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Theorem fo1stresm 5711
 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 y B → (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 2082 . . 3 (v = y → (v By B))
21cbvexv 1777 . 2 (v v By y B)
3 opelxp 4301 . . . . . . . . . 10 (⟨u, v (A × B) ↔ (u A v B))
4 fvres 5123 . . . . . . . . . . . 12 (⟨u, v (A × B) → ((1st ↾ (A × B))‘⟨u, v⟩) = (1st ‘⟨u, v⟩))
5 vex 2538 . . . . . . . . . . . . 13 u V
6 vex 2538 . . . . . . . . . . . . 13 v V
75, 6op1st 5696 . . . . . . . . . . . 12 (1st ‘⟨u, v⟩) = u
84, 7syl6req 2071 . . . . . . . . . . 11 (⟨u, v (A × B) → u = ((1st ↾ (A × B))‘⟨u, v⟩))
9 f1stres 5709 . . . . . . . . . . . . 13 (1st ↾ (A × B)):(A × B)⟶A
10 ffn 4972 . . . . . . . . . . . . 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 5224 . . . . . . . . . . . 12 (((1st ↾ (A × B)) Fn (A × B) u, v (A × B)) → ((1st ↾ (A × B))‘⟨u, v⟩) ran (1st ↾ (A × B)))
1311, 12mpan 402 . . . . . . . . . . 11 (⟨u, v (A × B) → ((1st ↾ (A × B))‘⟨u, v⟩) ran (1st ↾ (A × B)))
148, 13eqeltrd 2096 . . . . . . . . . 10 (⟨u, v (A × B) → u ran (1st ↾ (A × B)))
153, 14sylbir 125 . . . . . . . . 9 ((u A v B) → u ran (1st ↾ (A × B)))
1615expcom 109 . . . . . . . 8 (v B → (u Au ran (1st ↾ (A × B))))
1716exlimiv 1471 . . . . . . 7 (v v B → (u Au ran (1st ↾ (A × B))))
1817ssrdv 2928 . . . . . 6 (v v BA ⊆ ran (1st ↾ (A × B)))
19 frn 4978 . . . . . . 7 ((1st ↾ (A × B)):(A × B)⟶A → ran (1st ↾ (A × B)) ⊆ A)
209, 19ax-mp 7 . . . . . 6 ran (1st ↾ (A × B)) ⊆ A
2118, 20jctil 295 . . . . 5 (v v B → (ran (1st ↾ (A × B)) ⊆ A A ⊆ ran (1st ↾ (A × B))))
22 eqss 2937 . . . . 5 (ran (1st ↾ (A × B)) = A ↔ (ran (1st ↾ (A × B)) ⊆ A A ⊆ ran (1st ↾ (A × B))))
2321, 22sylibr 137 . . . 4 (v v B → ran (1st ↾ (A × B)) = A)
2423, 9jctil 295 . . 3 (v v B → ((1st ↾ (A × B)):(A × B)⟶A ran (1st ↾ (A × B)) = A))
25 dffo2 5035 . . 3 ((1st ↾ (A × B)):(A × B)–ontoA ↔ ((1st ↾ (A × B)):(A × B)⟶A ran (1st ↾ (A × B)) = A))
2624, 25sylibr 137 . 2 (v v B → (1st ↾ (A × B)):(A × B)–ontoA)
272, 26sylbir 125 1 (y y B → (1st ↾ (A × B)):(A × B)–ontoA)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 97   = wceq 1228  ∃wex 1362   ∈ wcel 1374   ⊆ wss 2894  ⟨cop 3353   × cxp 4270  ran crn 4273   ↾ cres 4274   Fn wfn 4824  ⟶wf 4825  –onto→wfo 4827  ‘cfv 4829  1st c1st 5688 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-13 1385  ax-14 1386  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004  ax-sep 3849  ax-pow 3901  ax-pr 3918  ax-un 4120 This theorem depends on definitions:  df-bi 110  df-3an 875  df-tru 1231  df-nf 1330  df-sb 1628  df-eu 1885  df-mo 1886  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-ral 2289  df-rex 2290  df-rab 2293  df-v 2537  df-sbc 2742  df-csb 2830  df-un 2899  df-in 2901  df-ss 2908  df-pw 3336  df-sn 3356  df-pr 3357  df-op 3359  df-uni 3555  df-iun 3633  df-br 3739  df-opab 3793  df-mpt 3794  df-id 4004  df-xp 4278  df-rel 4279  df-cnv 4280  df-co 4281  df-dm 4282  df-rn 4283  df-res 4284  df-ima 4285  df-iota 4794  df-fun 4831  df-fn 4832  df-f 4833  df-fo 4835  df-fv 4837  df-1st 5690 This theorem is referenced by:  1stconst  5765
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