrn 5592 . . . . . . . . . . . 12 (((1st ↾ (𝐴 × 𝐵)) Fn (𝐴 × 𝐵) ∧ ⟨𝑢, 𝑣⟩ ∈ (𝐴 × 𝐵)) → ((1st ↾ (𝐴 × 𝐵))‘⟨𝑢, 𝑣⟩) ∈ ran (1st ↾ (𝐴 × 𝐵)))
1311, 12mpan 421 . . . . . . . . . . 11 (⟨𝑢, 𝑣⟩ ∈ (𝐴 × 𝐵) → ((1st ↾ (𝐴 × 𝐵))‘⟨𝑢, 𝑣⟩) ∈ ran (1st ↾ (𝐴 × 𝐵)))
148, 13eqeltrd 2231 . . . . . . . . . 10 (⟨𝑢, 𝑣⟩ ∈ (𝐴 × 𝐵) → 𝑢 ∈ ran (1st ↾ (𝐴 × 𝐵)))
153, 14sylbir 134 . . . . . . . . 9 ((𝑢𝐴𝑣𝐵) → 𝑢 ∈ ran (1st ↾ (𝐴 × 𝐵)))
1615expcom 115 . . . . . . . 8 (𝑣𝐵 → (𝑢𝐴𝑢 ∈ ran (1st ↾ (𝐴 × 𝐵))))
1716exlimiv 1575 . . . . . . 7 (∃𝑣 𝑣𝐵 → (𝑢𝐴𝑢 ∈ ran (1st ↾ (𝐴 × 𝐵))))
1817ssrdv 3130 . . . . . 6 (∃𝑣 𝑣𝐵𝐴 ⊆ ran (1st ↾ (𝐴 × 𝐵)))
19 frn 5321 . . . . . . 7 ((1st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐴 → ran (1st ↾ (𝐴 × 𝐵)) ⊆ 𝐴)
209, 19ax-mp 5 . . . . . 6 ran (1st ↾ (𝐴 × 𝐵)) ⊆ 𝐴
2118, 20jctil 310 . . . . 5 (∃𝑣 𝑣𝐵 → (ran (1st ↾ (𝐴 × 𝐵)) ⊆ 𝐴𝐴 ⊆ ran (1st ↾ (𝐴 × 𝐵))))
22 eqss 3139 . . . . 5 (ran (1st ↾ (𝐴 × 𝐵)) = 𝐴 ↔ (ran (1st ↾ (𝐴 × 𝐵)) ⊆ 𝐴𝐴 ⊆ ran (1st ↾ (𝐴 × 𝐵))))
2321, 22sylibr 133 . . . 4 (∃𝑣 𝑣𝐵 → ran (1st ↾ (𝐴 × 𝐵)) = 𝐴)
2423, 9jctil 310 . . 3 (∃𝑣 𝑣𝐵 → ((1st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐴 ∧ ran (1st ↾ (𝐴 × 𝐵)) = 𝐴))
25 dffo2 5389 . . 3 ((1st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)–onto𝐴 ↔ ((1st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐴 ∧ ran (1st ↾ (𝐴 × 𝐵)) = 𝐴))
2624, 25sylibr 133 . 2 (∃𝑣 𝑣𝐵 → (1st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)–onto𝐴)
272, 26sylbir 134 1 (∃𝑦 𝑦𝐵 → (1st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)–onto𝐴)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   = wceq 1332  ∃wex 1469   ∈ wcel 2125   ⊆ wss 3098  ⟨cop 3559   × cxp 4577  ran crn 4580   ↾ cres 4581   Fn wfn 5158  ⟶wf 5159  –onto→wfo 5161  ‘cfv 5163  1st c1st 6076 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1481  ax-10 1482  ax-11 1483  ax-i12 1484  ax-bndl 1486  ax-4 1487  ax-17 1503  ax-i9 1507  ax-ial 1511  ax-i5r 1512  ax-13 2127  ax-14 2128  ax-ext 2136  ax-sep 4078  ax-pow 4130  ax-pr 4164  ax-un 4388 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1740  df-eu 2006  df-mo 2007  df-clab 2141  df-cleq 2147  df-clel 2150  df-nfc 2285  df-ral 2437  df-rex 2438  df-rab 2441  df-v 2711  df-sbc 2934  df-csb 3028  df-un 3102  df-in 3104  df-ss 3111  df-pw 3541  df-sn 3562  df-pr 3563  df-op 3565  df-uni 3769  df-iun 3847  df-br 3962  df-opab 4022  df-mpt 4023  df-id 4248  df-xp 4585  df-rel 4586  df-cnv 4587  df-co 4588  df-dm 4589  df-rn 4590  df-res 4591  df-ima 4592  df-iota 5128  df-fun 5165  df-fn 5166  df-f 5167  df-fo 5169  df-fv 5171  df-1st 6078 This theorem is referenced by:  1stconst  6158
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