ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  fo2ndresm Structured version   GIF version

Theorem fo2ndresm 5731
Description: Onto mapping of a restriction of the 2nd (second member of an ordered pair) function. (Contributed by Jim Kingdon, 24-Jan-2019.)
Assertion
Ref Expression
fo2ndresm (x x A → (2nd ↾ (A × B)):(A × B)–ontoB)
Distinct variable group:   x,A
Allowed substitution hint:   B(x)

Proof of Theorem fo2ndresm
Dummy variables v u are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq1 2097 . . 3 (u = x → (u Ax A))
21cbvexv 1792 . 2 (u u Ax x A)
3 opelxp 4317 . . . . . . . . . 10 (⟨u, v (A × B) ↔ (u A v B))
4 fvres 5141 . . . . . . . . . . . 12 (⟨u, v (A × B) → ((2nd ↾ (A × B))‘⟨u, v⟩) = (2nd ‘⟨u, v⟩))
5 vex 2554 . . . . . . . . . . . . 13 u V
6 vex 2554 . . . . . . . . . . . . 13 v V
75, 6op2nd 5716 . . . . . . . . . . . 12 (2nd ‘⟨u, v⟩) = v
84, 7syl6req 2086 . . . . . . . . . . 11 (⟨u, v (A × B) → v = ((2nd ↾ (A × B))‘⟨u, v⟩))
9 f2ndres 5729 . . . . . . . . . . . . 13 (2nd ↾ (A × B)):(A × B)⟶B
10 ffn 4989 . . . . . . . . . . . . 13 ((2nd ↾ (A × B)):(A × B)⟶B → (2nd ↾ (A × B)) Fn (A × B))
119, 10ax-mp 7 . . . . . . . . . . . 12 (2nd ↾ (A × B)) Fn (A × B)
12 fnfvelrn 5242 . . . . . . . . . . . 12 (((2nd ↾ (A × B)) Fn (A × B) u, v (A × B)) → ((2nd ↾ (A × B))‘⟨u, v⟩) ran (2nd ↾ (A × B)))
1311, 12mpan 400 . . . . . . . . . . 11 (⟨u, v (A × B) → ((2nd ↾ (A × B))‘⟨u, v⟩) ran (2nd ↾ (A × B)))
148, 13eqeltrd 2111 . . . . . . . . . 10 (⟨u, v (A × B) → v ran (2nd ↾ (A × B)))
153, 14sylbir 125 . . . . . . . . 9 ((u A v B) → v ran (2nd ↾ (A × B)))
1615ex 108 . . . . . . . 8 (u A → (v Bv ran (2nd ↾ (A × B))))
1716exlimiv 1486 . . . . . . 7 (u u A → (v Bv ran (2nd ↾ (A × B))))
1817ssrdv 2945 . . . . . 6 (u u AB ⊆ ran (2nd ↾ (A × B)))
19 frn 4995 . . . . . . 7 ((2nd ↾ (A × B)):(A × B)⟶B → ran (2nd ↾ (A × B)) ⊆ B)
209, 19ax-mp 7 . . . . . 6 ran (2nd ↾ (A × B)) ⊆ B
2118, 20jctil 295 . . . . 5 (u u A → (ran (2nd ↾ (A × B)) ⊆ B B ⊆ ran (2nd ↾ (A × B))))
22 eqss 2954 . . . . 5 (ran (2nd ↾ (A × B)) = B ↔ (ran (2nd ↾ (A × B)) ⊆ B B ⊆ ran (2nd ↾ (A × B))))
2321, 22sylibr 137 . . . 4 (u u A → ran (2nd ↾ (A × B)) = B)
2423, 9jctil 295 . . 3 (u u A → ((2nd ↾ (A × B)):(A × B)⟶B ran (2nd ↾ (A × B)) = B))
25 dffo2 5053 . . 3 ((2nd ↾ (A × B)):(A × B)–ontoB ↔ ((2nd ↾ (A × B)):(A × B)⟶B ran (2nd ↾ (A × B)) = B))
2624, 25sylibr 137 . 2 (u u A → (2nd ↾ (A × B)):(A × B)–ontoB)
272, 26sylbir 125 1 (x x A → (2nd ↾ (A × B)):(A × B)–ontoB)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97   = wceq 1242  wex 1378   wcel 1390  wss 2911  cop 3370   × cxp 4286  ran crn 4289  cres 4290   Fn wfn 4840  wf 4841  ontowfo 4843  cfv 4845  2nd c2nd 5708
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 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935  ax-un 4136
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-rab 2309  df-v 2553  df-sbc 2759  df-csb 2847  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-iun 3650  df-br 3756  df-opab 3810  df-mpt 3811  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-res 4300  df-ima 4301  df-iota 4810  df-fun 4847  df-fn 4848  df-f 4849  df-fo 4851  df-fv 4853  df-2nd 5710
This theorem is referenced by:  2ndconst  5785
  Copyright terms: Public domain W3C validator