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Theorem fv2ndcnv 35282
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 4657 . . . 4 (𝑋𝑉𝑋 ∈ {𝑋})
21anim1i 614 . . 3 ((𝑋𝑉𝑌𝐴) → (𝑋 ∈ {𝑋} ∧ 𝑌𝐴))
3 eqid 2726 . . 3 𝑌 = 𝑌
42, 3jctir 520 . 2 ((𝑋𝑉𝑌𝐴) → ((𝑋 ∈ {𝑋} ∧ 𝑌𝐴) ∧ 𝑌 = 𝑌))
5 2ndconst 8087 . . . . . 6 (𝑋𝑉 → (2nd ↾ ({𝑋} × 𝐴)):({𝑋} × 𝐴)–1-1-onto𝐴)
65adantr 480 . . . . 5 ((𝑋𝑉𝑌𝐴) → (2nd ↾ ({𝑋} × 𝐴)):({𝑋} × 𝐴)–1-1-onto𝐴)
7 f1ocnv 6839 . . . . 5 ((2nd ↾ ({𝑋} × 𝐴)):({𝑋} × 𝐴)–1-1-onto𝐴(2nd ↾ ({𝑋} × 𝐴)):𝐴1-1-onto→({𝑋} × 𝐴))
8 f1ofn 6828 . . . . 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 6938 . . . 4 (((2nd ↾ ({𝑋} × 𝐴)) Fn 𝐴𝑌𝐴) → (((2nd ↾ ({𝑋} × 𝐴))‘𝑌) = ⟨𝑋, 𝑌⟩ ↔ 𝑌(2nd ↾ ({𝑋} × 𝐴))⟨𝑋, 𝑌⟩))
119, 10sylancom 587 . . 3 ((𝑋𝑉𝑌𝐴) → (((2nd ↾ ({𝑋} × 𝐴))‘𝑌) = ⟨𝑋, 𝑌⟩ ↔ 𝑌(2nd ↾ ({𝑋} × 𝐴))⟨𝑋, 𝑌⟩))
12 opex 5457 . . . . . 6 𝑋, 𝑌⟩ ∈ V
13 brcnvg 5873 . . . . . 6 ((𝑌𝐴 ∧ ⟨𝑋, 𝑌⟩ ∈ V) → (𝑌(2nd ↾ ({𝑋} × 𝐴))⟨𝑋, 𝑌⟩ ↔ ⟨𝑋, 𝑌⟩(2nd ↾ ({𝑋} × 𝐴))𝑌))
1412, 13mpan2 688 . . . . 5 (𝑌𝐴 → (𝑌(2nd ↾ ({𝑋} × 𝐴))⟨𝑋, 𝑌⟩ ↔ ⟨𝑋, 𝑌⟩(2nd ↾ ({𝑋} × 𝐴))𝑌))
1514adantl 481 . . . 4 ((𝑋𝑉𝑌𝐴) → (𝑌(2nd ↾ ({𝑋} × 𝐴))⟨𝑋, 𝑌⟩ ↔ ⟨𝑋, 𝑌⟩(2nd ↾ ({𝑋} × 𝐴))𝑌))
16 brres 5982 . . . . . 6 (𝑌𝐴 → (⟨𝑋, 𝑌⟩(2nd ↾ ({𝑋} × 𝐴))𝑌 ↔ (⟨𝑋, 𝑌⟩ ∈ ({𝑋} × 𝐴) ∧ ⟨𝑋, 𝑌⟩2nd 𝑌)))
1716adantl 481 . . . . 5 ((𝑋𝑉𝑌𝐴) → (⟨𝑋, 𝑌⟩(2nd ↾ ({𝑋} × 𝐴))𝑌 ↔ (⟨𝑋, 𝑌⟩ ∈ ({𝑋} × 𝐴) ∧ ⟨𝑋, 𝑌⟩2nd 𝑌)))
18 opelxp 5705 . . . . . . 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 7997 . . . . . . 7 ((𝑋𝑉𝑌𝐴) → (⟨𝑋, 𝑌⟩2nd 𝑌𝑌 = 𝑌))
2120anbi2d 628 . . . . . 6 ((𝑋𝑉𝑌𝐴) → (((𝑋 ∈ {𝑋} ∧ 𝑌𝐴) ∧ ⟨𝑋, 𝑌⟩2nd 𝑌) ↔ ((𝑋 ∈ {𝑋} ∧ 𝑌𝐴) ∧ 𝑌 = 𝑌)))
2219, 21bitrid 283 . . . . 5 ((𝑋𝑉𝑌𝐴) → ((⟨𝑋, 𝑌⟩ ∈ ({𝑋} × 𝐴) ∧ ⟨𝑋, 𝑌⟩2nd 𝑌) ↔ ((𝑋 ∈ {𝑋} ∧ 𝑌𝐴) ∧ 𝑌 = 𝑌)))
2317, 22bitrd 279 . . . 4 ((𝑋𝑉𝑌𝐴) → (⟨𝑋, 𝑌⟩(2nd ↾ ({𝑋} × 𝐴))𝑌 ↔ ((𝑋 ∈ {𝑋} ∧ 𝑌𝐴) ∧ 𝑌 = 𝑌)))
2415, 23bitrd 279 . . 3 ((𝑋𝑉𝑌𝐴) → (𝑌(2nd ↾ ({𝑋} × 𝐴))⟨𝑋, 𝑌⟩ ↔ ((𝑋 ∈ {𝑋} ∧ 𝑌𝐴) ∧ 𝑌 = 𝑌)))
2511, 24bitrd 279 . 2 ((𝑋𝑉𝑌𝐴) → (((2nd ↾ ({𝑋} × 𝐴))‘𝑌) = ⟨𝑋, 𝑌⟩ ↔ ((𝑋 ∈ {𝑋} ∧ 𝑌𝐴) ∧ 𝑌 = 𝑌)))
264, 25mpbird 257 1 ((𝑋𝑉𝑌𝐴) → ((2nd ↾ ({𝑋} × 𝐴))‘𝑌) = ⟨𝑋, 𝑌⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 395   = wceq 1533  wcel 2098  Vcvv 3468  {csn 4623  cop 4629   class class class wbr 5141   × cxp 5667  ccnv 5668  cres 5671   Fn wfn 6532  1-1-ontowf1o 6536  cfv 6537  2nd c2nd 7973
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-1st 7974  df-2nd 7975
This theorem is referenced by: (None)
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