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Theorem fo1stresm 6161
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 2240 . . 3 (𝑣 = 𝑦 → (𝑣𝐵𝑦𝐵))
21cbvexv 1918 . 2 (∃𝑣 𝑣𝐵 ↔ ∃𝑦 𝑦𝐵)
3 opelxp 4656 . . . . . . . . . 10 (⟨𝑢, 𝑣⟩ ∈ (𝐴 × 𝐵) ↔ (𝑢𝐴𝑣𝐵))
4 fvres 5539 . . . . . . . . . . . 12 (⟨𝑢, 𝑣⟩ ∈ (𝐴 × 𝐵) → ((1st ↾ (𝐴 × 𝐵))‘⟨𝑢, 𝑣⟩) = (1st ‘⟨𝑢, 𝑣⟩))
5 vex 2740 . . . . . . . . . . . . 13 𝑢 ∈ V
6 vex 2740 . . . . . . . . . . . . 13 𝑣 ∈ V
75, 6op1st 6146 . . . . . . . . . . . 12 (1st ‘⟨𝑢, 𝑣⟩) = 𝑢
84, 7eqtr2di 2227 . . . . . . . . . . 11 (⟨𝑢, 𝑣⟩ ∈ (𝐴 × 𝐵) → 𝑢 = ((1st ↾ (𝐴 × 𝐵))‘⟨𝑢, 𝑣⟩))
9 f1stres 6159 . . . . . . . . . . . . 13 (1st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐴
10 ffn 5365 . . . . . . . . . . . . 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 5648 . . . . . . . . . . . 12 (((1st ↾ (𝐴 × 𝐵)) Fn (𝐴 × 𝐵) ∧ ⟨𝑢, 𝑣⟩ ∈ (𝐴 × 𝐵)) → ((1st ↾ (𝐴 × 𝐵))‘⟨𝑢, 𝑣⟩) ∈ ran (1st ↾ (𝐴 × 𝐵)))
1311, 12mpan 424 . . . . . . . . . . 11 (⟨𝑢, 𝑣⟩ ∈ (𝐴 × 𝐵) → ((1st ↾ (𝐴 × 𝐵))‘⟨𝑢, 𝑣⟩) ∈ ran (1st ↾ (𝐴 × 𝐵)))
148, 13eqeltrd 2254 . . . . . . . . . 10 (⟨𝑢, 𝑣⟩ ∈ (𝐴 × 𝐵) → 𝑢 ∈ ran (1st ↾ (𝐴 × 𝐵)))
153, 14sylbir 135 . . . . . . . . 9 ((𝑢𝐴𝑣𝐵) → 𝑢 ∈ ran (1st ↾ (𝐴 × 𝐵)))
1615expcom 116 . . . . . . . 8 (𝑣𝐵 → (𝑢𝐴𝑢 ∈ ran (1st ↾ (𝐴 × 𝐵))))
1716exlimiv 1598 . . . . . . 7 (∃𝑣 𝑣𝐵 → (𝑢𝐴𝑢 ∈ ran (1st ↾ (𝐴 × 𝐵))))
1817ssrdv 3161 . . . . . 6 (∃𝑣 𝑣𝐵𝐴 ⊆ ran (1st ↾ (𝐴 × 𝐵)))
19 frn 5374 . . . . . . 7 ((1st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐴 → ran (1st ↾ (𝐴 × 𝐵)) ⊆ 𝐴)
209, 19ax-mp 5 . . . . . 6 ran (1st ↾ (𝐴 × 𝐵)) ⊆ 𝐴
2118, 20jctil 312 . . . . 5 (∃𝑣 𝑣𝐵 → (ran (1st ↾ (𝐴 × 𝐵)) ⊆ 𝐴𝐴 ⊆ ran (1st ↾ (𝐴 × 𝐵))))
22 eqss 3170 . . . . 5 (ran (1st ↾ (𝐴 × 𝐵)) = 𝐴 ↔ (ran (1st ↾ (𝐴 × 𝐵)) ⊆ 𝐴𝐴 ⊆ ran (1st ↾ (𝐴 × 𝐵))))
2321, 22sylibr 134 . . . 4 (∃𝑣 𝑣𝐵 → ran (1st ↾ (𝐴 × 𝐵)) = 𝐴)
2423, 9jctil 312 . . 3 (∃𝑣 𝑣𝐵 → ((1st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐴 ∧ ran (1st ↾ (𝐴 × 𝐵)) = 𝐴))
25 dffo2 5442 . . 3 ((1st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)–onto𝐴 ↔ ((1st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐴 ∧ ran (1st ↾ (𝐴 × 𝐵)) = 𝐴))
2624, 25sylibr 134 . 2 (∃𝑣 𝑣𝐵 → (1st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)–onto𝐴)
272, 26sylbir 135 1 (∃𝑦 𝑦𝐵 → (1st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)–onto𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104   = wceq 1353  wex 1492  wcel 2148  wss 3129  cop 3595   × cxp 4624  ran crn 4627  cres 4628   Fn wfn 5211  wf 5212  ontowfo 5214  cfv 5216  1st c1st 6138
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-un 3133  df-in 3135  df-ss 3142  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-id 4293  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-fo 5222  df-fv 5224  df-1st 6140
This theorem is referenced by:  1stconst  6221
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