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Theorem 2ndconst 5785
Description: The mapping of a restriction of the 2nd function to a converse constant function. (Contributed by NM, 27-Mar-2008.)
Assertion
Ref Expression
2ndconst (A 𝑉 → (2nd ↾ ({A} × B)):({A} × B)–1-1-ontoB)

Proof of Theorem 2ndconst
Dummy variables x y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 snmg 3477 . . 3 (A 𝑉x x {A})
2 fo2ndresm 5731 . . 3 (x x {A} → (2nd ↾ ({A} × B)):({A} × B)–ontoB)
31, 2syl 14 . 2 (A 𝑉 → (2nd ↾ ({A} × B)):({A} × B)–ontoB)
4 moeq 2710 . . . . . 6 ∃*x x = ⟨A, y
54moani 1967 . . . . 5 ∃*x(y B x = ⟨A, y⟩)
6 vex 2554 . . . . . . . 8 y V
76brres 4561 . . . . . . 7 (x(2nd ↾ ({A} × B))y ↔ (x2nd y x ({A} × B)))
8 fo2nd 5727 . . . . . . . . . . 11 2nd :V–onto→V
9 fofn 5051 . . . . . . . . . . 11 (2nd :V–onto→V → 2nd Fn V)
108, 9ax-mp 7 . . . . . . . . . 10 2nd Fn V
11 vex 2554 . . . . . . . . . 10 x V
12 fnbrfvb 5157 . . . . . . . . . 10 ((2nd Fn V x V) → ((2ndx) = yx2nd y))
1310, 11, 12mp2an 402 . . . . . . . . 9 ((2ndx) = yx2nd y)
1413anbi1i 431 . . . . . . . 8 (((2ndx) = y x ({A} × B)) ↔ (x2nd y x ({A} × B)))
15 elxp7 5739 . . . . . . . . . . 11 (x ({A} × B) ↔ (x (V × V) ((1stx) {A} (2ndx) B)))
16 eleq1 2097 . . . . . . . . . . . . . . 15 ((2ndx) = y → ((2ndx) By B))
1716biimpa 280 . . . . . . . . . . . . . 14 (((2ndx) = y (2ndx) B) → y B)
1817adantrl 447 . . . . . . . . . . . . 13 (((2ndx) = y ((1stx) {A} (2ndx) B)) → y B)
1918adantrl 447 . . . . . . . . . . . 12 (((2ndx) = y (x (V × V) ((1stx) {A} (2ndx) B))) → y B)
20 elsni 3391 . . . . . . . . . . . . . 14 ((1stx) {A} → (1stx) = A)
21 eqopi 5740 . . . . . . . . . . . . . . . 16 ((x (V × V) ((1stx) = A (2ndx) = y)) → x = ⟨A, y⟩)
2221ancom2s 500 . . . . . . . . . . . . . . 15 ((x (V × V) ((2ndx) = y (1stx) = A)) → x = ⟨A, y⟩)
2322an12s 499 . . . . . . . . . . . . . 14 (((2ndx) = y (x (V × V) (1stx) = A)) → x = ⟨A, y⟩)
2420, 23sylanr2 385 . . . . . . . . . . . . 13 (((2ndx) = y (x (V × V) (1stx) {A})) → x = ⟨A, y⟩)
2524adantrrr 456 . . . . . . . . . . . 12 (((2ndx) = y (x (V × V) ((1stx) {A} (2ndx) B))) → x = ⟨A, y⟩)
2619, 25jca 290 . . . . . . . . . . 11 (((2ndx) = y (x (V × V) ((1stx) {A} (2ndx) B))) → (y B x = ⟨A, y⟩))
2715, 26sylan2b 271 . . . . . . . . . 10 (((2ndx) = y x ({A} × B)) → (y B x = ⟨A, y⟩))
2827adantl 262 . . . . . . . . 9 ((A 𝑉 ((2ndx) = y x ({A} × B))) → (y B x = ⟨A, y⟩))
29 fveq2 5121 . . . . . . . . . . . 12 (x = ⟨A, y⟩ → (2ndx) = (2nd ‘⟨A, y⟩))
30 op2ndg 5720 . . . . . . . . . . . . 13 ((A 𝑉 y V) → (2nd ‘⟨A, y⟩) = y)
316, 30mpan2 401 . . . . . . . . . . . 12 (A 𝑉 → (2nd ‘⟨A, y⟩) = y)
3229, 31sylan9eqr 2091 . . . . . . . . . . 11 ((A 𝑉 x = ⟨A, y⟩) → (2ndx) = y)
3332adantrl 447 . . . . . . . . . 10 ((A 𝑉 (y B x = ⟨A, y⟩)) → (2ndx) = y)
34 simprr 484 . . . . . . . . . . 11 ((A 𝑉 (y B x = ⟨A, y⟩)) → x = ⟨A, y⟩)
35 snidg 3392 . . . . . . . . . . . . 13 (A 𝑉A {A})
3635adantr 261 . . . . . . . . . . . 12 ((A 𝑉 (y B x = ⟨A, y⟩)) → A {A})
37 simprl 483 . . . . . . . . . . . 12 ((A 𝑉 (y B x = ⟨A, y⟩)) → y B)
38 opelxpi 4319 . . . . . . . . . . . 12 ((A {A} y B) → ⟨A, y ({A} × B))
3936, 37, 38syl2anc 391 . . . . . . . . . . 11 ((A 𝑉 (y B x = ⟨A, y⟩)) → ⟨A, y ({A} × B))
4034, 39eqeltrd 2111 . . . . . . . . . 10 ((A 𝑉 (y B x = ⟨A, y⟩)) → x ({A} × B))
4133, 40jca 290 . . . . . . . . 9 ((A 𝑉 (y B x = ⟨A, y⟩)) → ((2ndx) = y x ({A} × B)))
4228, 41impbida 528 . . . . . . . 8 (A 𝑉 → (((2ndx) = y x ({A} × B)) ↔ (y B x = ⟨A, y⟩)))
4314, 42syl5bbr 183 . . . . . . 7 (A 𝑉 → ((x2nd y x ({A} × B)) ↔ (y B x = ⟨A, y⟩)))
447, 43syl5bb 181 . . . . . 6 (A 𝑉 → (x(2nd ↾ ({A} × B))y ↔ (y B x = ⟨A, y⟩)))
4544mobidv 1933 . . . . 5 (A 𝑉 → (∃*x x(2nd ↾ ({A} × B))y∃*x(y B x = ⟨A, y⟩)))
465, 45mpbiri 157 . . . 4 (A 𝑉∃*x x(2nd ↾ ({A} × B))y)
4746alrimiv 1751 . . 3 (A 𝑉y∃*x x(2nd ↾ ({A} × B))y)
48 funcnv2 4902 . . 3 (Fun (2nd ↾ ({A} × B)) ↔ y∃*x x(2nd ↾ ({A} × B))y)
4947, 48sylibr 137 . 2 (A 𝑉 → Fun (2nd ↾ ({A} × B)))
50 dff1o3 5075 . 2 ((2nd ↾ ({A} × B)):({A} × B)–1-1-ontoB ↔ ((2nd ↾ ({A} × B)):({A} × B)–ontoB Fun (2nd ↾ ({A} × B))))
513, 49, 50sylanbrc 394 1 (A 𝑉 → (2nd ↾ ({A} × B)):({A} × B)–1-1-ontoB)
Colors of variables: wff set class
Syntax hints:  wi 4   wa 97  wb 98  wal 1240   = wceq 1242  wex 1378   wcel 1390  ∃*wmo 1898  Vcvv 2551  {csn 3367  cop 3370   class class class wbr 3755   × cxp 4286  ccnv 4287  cres 4290  Fun wfun 4839   Fn wfn 4840  ontowfo 4843  1-1-ontowf1o 4844  cfv 4845  1st c1st 5707  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-f1 4850  df-fo 4851  df-f1o 4852  df-fv 4853  df-1st 5709  df-2nd 5710
This theorem is referenced by: (None)
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