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Theorem fv2ndcnv 34391
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 4625 . . . 4 (𝑋𝑉𝑋 ∈ {𝑋})
21anim1i 616 . . 3 ((𝑋𝑉𝑌𝐴) → (𝑋 ∈ {𝑋} ∧ 𝑌𝐴))
3 eqid 2737 . . 3 𝑌 = 𝑌
42, 3jctir 522 . 2 ((𝑋𝑉𝑌𝐴) → ((𝑋 ∈ {𝑋} ∧ 𝑌𝐴) ∧ 𝑌 = 𝑌))
5 2ndconst 8038 . . . . . 6 (𝑋𝑉 → (2nd ↾ ({𝑋} × 𝐴)):({𝑋} × 𝐴)–1-1-onto𝐴)
65adantr 482 . . . . 5 ((𝑋𝑉𝑌𝐴) → (2nd ↾ ({𝑋} × 𝐴)):({𝑋} × 𝐴)–1-1-onto𝐴)
7 f1ocnv 6801 . . . . 5 ((2nd ↾ ({𝑋} × 𝐴)):({𝑋} × 𝐴)–1-1-onto𝐴(2nd ↾ ({𝑋} × 𝐴)):𝐴1-1-onto→({𝑋} × 𝐴))
8 f1ofn 6790 . . . . 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 6900 . . . 4 (((2nd ↾ ({𝑋} × 𝐴)) Fn 𝐴𝑌𝐴) → (((2nd ↾ ({𝑋} × 𝐴))‘𝑌) = ⟨𝑋, 𝑌⟩ ↔ 𝑌(2nd ↾ ({𝑋} × 𝐴))⟨𝑋, 𝑌⟩))
119, 10sylancom 589 . . 3 ((𝑋𝑉𝑌𝐴) → (((2nd ↾ ({𝑋} × 𝐴))‘𝑌) = ⟨𝑋, 𝑌⟩ ↔ 𝑌(2nd ↾ ({𝑋} × 𝐴))⟨𝑋, 𝑌⟩))
12 opex 5426 . . . . . 6 𝑋, 𝑌⟩ ∈ V
13 brcnvg 5840 . . . . . 6 ((𝑌𝐴 ∧ ⟨𝑋, 𝑌⟩ ∈ V) → (𝑌(2nd ↾ ({𝑋} × 𝐴))⟨𝑋, 𝑌⟩ ↔ ⟨𝑋, 𝑌⟩(2nd ↾ ({𝑋} × 𝐴))𝑌))
1412, 13mpan2 690 . . . . 5 (𝑌𝐴 → (𝑌(2nd ↾ ({𝑋} × 𝐴))⟨𝑋, 𝑌⟩ ↔ ⟨𝑋, 𝑌⟩(2nd ↾ ({𝑋} × 𝐴))𝑌))
1514adantl 483 . . . 4 ((𝑋𝑉𝑌𝐴) → (𝑌(2nd ↾ ({𝑋} × 𝐴))⟨𝑋, 𝑌⟩ ↔ ⟨𝑋, 𝑌⟩(2nd ↾ ({𝑋} × 𝐴))𝑌))
16 brres 5949 . . . . . 6 (𝑌𝐴 → (⟨𝑋, 𝑌⟩(2nd ↾ ({𝑋} × 𝐴))𝑌 ↔ (⟨𝑋, 𝑌⟩ ∈ ({𝑋} × 𝐴) ∧ ⟨𝑋, 𝑌⟩2nd 𝑌)))
1716adantl 483 . . . . 5 ((𝑋𝑉𝑌𝐴) → (⟨𝑋, 𝑌⟩(2nd ↾ ({𝑋} × 𝐴))𝑌 ↔ (⟨𝑋, 𝑌⟩ ∈ ({𝑋} × 𝐴) ∧ ⟨𝑋, 𝑌⟩2nd 𝑌)))
18 opelxp 5674 . . . . . . 7 (⟨𝑋, 𝑌⟩ ∈ ({𝑋} × 𝐴) ↔ (𝑋 ∈ {𝑋} ∧ 𝑌𝐴))
1918anbi1i 625 . . . . . 6 ((⟨𝑋, 𝑌⟩ ∈ ({𝑋} × 𝐴) ∧ ⟨𝑋, 𝑌⟩2nd 𝑌) ↔ ((𝑋 ∈ {𝑋} ∧ 𝑌𝐴) ∧ ⟨𝑋, 𝑌⟩2nd 𝑌))
20 br2ndeqg 7949 . . . . . . 7 ((𝑋𝑉𝑌𝐴) → (⟨𝑋, 𝑌⟩2nd 𝑌𝑌 = 𝑌))
2120anbi2d 630 . . . . . 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 397   = wceq 1542  wcel 2107  Vcvv 3448  {csn 4591  cop 4597   class class class wbr 5110   × cxp 5636  ccnv 5637  cres 5640   Fn wfn 6496  1-1-ontowf1o 6500  cfv 6501  2nd c2nd 7925
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389  ax-un 7677
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2890  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3411  df-v 3450  df-sbc 3745  df-csb 3861  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-iun 4961  df-br 5111  df-opab 5173  df-mpt 5194  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-1st 7926  df-2nd 7927
This theorem is referenced by: (None)
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