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Theorem fo2ndresm 5708
 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 2078 . . 3 (u = x → (u Ax A))
21cbvexv 1773 . 2 (u u Ax x A)
3 opelxp 4297 . . . . . . . . . 10 (⟨u, v (A × B) ↔ (u A v B))
4 fvres 5119 . . . . . . . . . . . 12 (⟨u, v (A × B) → ((2nd ↾ (A × B))‘⟨u, v⟩) = (2nd ‘⟨u, v⟩))
5 vex 2534 . . . . . . . . . . . . 13 u V
6 vex 2534 . . . . . . . . . . . . 13 v V
75, 6op2nd 5693 . . . . . . . . . . . 12 (2nd ‘⟨u, v⟩) = v
84, 7syl6req 2067 . . . . . . . . . . 11 (⟨u, v (A × B) → v = ((2nd ↾ (A × B))‘⟨u, v⟩))
9 f2ndres 5706 . . . . . . . . . . . . 13 (2nd ↾ (A × B)):(A × B)⟶B
10 ffn 4968 . . . . . . . . . . . . 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 5220 . . . . . . . . . . . 12 (((2nd ↾ (A × B)) Fn (A × B) u, v (A × B)) → ((2nd ↾ (A × B))‘⟨u, v⟩) ran (2nd ↾ (A × B)))
1311, 12mpan 402 . . . . . . . . . . 11 (⟨u, v (A × B) → ((2nd ↾ (A × B))‘⟨u, v⟩) ran (2nd ↾ (A × B)))
148, 13eqeltrd 2092 . . . . . . . . . 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 1467 . . . . . . 7 (u u A → (v Bv ran (2nd ↾ (A × B))))
1817ssrdv 2924 . . . . . 6 (u u AB ⊆ ran (2nd ↾ (A × B)))
19 frn 4974 . . . . . . 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 2933 . . . . 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 5031 . . 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 1226  ∃wex 1358   ∈ wcel 1370   ⊆ wss 2890  ⟨cop 3349   × cxp 4266  ran crn 4269   ↾ cres 4270   Fn wfn 4820  ⟶wf 4821  –onto→wfo 4823  ‘cfv 4825  2nd c2nd 5685 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 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-13 1381  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914  ax-un 4116 This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-eu 1881  df-mo 1882  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-rab 2289  df-v 2533  df-sbc 2738  df-csb 2826  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-uni 3551  df-iun 3629  df-br 3735  df-opab 3789  df-mpt 3790  df-id 4000  df-xp 4274  df-rel 4275  df-cnv 4276  df-co 4277  df-dm 4278  df-rn 4279  df-res 4280  df-ima 4281  df-iota 4790  df-fun 4827  df-fn 4828  df-f 4829  df-fo 4831  df-fv 4833  df-2nd 5687 This theorem is referenced by:  2ndconst  5762
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