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Theorem 2ndconst 6169
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 3677 . . 3 (𝐴𝑉 → ∃𝑥 𝑥 ∈ {𝐴})
2 fo2ndresm 6110 . . 3 (∃𝑥 𝑥 ∈ {𝐴} → (2nd ↾ ({𝐴} × 𝐵)):({𝐴} × 𝐵)–onto𝐵)
31, 2syl 14 . 2 (𝐴𝑉 → (2nd ↾ ({𝐴} × 𝐵)):({𝐴} × 𝐵)–onto𝐵)
4 moeq 2887 . . . . . 6 ∃*𝑥 𝑥 = ⟨𝐴, 𝑦
54moani 2076 . . . . 5 ∃*𝑥(𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)
6 vex 2715 . . . . . . . 8 𝑦 ∈ V
76brres 4872 . . . . . . 7 (𝑥(2nd ↾ ({𝐴} × 𝐵))𝑦 ↔ (𝑥2nd 𝑦𝑥 ∈ ({𝐴} × 𝐵)))
8 fo2nd 6106 . . . . . . . . . . 11 2nd :V–onto→V
9 fofn 5394 . . . . . . . . . . 11 (2nd :V–onto→V → 2nd Fn V)
108, 9ax-mp 5 . . . . . . . . . 10 2nd Fn V
11 vex 2715 . . . . . . . . . 10 𝑥 ∈ V
12 fnbrfvb 5509 . . . . . . . . . 10 ((2nd Fn V ∧ 𝑥 ∈ V) → ((2nd𝑥) = 𝑦𝑥2nd 𝑦))
1310, 11, 12mp2an 423 . . . . . . . . 9 ((2nd𝑥) = 𝑦𝑥2nd 𝑦)
1413anbi1i 454 . . . . . . . 8 (((2nd𝑥) = 𝑦𝑥 ∈ ({𝐴} × 𝐵)) ↔ (𝑥2nd 𝑦𝑥 ∈ ({𝐴} × 𝐵)))
15 elxp7 6118 . . . . . . . . . . 11 (𝑥 ∈ ({𝐴} × 𝐵) ↔ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ {𝐴} ∧ (2nd𝑥) ∈ 𝐵)))
16 eleq1 2220 . . . . . . . . . . . . . . 15 ((2nd𝑥) = 𝑦 → ((2nd𝑥) ∈ 𝐵𝑦𝐵))
1716biimpa 294 . . . . . . . . . . . . . 14 (((2nd𝑥) = 𝑦 ∧ (2nd𝑥) ∈ 𝐵) → 𝑦𝐵)
1817adantrl 470 . . . . . . . . . . . . 13 (((2nd𝑥) = 𝑦 ∧ ((1st𝑥) ∈ {𝐴} ∧ (2nd𝑥) ∈ 𝐵)) → 𝑦𝐵)
1918adantrl 470 . . . . . . . . . . . 12 (((2nd𝑥) = 𝑦 ∧ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ {𝐴} ∧ (2nd𝑥) ∈ 𝐵))) → 𝑦𝐵)
20 elsni 3578 . . . . . . . . . . . . . 14 ((1st𝑥) ∈ {𝐴} → (1st𝑥) = 𝐴)
21 eqopi 6120 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ (V × V) ∧ ((1st𝑥) = 𝐴 ∧ (2nd𝑥) = 𝑦)) → 𝑥 = ⟨𝐴, 𝑦⟩)
2221ancom2s 556 . . . . . . . . . . . . . . 15 ((𝑥 ∈ (V × V) ∧ ((2nd𝑥) = 𝑦 ∧ (1st𝑥) = 𝐴)) → 𝑥 = ⟨𝐴, 𝑦⟩)
2322an12s 555 . . . . . . . . . . . . . 14 (((2nd𝑥) = 𝑦 ∧ (𝑥 ∈ (V × V) ∧ (1st𝑥) = 𝐴)) → 𝑥 = ⟨𝐴, 𝑦⟩)
2420, 23sylanr2 403 . . . . . . . . . . . . 13 (((2nd𝑥) = 𝑦 ∧ (𝑥 ∈ (V × V) ∧ (1st𝑥) ∈ {𝐴})) → 𝑥 = ⟨𝐴, 𝑦⟩)
2524adantrrr 479 . . . . . . . . . . . 12 (((2nd𝑥) = 𝑦 ∧ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ {𝐴} ∧ (2nd𝑥) ∈ 𝐵))) → 𝑥 = ⟨𝐴, 𝑦⟩)
2619, 25jca 304 . . . . . . . . . . 11 (((2nd𝑥) = 𝑦 ∧ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ {𝐴} ∧ (2nd𝑥) ∈ 𝐵))) → (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩))
2715, 26sylan2b 285 . . . . . . . . . 10 (((2nd𝑥) = 𝑦𝑥 ∈ ({𝐴} × 𝐵)) → (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩))
2827adantl 275 . . . . . . . . 9 ((𝐴𝑉 ∧ ((2nd𝑥) = 𝑦𝑥 ∈ ({𝐴} × 𝐵))) → (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩))
29 fveq2 5468 . . . . . . . . . . . 12 (𝑥 = ⟨𝐴, 𝑦⟩ → (2nd𝑥) = (2nd ‘⟨𝐴, 𝑦⟩))
30 op2ndg 6099 . . . . . . . . . . . . 13 ((𝐴𝑉𝑦 ∈ V) → (2nd ‘⟨𝐴, 𝑦⟩) = 𝑦)
316, 30mpan2 422 . . . . . . . . . . . 12 (𝐴𝑉 → (2nd ‘⟨𝐴, 𝑦⟩) = 𝑦)
3229, 31sylan9eqr 2212 . . . . . . . . . . 11 ((𝐴𝑉𝑥 = ⟨𝐴, 𝑦⟩) → (2nd𝑥) = 𝑦)
3332adantrl 470 . . . . . . . . . 10 ((𝐴𝑉 ∧ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)) → (2nd𝑥) = 𝑦)
34 simprr 522 . . . . . . . . . . 11 ((𝐴𝑉 ∧ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)) → 𝑥 = ⟨𝐴, 𝑦⟩)
35 snidg 3589 . . . . . . . . . . . . 13 (𝐴𝑉𝐴 ∈ {𝐴})
3635adantr 274 . . . . . . . . . . . 12 ((𝐴𝑉 ∧ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)) → 𝐴 ∈ {𝐴})
37 simprl 521 . . . . . . . . . . . 12 ((𝐴𝑉 ∧ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)) → 𝑦𝐵)
38 opelxpi 4618 . . . . . . . . . . . 12 ((𝐴 ∈ {𝐴} ∧ 𝑦𝐵) → ⟨𝐴, 𝑦⟩ ∈ ({𝐴} × 𝐵))
3936, 37, 38syl2anc 409 . . . . . . . . . . 11 ((𝐴𝑉 ∧ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)) → ⟨𝐴, 𝑦⟩ ∈ ({𝐴} × 𝐵))
4034, 39eqeltrd 2234 . . . . . . . . . 10 ((𝐴𝑉 ∧ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)) → 𝑥 ∈ ({𝐴} × 𝐵))
4133, 40jca 304 . . . . . . . . 9 ((𝐴𝑉 ∧ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)) → ((2nd𝑥) = 𝑦𝑥 ∈ ({𝐴} × 𝐵)))
4228, 41impbida 586 . . . . . . . 8 (𝐴𝑉 → (((2nd𝑥) = 𝑦𝑥 ∈ ({𝐴} × 𝐵)) ↔ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)))
4314, 42bitr3id 193 . . . . . . 7 (𝐴𝑉 → ((𝑥2nd 𝑦𝑥 ∈ ({𝐴} × 𝐵)) ↔ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)))
447, 43syl5bb 191 . . . . . 6 (𝐴𝑉 → (𝑥(2nd ↾ ({𝐴} × 𝐵))𝑦 ↔ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)))
4544mobidv 2042 . . . . 5 (𝐴𝑉 → (∃*𝑥 𝑥(2nd ↾ ({𝐴} × 𝐵))𝑦 ↔ ∃*𝑥(𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)))
465, 45mpbiri 167 . . . 4 (𝐴𝑉 → ∃*𝑥 𝑥(2nd ↾ ({𝐴} × 𝐵))𝑦)
4746alrimiv 1854 . . 3 (𝐴𝑉 → ∀𝑦∃*𝑥 𝑥(2nd ↾ ({𝐴} × 𝐵))𝑦)
48 funcnv2 5230 . . 3 (Fun (2nd ↾ ({𝐴} × 𝐵)) ↔ ∀𝑦∃*𝑥 𝑥(2nd ↾ ({𝐴} × 𝐵))𝑦)
4947, 48sylibr 133 . 2 (𝐴𝑉 → Fun (2nd ↾ ({𝐴} × 𝐵)))
50 dff1o3 5420 . 2 ((2nd ↾ ({𝐴} × 𝐵)):({𝐴} × 𝐵)–1-1-onto𝐵 ↔ ((2nd ↾ ({𝐴} × 𝐵)):({𝐴} × 𝐵)–onto𝐵 ∧ Fun (2nd ↾ ({𝐴} × 𝐵))))
513, 49, 50sylanbrc 414 1 (𝐴𝑉 → (2nd ↾ ({𝐴} × 𝐵)):({𝐴} × 𝐵)–1-1-onto𝐵)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wb 104  wal 1333   = wceq 1335  wex 1472  ∃*wmo 2007  wcel 2128  Vcvv 2712  {csn 3560  cop 3563   class class class wbr 3965   × cxp 4584  ccnv 4585  cres 4588  Fun wfun 5164   Fn wfn 5165  ontowfo 5168  1-1-ontowf1o 5169  cfv 5170  1st c1st 6086  2nd c2nd 6087
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135  ax-pr 4169  ax-un 4393
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1338  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-rab 2444  df-v 2714  df-sbc 2938  df-csb 3032  df-un 3106  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-iun 3851  df-br 3966  df-opab 4026  df-mpt 4027  df-id 4253  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-f1 5175  df-fo 5176  df-f1o 5177  df-fv 5178  df-1st 6088  df-2nd 6089
This theorem is referenced by:  xpfi  6874  fsum2dlemstep  11331  fprod2dlemstep  11519
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