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Theorem 2ndconst 7790
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 4708 . . 3 (𝐴𝑉 → {𝐴} ≠ ∅)
2 fo2ndres 7710 . . 3 ({𝐴} ≠ ∅ → (2nd ↾ ({𝐴} × 𝐵)):({𝐴} × 𝐵)–onto𝐵)
31, 2syl 17 . 2 (𝐴𝑉 → (2nd ↾ ({𝐴} × 𝐵)):({𝐴} × 𝐵)–onto𝐵)
4 moeq 3701 . . . . . 6 ∃*𝑥 𝑥 = ⟨𝐴, 𝑦
54moani 2634 . . . . 5 ∃*𝑥(𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)
6 vex 3502 . . . . . . . 8 𝑦 ∈ V
76brresi 5860 . . . . . . 7 (𝑥(2nd ↾ ({𝐴} × 𝐵))𝑦 ↔ (𝑥 ∈ ({𝐴} × 𝐵) ∧ 𝑥2nd 𝑦))
8 fo2nd 7704 . . . . . . . . . . 11 2nd :V–onto→V
9 fofn 6588 . . . . . . . . . . 11 (2nd :V–onto→V → 2nd Fn V)
108, 9ax-mp 5 . . . . . . . . . 10 2nd Fn V
11 vex 3502 . . . . . . . . . 10 𝑥 ∈ V
12 fnbrfvb 6714 . . . . . . . . . 10 ((2nd Fn V ∧ 𝑥 ∈ V) → ((2nd𝑥) = 𝑦𝑥2nd 𝑦))
1310, 11, 12mp2an 688 . . . . . . . . 9 ((2nd𝑥) = 𝑦𝑥2nd 𝑦)
1413anbi2i 622 . . . . . . . 8 ((𝑥 ∈ ({𝐴} × 𝐵) ∧ (2nd𝑥) = 𝑦) ↔ (𝑥 ∈ ({𝐴} × 𝐵) ∧ 𝑥2nd 𝑦))
15 elxp7 7718 . . . . . . . . . . 11 (𝑥 ∈ ({𝐴} × 𝐵) ↔ (𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ {𝐴} ∧ (2nd𝑥) ∈ 𝐵)))
16 eleq1 2904 . . . . . . . . . . . . . . 15 ((2nd𝑥) = 𝑦 → ((2nd𝑥) ∈ 𝐵𝑦𝐵))
1716biimpac 479 . . . . . . . . . . . . . 14 (((2nd𝑥) ∈ 𝐵 ∧ (2nd𝑥) = 𝑦) → 𝑦𝐵)
1817adantll 710 . . . . . . . . . . . . 13 ((((1st𝑥) ∈ {𝐴} ∧ (2nd𝑥) ∈ 𝐵) ∧ (2nd𝑥) = 𝑦) → 𝑦𝐵)
1918adantll 710 . . . . . . . . . . . 12 (((𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ {𝐴} ∧ (2nd𝑥) ∈ 𝐵)) ∧ (2nd𝑥) = 𝑦) → 𝑦𝐵)
20 elsni 4580 . . . . . . . . . . . . . 14 ((1st𝑥) ∈ {𝐴} → (1st𝑥) = 𝐴)
21 eqopi 7719 . . . . . . . . . . . . . . 15 ((𝑥 ∈ (V × V) ∧ ((1st𝑥) = 𝐴 ∧ (2nd𝑥) = 𝑦)) → 𝑥 = ⟨𝐴, 𝑦⟩)
2221anassrs 468 . . . . . . . . . . . . . 14 (((𝑥 ∈ (V × V) ∧ (1st𝑥) = 𝐴) ∧ (2nd𝑥) = 𝑦) → 𝑥 = ⟨𝐴, 𝑦⟩)
2320, 22sylanl2 677 . . . . . . . . . . . . 13 (((𝑥 ∈ (V × V) ∧ (1st𝑥) ∈ {𝐴}) ∧ (2nd𝑥) = 𝑦) → 𝑥 = ⟨𝐴, 𝑦⟩)
2423adantlrr 717 . . . . . . . . . . . 12 (((𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ {𝐴} ∧ (2nd𝑥) ∈ 𝐵)) ∧ (2nd𝑥) = 𝑦) → 𝑥 = ⟨𝐴, 𝑦⟩)
2519, 24jca 512 . . . . . . . . . . 11 (((𝑥 ∈ (V × V) ∧ ((1st𝑥) ∈ {𝐴} ∧ (2nd𝑥) ∈ 𝐵)) ∧ (2nd𝑥) = 𝑦) → (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩))
2615, 25sylanb 581 . . . . . . . . . 10 ((𝑥 ∈ ({𝐴} × 𝐵) ∧ (2nd𝑥) = 𝑦) → (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩))
2726adantl 482 . . . . . . . . 9 ((𝐴𝑉 ∧ (𝑥 ∈ ({𝐴} × 𝐵) ∧ (2nd𝑥) = 𝑦)) → (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩))
28 simprr 769 . . . . . . . . . . 11 ((𝐴𝑉 ∧ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)) → 𝑥 = ⟨𝐴, 𝑦⟩)
29 snidg 4595 . . . . . . . . . . . . 13 (𝐴𝑉𝐴 ∈ {𝐴})
3029adantr 481 . . . . . . . . . . . 12 ((𝐴𝑉 ∧ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)) → 𝐴 ∈ {𝐴})
31 simprl 767 . . . . . . . . . . . 12 ((𝐴𝑉 ∧ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)) → 𝑦𝐵)
3230, 31opelxpd 5591 . . . . . . . . . . 11 ((𝐴𝑉 ∧ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)) → ⟨𝐴, 𝑦⟩ ∈ ({𝐴} × 𝐵))
3328, 32eqeltrd 2917 . . . . . . . . . 10 ((𝐴𝑉 ∧ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)) → 𝑥 ∈ ({𝐴} × 𝐵))
34 fveq2 6666 . . . . . . . . . . . 12 (𝑥 = ⟨𝐴, 𝑦⟩ → (2nd𝑥) = (2nd ‘⟨𝐴, 𝑦⟩))
35 op2ndg 7696 . . . . . . . . . . . . 13 ((𝐴𝑉𝑦 ∈ V) → (2nd ‘⟨𝐴, 𝑦⟩) = 𝑦)
3635elvd 3505 . . . . . . . . . . . 12 (𝐴𝑉 → (2nd ‘⟨𝐴, 𝑦⟩) = 𝑦)
3734, 36sylan9eqr 2882 . . . . . . . . . . 11 ((𝐴𝑉𝑥 = ⟨𝐴, 𝑦⟩) → (2nd𝑥) = 𝑦)
3837adantrl 712 . . . . . . . . . 10 ((𝐴𝑉 ∧ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)) → (2nd𝑥) = 𝑦)
3933, 38jca 512 . . . . . . . . 9 ((𝐴𝑉 ∧ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)) → (𝑥 ∈ ({𝐴} × 𝐵) ∧ (2nd𝑥) = 𝑦))
4027, 39impbida 797 . . . . . . . 8 (𝐴𝑉 → ((𝑥 ∈ ({𝐴} × 𝐵) ∧ (2nd𝑥) = 𝑦) ↔ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)))
4114, 40syl5bbr 286 . . . . . . 7 (𝐴𝑉 → ((𝑥 ∈ ({𝐴} × 𝐵) ∧ 𝑥2nd 𝑦) ↔ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)))
427, 41syl5bb 284 . . . . . 6 (𝐴𝑉 → (𝑥(2nd ↾ ({𝐴} × 𝐵))𝑦 ↔ (𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)))
4342mobidv 2630 . . . . 5 (𝐴𝑉 → (∃*𝑥 𝑥(2nd ↾ ({𝐴} × 𝐵))𝑦 ↔ ∃*𝑥(𝑦𝐵𝑥 = ⟨𝐴, 𝑦⟩)))
445, 43mpbiri 259 . . . 4 (𝐴𝑉 → ∃*𝑥 𝑥(2nd ↾ ({𝐴} × 𝐵))𝑦)
4544alrimiv 1921 . . 3 (𝐴𝑉 → ∀𝑦∃*𝑥 𝑥(2nd ↾ ({𝐴} × 𝐵))𝑦)
46 funcnv2 6418 . . 3 (Fun (2nd ↾ ({𝐴} × 𝐵)) ↔ ∀𝑦∃*𝑥 𝑥(2nd ↾ ({𝐴} × 𝐵))𝑦)
4745, 46sylibr 235 . 2 (𝐴𝑉 → Fun (2nd ↾ ({𝐴} × 𝐵)))
48 dff1o3 6617 . 2 ((2nd ↾ ({𝐴} × 𝐵)):({𝐴} × 𝐵)–1-1-onto𝐵 ↔ ((2nd ↾ ({𝐴} × 𝐵)):({𝐴} × 𝐵)–onto𝐵 ∧ Fun (2nd ↾ ({𝐴} × 𝐵))))
493, 47, 48sylanbrc 583 1 (𝐴𝑉 → (2nd ↾ ({𝐴} × 𝐵)):({𝐴} × 𝐵)–1-1-onto𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396  wal 1528   = wceq 1530  wcel 2107  ∃*wmo 2617  wne 3020  Vcvv 3499  c0 4294  {csn 4563  cop 4569   class class class wbr 5062   × cxp 5551  ccnv 5552  cres 5555  Fun wfun 6345   Fn wfn 6346  ontowfo 6349  1-1-ontowf1o 6350  cfv 6351  1st c1st 7681  2nd c2nd 7682
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2797  ax-sep 5199  ax-nul 5206  ax-pow 5262  ax-pr 5325  ax-un 7454
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2619  df-eu 2651  df-clab 2804  df-cleq 2818  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-rab 3151  df-v 3501  df-sbc 3776  df-csb 3887  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-uni 4837  df-iun 4918  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-res 5565  df-ima 5566  df-iota 6311  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358  df-fv 6359  df-1st 7683  df-2nd 7684
This theorem is referenced by:  curry1  7793  xpfi  8781  fsum2dlem  15117  fprod2dlem  15326  gsum2dlem2  19013  ovoliunlem1  24020  gsummpt2d  30603  fv2ndcnv  32907
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