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Theorem 2ndconst 8142
Description: The mapping of a restriction of the 2nd function to a converse constant function. (Contributed by NM, 27-Mar-2008.) (Proof shortened by Peter Mazsa, 2-Oct-2022.)
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 snnzg 4799 . . 3 (𝐴𝑉 → {𝐴} ≠ ∅)
2 fo2ndres 8057 . . 3 ({𝐴} ≠ ∅ → (2nd ↾ ({𝐴} × 𝐵)):({𝐴} × 𝐵)–onto𝐵)
31, 2syl 17 . 2 (𝐴𝑉 → (2nd ↾ ({𝐴} × 𝐵)):({𝐴} × 𝐵)–onto𝐵)
4 moeq 3729 . . . . . 6 ∃*𝑥 𝑥 = ⟨𝐴, 𝑦
54moani 2556 . . . . 5 ∃*𝑥(𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)
6 vex 3492 . . . . . . . 8 𝑦 ∈ V
76brresi 6018 . . . . . . 7 (𝑥(2nd ↾ ({𝐴} × 𝐵))𝑦 ↔ (𝑥 ∈ ({𝐴} × 𝐵) ∧ 𝑥2nd 𝑦))
8 fo2nd 8051 . . . . . . . . . . 11 2nd :V–onto→V
9 fofn 6836 . . . . . . . . . . 11 (2nd :V–onto→V → 2nd Fn V)
108, 9ax-mp 5 . . . . . . . . . 10 2nd Fn V
11 vex 3492 . . . . . . . . . 10 𝑥 ∈ V
12 fnbrfvb 6973 . . . . . . . . . 10 ((2nd Fn V ∧ 𝑥 ∈ V) → ((2nd𝑥) = 𝑦𝑥2nd 𝑦))
1310, 11, 12mp2an 691 . . . . . . . . 9 ((2nd𝑥) = 𝑦𝑥2nd 𝑦)
1413anbi2i 622 . . . . . . . 8 ((𝑥 ∈ ({𝐴} × 𝐵) ∧ (2nd𝑥) = 𝑦) ↔ (𝑥 ∈ ({𝐴} × 𝐵) ∧ 𝑥2nd 𝑦))
15 elxp7 8065 . . . . . . . . . . 11 (𝑥 ∈ ({𝐴} × 𝐵) ↔ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ {𝐴} ∧ (2nd𝑥) ∈ 𝐵)))
16 eleq1 2832 . . . . . . . . . . . . . . 15 ((2nd𝑥) = 𝑦 → ((2nd𝑥) ∈ 𝐵𝑦𝐵))
1716biimpac 478 . . . . . . . . . . . . . 14 (((2nd𝑥) ∈ 𝐵 ∧ (2nd𝑥) = 𝑦) → 𝑦𝐵)
1817adantll 713 . . . . . . . . . . . . 13 ((((1st𝑥) ∈ {𝐴} ∧ (2nd𝑥) ∈ 𝐵) ∧ (2nd𝑥) = 𝑦) → 𝑦𝐵)
1918adantll 713 . . . . . . . . . . . 12 (((𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ {𝐴} ∧ (2nd𝑥) ∈ 𝐵)) ∧ (2nd𝑥) = 𝑦) → 𝑦𝐵)
20 elsni 4665 . . . . . . . . . . . . . 14 ((1st𝑥) ∈ {𝐴} → (1st𝑥) = 𝐴)
21 eqopi 8066 . . . . . . . . . . . . . . 15 ((𝑥 ∈ (V × V) ∧ ((1st𝑥) = 𝐴 ∧ (2nd𝑥) = 𝑦)) → 𝑥 = ⟨𝐴, 𝑦⟩)
2221anassrs 467 . . . . . . . . . . . . . 14 (((𝑥 ∈ (V × V) ∧ (1st𝑥) = 𝐴) ∧ (2nd𝑥) = 𝑦) → 𝑥 = ⟨𝐴, 𝑦⟩)
2320, 22sylanl2 680 . . . . . . . . . . . . 13 (((𝑥 ∈ (V × V) ∧ (1st𝑥) ∈ {𝐴}) ∧ (2nd𝑥) = 𝑦) → 𝑥 = ⟨𝐴, 𝑦⟩)
2423adantlrr 720 . . . . . . . . . . . 12 (((𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ {𝐴} ∧ (2nd𝑥) ∈ 𝐵)) ∧ (2nd𝑥) = 𝑦) → 𝑥 = ⟨𝐴, 𝑦⟩)
2519, 24jca 511 . . . . . . . . . . 11 (((𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ {𝐴} ∧ (2nd𝑥) ∈ 𝐵)) ∧ (2nd𝑥) = 𝑦) → (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩))
2615, 25sylanb 580 . . . . . . . . . 10 ((𝑥 ∈ ({𝐴} × 𝐵) ∧ (2nd𝑥) = 𝑦) → (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩))
2726adantl 481 . . . . . . . . 9 ((𝐴𝑉 ∧ (𝑥 ∈ ({𝐴} × 𝐵) ∧ (2nd𝑥) = 𝑦)) → (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩))
28 simprr 772 . . . . . . . . . . 11 ((𝐴𝑉 ∧ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)) → 𝑥 = ⟨𝐴, 𝑦⟩)
29 snidg 4682 . . . . . . . . . . . . 13 (𝐴𝑉𝐴 ∈ {𝐴})
3029adantr 480 . . . . . . . . . . . 12 ((𝐴𝑉 ∧ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)) → 𝐴 ∈ {𝐴})
31 simprl 770 . . . . . . . . . . . 12 ((𝐴𝑉 ∧ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)) → 𝑦𝐵)
3230, 31opelxpd 5739 . . . . . . . . . . 11 ((𝐴𝑉 ∧ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)) → ⟨𝐴, 𝑦⟩ ∈ ({𝐴} × 𝐵))
3328, 32eqeltrd 2844 . . . . . . . . . 10 ((𝐴𝑉 ∧ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)) → 𝑥 ∈ ({𝐴} × 𝐵))
34 fveq2 6920 . . . . . . . . . . . 12 (𝑥 = ⟨𝐴, 𝑦⟩ → (2nd𝑥) = (2nd ‘⟨𝐴, 𝑦⟩))
35 op2ndg 8043 . . . . . . . . . . . . 13 ((𝐴𝑉𝑦 ∈ V) → (2nd ‘⟨𝐴, 𝑦⟩) = 𝑦)
3635elvd 3494 . . . . . . . . . . . 12 (𝐴𝑉 → (2nd ‘⟨𝐴, 𝑦⟩) = 𝑦)
3734, 36sylan9eqr 2802 . . . . . . . . . . 11 ((𝐴𝑉𝑥 = ⟨𝐴, 𝑦⟩) → (2nd𝑥) = 𝑦)
3837adantrl 715 . . . . . . . . . 10 ((𝐴𝑉 ∧ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)) → (2nd𝑥) = 𝑦)
3933, 38jca 511 . . . . . . . . 9 ((𝐴𝑉 ∧ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)) → (𝑥 ∈ ({𝐴} × 𝐵) ∧ (2nd𝑥) = 𝑦))
4027, 39impbida 800 . . . . . . . 8 (𝐴𝑉 → ((𝑥 ∈ ({𝐴} × 𝐵) ∧ (2nd𝑥) = 𝑦) ↔ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)))
4114, 40bitr3id 285 . . . . . . 7 (𝐴𝑉 → ((𝑥 ∈ ({𝐴} × 𝐵) ∧ 𝑥2nd 𝑦) ↔ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)))
427, 41bitrid 283 . . . . . 6 (𝐴𝑉 → (𝑥(2nd ↾ ({𝐴} × 𝐵))𝑦 ↔ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)))
4342mobidv 2552 . . . . 5 (𝐴𝑉 → (∃*𝑥 𝑥(2nd ↾ ({𝐴} × 𝐵))𝑦 ↔ ∃*𝑥(𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)))
445, 43mpbiri 258 . . . 4 (𝐴𝑉 → ∃*𝑥 𝑥(2nd ↾ ({𝐴} × 𝐵))𝑦)
4544alrimiv 1926 . . 3 (𝐴𝑉 → ∀𝑦∃*𝑥 𝑥(2nd ↾ ({𝐴} × 𝐵))𝑦)
46 funcnv2 6646 . . 3 (Fun (2nd ↾ ({𝐴} × 𝐵)) ↔ ∀𝑦∃*𝑥 𝑥(2nd ↾ ({𝐴} × 𝐵))𝑦)
4745, 46sylibr 234 . 2 (𝐴𝑉 → Fun (2nd ↾ ({𝐴} × 𝐵)))
48 dff1o3 6868 . 2 ((2nd ↾ ({𝐴} × 𝐵)):({𝐴} × 𝐵)–1-1-onto𝐵 ↔ ((2nd ↾ ({𝐴} × 𝐵)):({𝐴} × 𝐵)–onto𝐵 ∧ Fun (2nd ↾ ({𝐴} × 𝐵))))
493, 47, 48sylanbrc 582 1 (𝐴𝑉 → (2nd ↾ ({𝐴} × 𝐵)):({𝐴} × 𝐵)–1-1-onto𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395  wal 1535   = wceq 1537  wcel 2108  ∃*wmo 2541  wne 2946  Vcvv 3488  c0 4352  {csn 4648  cop 4654   class class class wbr 5166   × cxp 5698  ccnv 5699  cres 5702  Fun wfun 6567   Fn wfn 6568  ontowfo 6571  1-1-ontowf1o 6572  cfv 6573  1st c1st 8028  2nd c2nd 8029
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447  ax-un 7770
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-1st 8030  df-2nd 8031
This theorem is referenced by:  curry1  8145  xpfiOLD  9387  fsum2dlem  15818  fprod2dlem  16028  gsum2dlem2  20013  ovoliunlem1  25556  gsummpt2d  33032  fv2ndcnv  35741
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