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Theorem fv2ndcnv 31665
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 4204 . . . 4 (𝑋𝑉𝑋 ∈ {𝑋})
21anim1i 592 . . 3 ((𝑋𝑉𝑌𝐴) → (𝑋 ∈ {𝑋} ∧ 𝑌𝐴))
3 eqid 2621 . . 3 𝑌 = 𝑌
42, 3jctil 560 . 2 ((𝑋𝑉𝑌𝐴) → (𝑌 = 𝑌 ∧ (𝑋 ∈ {𝑋} ∧ 𝑌𝐴)))
5 2ndconst 7263 . . . . . 6 (𝑋𝑉 → (2nd ↾ ({𝑋} × 𝐴)):({𝑋} × 𝐴)–1-1-onto𝐴)
65adantr 481 . . . . 5 ((𝑋𝑉𝑌𝐴) → (2nd ↾ ({𝑋} × 𝐴)):({𝑋} × 𝐴)–1-1-onto𝐴)
7 f1ocnv 6147 . . . . 5 ((2nd ↾ ({𝑋} × 𝐴)):({𝑋} × 𝐴)–1-1-onto𝐴(2nd ↾ ({𝑋} × 𝐴)):𝐴1-1-onto→({𝑋} × 𝐴))
8 f1ofn 6136 . . . . 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 6234 . . . 4 (((2nd ↾ ({𝑋} × 𝐴)) Fn 𝐴𝑌𝐴) → (((2nd ↾ ({𝑋} × 𝐴))‘𝑌) = ⟨𝑋, 𝑌⟩ ↔ 𝑌(2nd ↾ ({𝑋} × 𝐴))⟨𝑋, 𝑌⟩))
119, 10sylancom 701 . . 3 ((𝑋𝑉𝑌𝐴) → (((2nd ↾ ({𝑋} × 𝐴))‘𝑌) = ⟨𝑋, 𝑌⟩ ↔ 𝑌(2nd ↾ ({𝑋} × 𝐴))⟨𝑋, 𝑌⟩))
12 opex 4930 . . . . . 6 𝑋, 𝑌⟩ ∈ V
13 brcnvg 5301 . . . . . 6 ((𝑌𝐴 ∧ ⟨𝑋, 𝑌⟩ ∈ V) → (𝑌(2nd ↾ ({𝑋} × 𝐴))⟨𝑋, 𝑌⟩ ↔ ⟨𝑋, 𝑌⟩(2nd ↾ ({𝑋} × 𝐴))𝑌))
1412, 13mpan2 707 . . . . 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 brresg 5403 . . . . . 6 (𝑌𝐴 → (⟨𝑋, 𝑌⟩(2nd ↾ ({𝑋} × 𝐴))𝑌 ↔ (⟨𝑋, 𝑌⟩2nd 𝑌 ∧ ⟨𝑋, 𝑌⟩ ∈ ({𝑋} × 𝐴))))
1716adantl 482 . . . . 5 ((𝑋𝑉𝑌𝐴) → (⟨𝑋, 𝑌⟩(2nd ↾ ({𝑋} × 𝐴))𝑌 ↔ (⟨𝑋, 𝑌⟩2nd 𝑌 ∧ ⟨𝑋, 𝑌⟩ ∈ ({𝑋} × 𝐴))))
18 opelxp 5144 . . . . . . 7 (⟨𝑋, 𝑌⟩ ∈ ({𝑋} × 𝐴) ↔ (𝑋 ∈ {𝑋} ∧ 𝑌𝐴))
1918anbi2i 730 . . . . . 6 ((⟨𝑋, 𝑌⟩2nd 𝑌 ∧ ⟨𝑋, 𝑌⟩ ∈ ({𝑋} × 𝐴)) ↔ (⟨𝑋, 𝑌⟩2nd 𝑌 ∧ (𝑋 ∈ {𝑋} ∧ 𝑌𝐴)))
20 br2ndeqg 31659 . . . . . . . 8 ((𝑋𝑉𝑌𝐴𝑌𝐴) → (⟨𝑋, 𝑌⟩2nd 𝑌𝑌 = 𝑌))
21203anidm23 1384 . . . . . . 7 ((𝑋𝑉𝑌𝐴) → (⟨𝑋, 𝑌⟩2nd 𝑌𝑌 = 𝑌))
2221anbi1d 741 . . . . . 6 ((𝑋𝑉𝑌𝐴) → ((⟨𝑋, 𝑌⟩2nd 𝑌 ∧ (𝑋 ∈ {𝑋} ∧ 𝑌𝐴)) ↔ (𝑌 = 𝑌 ∧ (𝑋 ∈ {𝑋} ∧ 𝑌𝐴))))
2319, 22syl5bb 272 . . . . 5 ((𝑋𝑉𝑌𝐴) → ((⟨𝑋, 𝑌⟩2nd 𝑌 ∧ ⟨𝑋, 𝑌⟩ ∈ ({𝑋} × 𝐴)) ↔ (𝑌 = 𝑌 ∧ (𝑋 ∈ {𝑋} ∧ 𝑌𝐴))))
2417, 23bitrd 268 . . . 4 ((𝑋𝑉𝑌𝐴) → (⟨𝑋, 𝑌⟩(2nd ↾ ({𝑋} × 𝐴))𝑌 ↔ (𝑌 = 𝑌 ∧ (𝑋 ∈ {𝑋} ∧ 𝑌𝐴))))
2515, 24bitrd 268 . . 3 ((𝑋𝑉𝑌𝐴) → (𝑌(2nd ↾ ({𝑋} × 𝐴))⟨𝑋, 𝑌⟩ ↔ (𝑌 = 𝑌 ∧ (𝑋 ∈ {𝑋} ∧ 𝑌𝐴))))
2611, 25bitrd 268 . 2 ((𝑋𝑉𝑌𝐴) → (((2nd ↾ ({𝑋} × 𝐴))‘𝑌) = ⟨𝑋, 𝑌⟩ ↔ (𝑌 = 𝑌 ∧ (𝑋 ∈ {𝑋} ∧ 𝑌𝐴))))
274, 26mpbird 247 1 ((𝑋𝑉𝑌𝐴) → ((2nd ↾ ({𝑋} × 𝐴))‘𝑌) = ⟨𝑋, 𝑌⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1482  wcel 1989  Vcvv 3198  {csn 4175  cop 4181   class class class wbr 4651   × cxp 5110  ccnv 5111  cres 5114   Fn wfn 5881  1-1-ontowf1o 5885  cfv 5886  2nd c2nd 7164
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-1st 7165  df-2nd 7166
This theorem is referenced by: (None)
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