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Theorem fo1stresm 5813
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 2114 . . 3 (𝑣 = 𝑦 → (𝑣𝐵𝑦𝐵))
21cbvexv 1809 . 2 (∃𝑣 𝑣𝐵 ↔ ∃𝑦 𝑦𝐵)
3 opelxp 4399 . . . . . . . . . 10 (⟨𝑢, 𝑣⟩ ∈ (𝐴 × 𝐵) ↔ (𝑢𝐴𝑣𝐵))
4 fvres 5223 . . . . . . . . . . . 12 (⟨𝑢, 𝑣⟩ ∈ (𝐴 × 𝐵) → ((1st ↾ (𝐴 × 𝐵))‘⟨𝑢, 𝑣⟩) = (1st ‘⟨𝑢, 𝑣⟩))
5 vex 2575 . . . . . . . . . . . . 13 𝑢 ∈ V
6 vex 2575 . . . . . . . . . . . . 13 𝑣 ∈ V
75, 6op1st 5798 . . . . . . . . . . . 12 (1st ‘⟨𝑢, 𝑣⟩) = 𝑢
84, 7syl6req 2103 . . . . . . . . . . 11 (⟨𝑢, 𝑣⟩ ∈ (𝐴 × 𝐵) → 𝑢 = ((1st ↾ (𝐴 × 𝐵))‘⟨𝑢, 𝑣⟩))
9 f1stres 5811 . . . . . . . . . . . . 13 (1st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐴
10 ffn 5071 . . . . . . . . . . . . 13 ((1st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐴 → (1st ↾ (𝐴 × 𝐵)) Fn (𝐴 × 𝐵))
119, 10ax-mp 7 . . . . . . . . . . . 12 (1st ↾ (𝐴 × 𝐵)) Fn (𝐴 × 𝐵)
12 fnfvelrn 5324 . . . . . . . . . . . 12 (((1st ↾ (𝐴 × 𝐵)) Fn (𝐴 × 𝐵) ∧ ⟨𝑢, 𝑣⟩ ∈ (𝐴 × 𝐵)) → ((1st ↾ (𝐴 × 𝐵))‘⟨𝑢, 𝑣⟩) ∈ ran (1st ↾ (𝐴 × 𝐵)))
1311, 12mpan 408 . . . . . . . . . . 11 (⟨𝑢, 𝑣⟩ ∈ (𝐴 × 𝐵) → ((1st ↾ (𝐴 × 𝐵))‘⟨𝑢, 𝑣⟩) ∈ ran (1st ↾ (𝐴 × 𝐵)))
148, 13eqeltrd 2128 . . . . . . . . . 10 (⟨𝑢, 𝑣⟩ ∈ (𝐴 × 𝐵) → 𝑢 ∈ ran (1st ↾ (𝐴 × 𝐵)))
153, 14sylbir 129 . . . . . . . . 9 ((𝑢𝐴𝑣𝐵) → 𝑢 ∈ ran (1st ↾ (𝐴 × 𝐵)))
1615expcom 113 . . . . . . . 8 (𝑣𝐵 → (𝑢𝐴𝑢 ∈ ran (1st ↾ (𝐴 × 𝐵))))
1716exlimiv 1503 . . . . . . 7 (∃𝑣 𝑣𝐵 → (𝑢𝐴𝑢 ∈ ran (1st ↾ (𝐴 × 𝐵))))
1817ssrdv 2976 . . . . . 6 (∃𝑣 𝑣𝐵𝐴 ⊆ ran (1st ↾ (𝐴 × 𝐵)))
19 frn 5077 . . . . . . 7 ((1st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐴 → ran (1st ↾ (𝐴 × 𝐵)) ⊆ 𝐴)
209, 19ax-mp 7 . . . . . 6 ran (1st ↾ (𝐴 × 𝐵)) ⊆ 𝐴
2118, 20jctil 299 . . . . 5 (∃𝑣 𝑣𝐵 → (ran (1st ↾ (𝐴 × 𝐵)) ⊆ 𝐴𝐴 ⊆ ran (1st ↾ (𝐴 × 𝐵))))
22 eqss 2985 . . . . 5 (ran (1st ↾ (𝐴 × 𝐵)) = 𝐴 ↔ (ran (1st ↾ (𝐴 × 𝐵)) ⊆ 𝐴𝐴 ⊆ ran (1st ↾ (𝐴 × 𝐵))))
2321, 22sylibr 141 . . . 4 (∃𝑣 𝑣𝐵 → ran (1st ↾ (𝐴 × 𝐵)) = 𝐴)
2423, 9jctil 299 . . 3 (∃𝑣 𝑣𝐵 → ((1st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐴 ∧ ran (1st ↾ (𝐴 × 𝐵)) = 𝐴))
25 dffo2 5135 . . 3 ((1st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)–onto𝐴 ↔ ((1st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)⟶𝐴 ∧ ran (1st ↾ (𝐴 × 𝐵)) = 𝐴))
2624, 25sylibr 141 . 2 (∃𝑣 𝑣𝐵 → (1st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)–onto𝐴)
272, 26sylbir 129 1 (∃𝑦 𝑦𝐵 → (1st ↾ (𝐴 × 𝐵)):(𝐴 × 𝐵)–onto𝐴)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101   = wceq 1257  wex 1395  wcel 1407  wss 2942  cop 3403   × cxp 4368  ran crn 4371  cres 4372   Fn wfn 4922  wf 4923  ontowfo 4925  cfv 4927  1st c1st 5790
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 638  ax-5 1350  ax-7 1351  ax-gen 1352  ax-ie1 1396  ax-ie2 1397  ax-8 1409  ax-10 1410  ax-11 1411  ax-i12 1412  ax-bndl 1413  ax-4 1414  ax-13 1418  ax-14 1419  ax-17 1433  ax-i9 1437  ax-ial 1441  ax-i5r 1442  ax-ext 2036  ax-sep 3900  ax-pow 3952  ax-pr 3969  ax-un 4195
This theorem depends on definitions:  df-bi 114  df-3an 896  df-tru 1260  df-nf 1364  df-sb 1660  df-eu 1917  df-mo 1918  df-clab 2041  df-cleq 2047  df-clel 2050  df-nfc 2181  df-ral 2326  df-rex 2327  df-rab 2330  df-v 2574  df-sbc 2785  df-csb 2878  df-un 2947  df-in 2949  df-ss 2956  df-pw 3386  df-sn 3406  df-pr 3407  df-op 3409  df-uni 3606  df-iun 3684  df-br 3790  df-opab 3844  df-mpt 3845  df-id 4055  df-xp 4376  df-rel 4377  df-cnv 4378  df-co 4379  df-dm 4380  df-rn 4381  df-res 4382  df-ima 4383  df-iota 4892  df-fun 4929  df-fn 4930  df-f 4931  df-fo 4933  df-fv 4935  df-1st 5792
This theorem is referenced by:  1stconst  5867
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