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Theorem fv2ndcnv 32905
Description: The value of the converse of 2nd restricted to a singleton. (Contributed by Scott Fenton, 2-Jul-2020.)
Assertion
Ref Expression
fv2ndcnv ((𝑋𝑉𝑌𝐴) → ((2nd ↾ ({𝑋} × 𝐴))‘𝑌) = ⟨𝑋, 𝑌⟩)

Proof of Theorem fv2ndcnv
StepHypRef Expression
1 snidg 4595 . . . 4 (𝑋𝑉𝑋 ∈ {𝑋})
21anim1i 614 . . 3 ((𝑋𝑉𝑌𝐴) → (𝑋 ∈ {𝑋} ∧ 𝑌𝐴))
3 eqid 2825 . . 3 𝑌 = 𝑌
42, 3jctir 521 . 2 ((𝑋𝑉𝑌𝐴) → ((𝑋 ∈ {𝑋} ∧ 𝑌𝐴) ∧ 𝑌 = 𝑌))
5 2ndconst 7790 . . . . . 6 (𝑋𝑉 → (2nd ↾ ({𝑋} × 𝐴)):({𝑋} × 𝐴)–1-1-onto𝐴)
65adantr 481 . . . . 5 ((𝑋𝑉𝑌𝐴) → (2nd ↾ ({𝑋} × 𝐴)):({𝑋} × 𝐴)–1-1-onto𝐴)
7 f1ocnv 6623 . . . . 5 ((2nd ↾ ({𝑋} × 𝐴)):({𝑋} × 𝐴)–1-1-onto𝐴(2nd ↾ ({𝑋} × 𝐴)):𝐴1-1-onto→({𝑋} × 𝐴))
8 f1ofn 6612 . . . . 5 ((2nd ↾ ({𝑋} × 𝐴)):𝐴1-1-onto→({𝑋} × 𝐴) → (2nd ↾ ({𝑋} × 𝐴)) Fn 𝐴)
96, 7, 83syl 18 . . . 4 ((𝑋𝑉𝑌𝐴) → (2nd ↾ ({𝑋} × 𝐴)) Fn 𝐴)
10 fnbrfvb 6714 . . . 4 (((2nd ↾ ({𝑋} × 𝐴)) Fn 𝐴𝑌𝐴) → (((2nd ↾ ({𝑋} × 𝐴))‘𝑌) = ⟨𝑋, 𝑌⟩ ↔ 𝑌(2nd ↾ ({𝑋} × 𝐴))⟨𝑋, 𝑌⟩))
119, 10sylancom 588 . . 3 ((𝑋𝑉𝑌𝐴) → (((2nd ↾ ({𝑋} × 𝐴))‘𝑌) = ⟨𝑋, 𝑌⟩ ↔ 𝑌(2nd ↾ ({𝑋} × 𝐴))⟨𝑋, 𝑌⟩))
12 opex 5352 . . . . . 6 𝑋, 𝑌⟩ ∈ V
13 brcnvg 5748 . . . . . 6 ((𝑌𝐴 ∧ ⟨𝑋, 𝑌⟩ ∈ V) → (𝑌(2nd ↾ ({𝑋} × 𝐴))⟨𝑋, 𝑌⟩ ↔ ⟨𝑋, 𝑌⟩(2nd ↾ ({𝑋} × 𝐴))𝑌))
1412, 13mpan2 687 . . . . 5 (𝑌𝐴 → (𝑌(2nd ↾ ({𝑋} × 𝐴))⟨𝑋, 𝑌⟩ ↔ ⟨𝑋, 𝑌⟩(2nd ↾ ({𝑋} × 𝐴))𝑌))
1514adantl 482 . . . 4 ((𝑋𝑉𝑌𝐴) → (𝑌(2nd ↾ ({𝑋} × 𝐴))⟨𝑋, 𝑌⟩ ↔ ⟨𝑋, 𝑌⟩(2nd ↾ ({𝑋} × 𝐴))𝑌))
16 brres 5858 . . . . . 6 (𝑌𝐴 → (⟨𝑋, 𝑌⟩(2nd ↾ ({𝑋} × 𝐴))𝑌 ↔ (⟨𝑋, 𝑌⟩ ∈ ({𝑋} × 𝐴) ∧ ⟨𝑋, 𝑌⟩2nd 𝑌)))
1716adantl 482 . . . . 5 ((𝑋𝑉𝑌𝐴) → (⟨𝑋, 𝑌⟩(2nd ↾ ({𝑋} × 𝐴))𝑌 ↔ (⟨𝑋, 𝑌⟩ ∈ ({𝑋} × 𝐴) ∧ ⟨𝑋, 𝑌⟩2nd 𝑌)))
18 opelxp 5589 . . . . . . 7 (⟨𝑋, 𝑌⟩ ∈ ({𝑋} × 𝐴) ↔ (𝑋 ∈ {𝑋} ∧ 𝑌𝐴))
1918anbi1i 623 . . . . . 6 ((⟨𝑋, 𝑌⟩ ∈ ({𝑋} × 𝐴) ∧ ⟨𝑋, 𝑌⟩2nd 𝑌) ↔ ((𝑋 ∈ {𝑋} ∧ 𝑌𝐴) ∧ ⟨𝑋, 𝑌⟩2nd 𝑌))
20 br2ndeqg 7706 . . . . . . 7 ((𝑋𝑉𝑌𝐴) → (⟨𝑋, 𝑌⟩2nd 𝑌𝑌 = 𝑌))
2120anbi2d 628 . . . . . 6 ((𝑋𝑉𝑌𝐴) → (((𝑋 ∈ {𝑋} ∧ 𝑌𝐴) ∧ ⟨𝑋, 𝑌⟩2nd 𝑌) ↔ ((𝑋 ∈ {𝑋} ∧ 𝑌𝐴) ∧ 𝑌 = 𝑌)))
2219, 21syl5bb 284 . . . . 5 ((𝑋𝑉𝑌𝐴) → ((⟨𝑋, 𝑌⟩ ∈ ({𝑋} × 𝐴) ∧ ⟨𝑋, 𝑌⟩2nd 𝑌) ↔ ((𝑋 ∈ {𝑋} ∧ 𝑌𝐴) ∧ 𝑌 = 𝑌)))
2317, 22bitrd 280 . . . 4 ((𝑋𝑉𝑌𝐴) → (⟨𝑋, 𝑌⟩(2nd ↾ ({𝑋} × 𝐴))𝑌 ↔ ((𝑋 ∈ {𝑋} ∧ 𝑌𝐴) ∧ 𝑌 = 𝑌)))
2415, 23bitrd 280 . . 3 ((𝑋𝑉𝑌𝐴) → (𝑌(2nd ↾ ({𝑋} × 𝐴))⟨𝑋, 𝑌⟩ ↔ ((𝑋 ∈ {𝑋} ∧ 𝑌𝐴) ∧ 𝑌 = 𝑌)))
2511, 24bitrd 280 . 2 ((𝑋𝑉𝑌𝐴) → (((2nd ↾ ({𝑋} × 𝐴))‘𝑌) = ⟨𝑋, 𝑌⟩ ↔ ((𝑋 ∈ {𝑋} ∧ 𝑌𝐴) ∧ 𝑌 = 𝑌)))
264, 25mpbird 258 1 ((𝑋𝑉𝑌𝐴) → ((2nd ↾ ({𝑋} × 𝐴))‘𝑌) = ⟨𝑋, 𝑌⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1530  wcel 2107  Vcvv 3499  {csn 4563  cop 4569   class class class wbr 5062   × cxp 5551  ccnv 5552  cres 5555   Fn wfn 6346  1-1-ontowf1o 6350  cfv 6351  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: (None)
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