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Theorem 2ndconst 6119
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 3641 . . 3 (𝐴𝑉 → ∃𝑥 𝑥 ∈ {𝐴})
2 fo2ndresm 6060 . . 3 (∃𝑥 𝑥 ∈ {𝐴} → (2nd ↾ ({𝐴} × 𝐵)):({𝐴} × 𝐵)–onto𝐵)
31, 2syl 14 . 2 (𝐴𝑉 → (2nd ↾ ({𝐴} × 𝐵)):({𝐴} × 𝐵)–onto𝐵)
4 moeq 2859 . . . . . 6 ∃*𝑥 𝑥 = ⟨𝐴, 𝑦
54moani 2069 . . . . 5 ∃*𝑥(𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)
6 vex 2689 . . . . . . . 8 𝑦 ∈ V
76brres 4825 . . . . . . 7 (𝑥(2nd ↾ ({𝐴} × 𝐵))𝑦 ↔ (𝑥2nd 𝑦𝑥 ∈ ({𝐴} × 𝐵)))
8 fo2nd 6056 . . . . . . . . . . 11 2nd :V–onto→V
9 fofn 5347 . . . . . . . . . . 11 (2nd :V–onto→V → 2nd Fn V)
108, 9ax-mp 5 . . . . . . . . . 10 2nd Fn V
11 vex 2689 . . . . . . . . . 10 𝑥 ∈ V
12 fnbrfvb 5462 . . . . . . . . . 10 ((2nd Fn V ∧ 𝑥 ∈ V) → ((2nd𝑥) = 𝑦𝑥2nd 𝑦))
1310, 11, 12mp2an 422 . . . . . . . . 9 ((2nd𝑥) = 𝑦𝑥2nd 𝑦)
1413anbi1i 453 . . . . . . . 8 (((2nd𝑥) = 𝑦𝑥 ∈ ({𝐴} × 𝐵)) ↔ (𝑥2nd 𝑦𝑥 ∈ ({𝐴} × 𝐵)))
15 elxp7 6068 . . . . . . . . . . 11 (𝑥 ∈ ({𝐴} × 𝐵) ↔ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ {𝐴} ∧ (2nd𝑥) ∈ 𝐵)))
16 eleq1 2202 . . . . . . . . . . . . . . 15 ((2nd𝑥) = 𝑦 → ((2nd𝑥) ∈ 𝐵𝑦𝐵))
1716biimpa 294 . . . . . . . . . . . . . 14 (((2nd𝑥) = 𝑦 ∧ (2nd𝑥) ∈ 𝐵) → 𝑦𝐵)
1817adantrl 469 . . . . . . . . . . . . 13 (((2nd𝑥) = 𝑦 ∧ ((1st𝑥) ∈ {𝐴} ∧ (2nd𝑥) ∈ 𝐵)) → 𝑦𝐵)
1918adantrl 469 . . . . . . . . . . . 12 (((2nd𝑥) = 𝑦 ∧ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ {𝐴} ∧ (2nd𝑥) ∈ 𝐵))) → 𝑦𝐵)
20 elsni 3545 . . . . . . . . . . . . . 14 ((1st𝑥) ∈ {𝐴} → (1st𝑥) = 𝐴)
21 eqopi 6070 . . . . . . . . . . . . . . . 16 ((𝑥 ∈ (V × V) ∧ ((1st𝑥) = 𝐴 ∧ (2nd𝑥) = 𝑦)) → 𝑥 = ⟨𝐴, 𝑦⟩)
2221ancom2s 555 . . . . . . . . . . . . . . 15 ((𝑥 ∈ (V × V) ∧ ((2nd𝑥) = 𝑦 ∧ (1st𝑥) = 𝐴)) → 𝑥 = ⟨𝐴, 𝑦⟩)
2322an12s 554 . . . . . . . . . . . . . 14 (((2nd𝑥) = 𝑦 ∧ (𝑥 ∈ (V × V) ∧ (1st𝑥) = 𝐴)) → 𝑥 = ⟨𝐴, 𝑦⟩)
2420, 23sylanr2 402 . . . . . . . . . . . . 13 (((2nd𝑥) = 𝑦 ∧ (𝑥 ∈ (V × V) ∧ (1st𝑥) ∈ {𝐴})) → 𝑥 = ⟨𝐴, 𝑦⟩)
2524adantrrr 478 . . . . . . . . . . . 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 5421 . . . . . . . . . . . 12 (𝑥 = ⟨𝐴, 𝑦⟩ → (2nd𝑥) = (2nd ‘⟨𝐴, 𝑦⟩))
30 op2ndg 6049 . . . . . . . . . . . . 13 ((𝐴𝑉𝑦 ∈ V) → (2nd ‘⟨𝐴, 𝑦⟩) = 𝑦)
316, 30mpan2 421 . . . . . . . . . . . 12 (𝐴𝑉 → (2nd ‘⟨𝐴, 𝑦⟩) = 𝑦)
3229, 31sylan9eqr 2194 . . . . . . . . . . 11 ((𝐴𝑉𝑥 = ⟨𝐴, 𝑦⟩) → (2nd𝑥) = 𝑦)
3332adantrl 469 . . . . . . . . . 10 ((𝐴𝑉 ∧ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)) → (2nd𝑥) = 𝑦)
34 simprr 521 . . . . . . . . . . 11 ((𝐴𝑉 ∧ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)) → 𝑥 = ⟨𝐴, 𝑦⟩)
35 snidg 3554 . . . . . . . . . . . . 13 (𝐴𝑉𝐴 ∈ {𝐴})
3635adantr 274 . . . . . . . . . . . 12 ((𝐴𝑉 ∧ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)) → 𝐴 ∈ {𝐴})
37 simprl 520 . . . . . . . . . . . 12 ((𝐴𝑉 ∧ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)) → 𝑦𝐵)
38 opelxpi 4571 . . . . . . . . . . . 12 ((𝐴 ∈ {𝐴} ∧ 𝑦𝐵) → ⟨𝐴, 𝑦⟩ ∈ ({𝐴} × 𝐵))
3936, 37, 38syl2anc 408 . . . . . . . . . . 11 ((𝐴𝑉 ∧ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)) → ⟨𝐴, 𝑦⟩ ∈ ({𝐴} × 𝐵))
4034, 39eqeltrd 2216 . . . . . . . . . 10 ((𝐴𝑉 ∧ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)) → 𝑥 ∈ ({𝐴} × 𝐵))
4133, 40jca 304 . . . . . . . . 9 ((𝐴𝑉 ∧ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)) → ((2nd𝑥) = 𝑦𝑥 ∈ ({𝐴} × 𝐵)))
4228, 41impbida 585 . . . . . . . 8 (𝐴𝑉 → (((2nd𝑥) = 𝑦𝑥 ∈ ({𝐴} × 𝐵)) ↔ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)))
4314, 42syl5bbr 193 . . . . . . 7 (𝐴𝑉 → ((𝑥2nd 𝑦𝑥 ∈ ({𝐴} × 𝐵)) ↔ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)))
447, 43syl5bb 191 . . . . . 6 (𝐴𝑉 → (𝑥(2nd ↾ ({𝐴} × 𝐵))𝑦 ↔ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)))
4544mobidv 2035 . . . . 5 (𝐴𝑉 → (∃*𝑥 𝑥(2nd ↾ ({𝐴} × 𝐵))𝑦 ↔ ∃*𝑥(𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)))
465, 45mpbiri 167 . . . 4 (𝐴𝑉 → ∃*𝑥 𝑥(2nd ↾ ({𝐴} × 𝐵))𝑦)
4746alrimiv 1846 . . 3 (𝐴𝑉 → ∀𝑦∃*𝑥 𝑥(2nd ↾ ({𝐴} × 𝐵))𝑦)
48 funcnv2 5183 . . 3 (Fun (2nd ↾ ({𝐴} × 𝐵)) ↔ ∀𝑦∃*𝑥 𝑥(2nd ↾ ({𝐴} × 𝐵))𝑦)
4947, 48sylibr 133 . 2 (𝐴𝑉 → Fun (2nd ↾ ({𝐴} × 𝐵)))
50 dff1o3 5373 . 2 ((2nd ↾ ({𝐴} × 𝐵)):({𝐴} × 𝐵)–1-1-onto𝐵 ↔ ((2nd ↾ ({𝐴} × 𝐵)):({𝐴} × 𝐵)–onto𝐵 ∧ Fun (2nd ↾ ({𝐴} × 𝐵))))
513, 49, 50sylanbrc 413 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 1329   = wceq 1331  wex 1468  wcel 1480  ∃*wmo 2000  Vcvv 2686  {csn 3527  cop 3530   class class class wbr 3929   × cxp 4537  ccnv 4538  cres 4541  Fun wfun 5117   Fn wfn 5118  ontowfo 5121  1-1-ontowf1o 5122  cfv 5123  1st c1st 6036  2nd c2nd 6037
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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-1st 6038  df-2nd 6039
This theorem is referenced by:  xpfi  6818  fsum2dlemstep  11203
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