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Theorem 2ndconst 6226
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 (𝐴𝑉 → (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 3712 . . 3 (𝐴𝑉 → ∃𝑥 𝑥 ∈ {𝐴})
2 fo2ndresm 6166 . . 3 (∃𝑥 𝑥 ∈ {𝐴} → (2nd ↾ ({𝐴} × 𝐵)):({𝐴} × 𝐵)–onto𝐵)
31, 2syl 14 . 2 (𝐴𝑉 → (2nd ↾ ({𝐴} × 𝐵)):({𝐴} × 𝐵)–onto𝐵)
4 moeq 2914 . . . . . 6 ∃*𝑥 𝑥 = ⟨𝐴, 𝑦
54moani 2096 . . . . 5 ∃*𝑥(𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)
6 vex 2742 . . . . . . . 8 𝑦 ∈ V
76brres 4915 . . . . . . 7 (𝑥(2nd ↾ ({𝐴} × 𝐵))𝑦 ↔ (𝑥2nd 𝑦𝑥 ∈ ({𝐴} × 𝐵)))
8 fo2nd 6162 . . . . . . . . . . 11 2nd :V–onto→V
9 fofn 5442 . . . . . . . . . . 11 (2nd :V–onto→V → 2nd Fn V)
108, 9ax-mp 5 . . . . . . . . . 10 2nd Fn V
11 vex 2742 . . . . . . . . . 10 𝑥 ∈ V
12 fnbrfvb 5559 . . . . . . . . . 10 ((2nd Fn V ∧ 𝑥 ∈ V) → ((2nd𝑥) = 𝑦𝑥2nd 𝑦))
1310, 11, 12mp2an 426 . . . . . . . . 9 ((2nd𝑥) = 𝑦𝑥2nd 𝑦)
1413anbi1i 458 . . . . . . . 8 (((2nd𝑥) = 𝑦𝑥 ∈ ({𝐴} × 𝐵)) ↔ (𝑥2nd 𝑦𝑥 ∈ ({𝐴} × 𝐵)))
15 elxp7 6174 . . . . . . . . . . 11 (𝑥 ∈ ({𝐴} × 𝐵) ↔ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ {𝐴} ∧ (2nd𝑥) ∈ 𝐵)))
16 eleq1 2240 . . . . . . . . . . . . . . 15 ((2nd𝑥) = 𝑦 → ((2nd𝑥) ∈ 𝐵𝑦𝐵))
1716biimpa 296 . . . . . . . . . . . . . 14 (((2nd𝑥) = 𝑦 ∧ (2nd𝑥) ∈ 𝐵) → 𝑦𝐵)
1817adantrl 478 . . . . . . . . . . . . 13 (((2nd𝑥) = 𝑦 ∧ ((1st𝑥) ∈ {𝐴} ∧ (2nd𝑥) ∈ 𝐵)) → 𝑦𝐵)
1918adantrl 478 . . . . . . . . . . . 12 (((2nd𝑥) = 𝑦 ∧ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ {𝐴} ∧ (2nd𝑥) ∈ 𝐵))) → 𝑦𝐵)
20 elsni 3612 . . . . . . . . . . . . . 14 ((1st𝑥) ∈ {𝐴} → (1st𝑥) = 𝐴)
21 eqopi 6176 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ (V × V) ∧ ((1st𝑥) = 𝐴 ∧ (2nd𝑥) = 𝑦)) → 𝑥 = ⟨𝐴, 𝑦⟩)
2221ancom2s 566 . . . . . . . . . . . . . . 15 ((𝑥 ∈ (V × V) ∧ ((2nd𝑥) = 𝑦 ∧ (1st𝑥) = 𝐴)) → 𝑥 = ⟨𝐴, 𝑦⟩)
2322an12s 565 . . . . . . . . . . . . . 14 (((2nd𝑥) = 𝑦 ∧ (𝑥 ∈ (V × V) ∧ (1st𝑥) = 𝐴)) → 𝑥 = ⟨𝐴, 𝑦⟩)
2420, 23sylanr2 405 . . . . . . . . . . . . 13 (((2nd𝑥) = 𝑦 ∧ (𝑥 ∈ (V × V) ∧ (1st𝑥) ∈ {𝐴})) → 𝑥 = ⟨𝐴, 𝑦⟩)
2524adantrrr 487 . . . . . . . . . . . 12 (((2nd𝑥) = 𝑦 ∧ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ {𝐴} ∧ (2nd𝑥) ∈ 𝐵))) → 𝑥 = ⟨𝐴, 𝑦⟩)
2619, 25jca 306 . . . . . . . . . . 11 (((2nd𝑥) = 𝑦 ∧ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ {𝐴} ∧ (2nd𝑥) ∈ 𝐵))) → (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩))
2715, 26sylan2b 287 . . . . . . . . . 10 (((2nd𝑥) = 𝑦𝑥 ∈ ({𝐴} × 𝐵)) → (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩))
2827adantl 277 . . . . . . . . 9 ((𝐴𝑉 ∧ ((2nd𝑥) = 𝑦𝑥 ∈ ({𝐴} × 𝐵))) → (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩))
29 fveq2 5517 . . . . . . . . . . . 12 (𝑥 = ⟨𝐴, 𝑦⟩ → (2nd𝑥) = (2nd ‘⟨𝐴, 𝑦⟩))
30 op2ndg 6155 . . . . . . . . . . . . 13 ((𝐴𝑉𝑦 ∈ V) → (2nd ‘⟨𝐴, 𝑦⟩) = 𝑦)
316, 30mpan2 425 . . . . . . . . . . . 12 (𝐴𝑉 → (2nd ‘⟨𝐴, 𝑦⟩) = 𝑦)
3229, 31sylan9eqr 2232 . . . . . . . . . . 11 ((𝐴𝑉𝑥 = ⟨𝐴, 𝑦⟩) → (2nd𝑥) = 𝑦)
3332adantrl 478 . . . . . . . . . 10 ((𝐴𝑉 ∧ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)) → (2nd𝑥) = 𝑦)
34 simprr 531 . . . . . . . . . . 11 ((𝐴𝑉 ∧ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)) → 𝑥 = ⟨𝐴, 𝑦⟩)
35 snidg 3623 . . . . . . . . . . . . 13 (𝐴𝑉𝐴 ∈ {𝐴})
3635adantr 276 . . . . . . . . . . . 12 ((𝐴𝑉 ∧ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)) → 𝐴 ∈ {𝐴})
37 simprl 529 . . . . . . . . . . . 12 ((𝐴𝑉 ∧ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)) → 𝑦𝐵)
38 opelxpi 4660 . . . . . . . . . . . 12 ((𝐴 ∈ {𝐴} ∧ 𝑦𝐵) → ⟨𝐴, 𝑦⟩ ∈ ({𝐴} × 𝐵))
3936, 37, 38syl2anc 411 . . . . . . . . . . 11 ((𝐴𝑉 ∧ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)) → ⟨𝐴, 𝑦⟩ ∈ ({𝐴} × 𝐵))
4034, 39eqeltrd 2254 . . . . . . . . . 10 ((𝐴𝑉 ∧ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)) → 𝑥 ∈ ({𝐴} × 𝐵))
4133, 40jca 306 . . . . . . . . 9 ((𝐴𝑉 ∧ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)) → ((2nd𝑥) = 𝑦𝑥 ∈ ({𝐴} × 𝐵)))
4228, 41impbida 596 . . . . . . . 8 (𝐴𝑉 → (((2nd𝑥) = 𝑦𝑥 ∈ ({𝐴} × 𝐵)) ↔ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)))
4314, 42bitr3id 194 . . . . . . 7 (𝐴𝑉 → ((𝑥2nd 𝑦𝑥 ∈ ({𝐴} × 𝐵)) ↔ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)))
447, 43bitrid 192 . . . . . 6 (𝐴𝑉 → (𝑥(2nd ↾ ({𝐴} × 𝐵))𝑦 ↔ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)))
4544mobidv 2062 . . . . 5 (𝐴𝑉 → (∃*𝑥 𝑥(2nd ↾ ({𝐴} × 𝐵))𝑦 ↔ ∃*𝑥(𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)))
465, 45mpbiri 168 . . . 4 (𝐴𝑉 → ∃*𝑥 𝑥(2nd ↾ ({𝐴} × 𝐵))𝑦)
4746alrimiv 1874 . . 3 (𝐴𝑉 → ∀𝑦∃*𝑥 𝑥(2nd ↾ ({𝐴} × 𝐵))𝑦)
48 funcnv2 5278 . . 3 (Fun (2nd ↾ ({𝐴} × 𝐵)) ↔ ∀𝑦∃*𝑥 𝑥(2nd ↾ ({𝐴} × 𝐵))𝑦)
4947, 48sylibr 134 . 2 (𝐴𝑉 → Fun (2nd ↾ ({𝐴} × 𝐵)))
50 dff1o3 5469 . 2 ((2nd ↾ ({𝐴} × 𝐵)):({𝐴} × 𝐵)–1-1-onto𝐵 ↔ ((2nd ↾ ({𝐴} × 𝐵)):({𝐴} × 𝐵)–onto𝐵 ∧ Fun (2nd ↾ ({𝐴} × 𝐵))))
513, 49, 50sylanbrc 417 1 (𝐴𝑉 → (2nd ↾ ({𝐴} × 𝐵)):({𝐴} × 𝐵)–1-1-onto𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  wal 1351   = wceq 1353  wex 1492  ∃*wmo 2027  wcel 2148  Vcvv 2739  {csn 3594  cop 3597   class class class wbr 4005   × cxp 4626  ccnv 4627  cres 4630  Fun wfun 5212   Fn wfn 5213  ontowfo 5216  1-1-ontowf1o 5217  cfv 5218  1st c1st 6142  2nd c2nd 6143
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-sbc 2965  df-csb 3060  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-iun 3890  df-br 4006  df-opab 4067  df-mpt 4068  df-id 4295  df-xp 4634  df-rel 4635  df-cnv 4636  df-co 4637  df-dm 4638  df-rn 4639  df-res 4640  df-ima 4641  df-iota 5180  df-fun 5220  df-fn 5221  df-f 5222  df-f1 5223  df-fo 5224  df-f1o 5225  df-fv 5226  df-1st 6144  df-2nd 6145
This theorem is referenced by:  xpfi  6932  fsum2dlemstep  11445  fprod2dlemstep  11633
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