Users' Mathboxes Mathbox for Scott Fenton < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  fv1stcnv Structured version   Visualization version   GIF version

Theorem fv1stcnv 35758
Description: The value of the converse of 1st restricted to a singleton. (Contributed by Scott Fenton, 2-Jul-2020.)
Assertion
Ref Expression
fv1stcnv ((𝑋𝐴𝑌𝑉) → ((1st ↾ (𝐴 × {𝑌}))‘𝑋) = ⟨𝑋, 𝑌⟩)

Proof of Theorem fv1stcnv
StepHypRef Expression
1 snidg 4665 . . . . 5 (𝑌𝑉𝑌 ∈ {𝑌})
21anim2i 617 . . . 4 ((𝑋𝐴𝑌𝑉) → (𝑋𝐴𝑌 ∈ {𝑌}))
3 eqid 2735 . . . 4 𝑋 = 𝑋
42, 3jctir 520 . . 3 ((𝑋𝐴𝑌𝑉) → ((𝑋𝐴𝑌 ∈ {𝑌}) ∧ 𝑋 = 𝑋))
5 opex 5475 . . . . . . 7 𝑋, 𝑌⟩ ∈ V
6 brcnvg 5893 . . . . . . 7 ((𝑋𝐴 ∧ ⟨𝑋, 𝑌⟩ ∈ V) → (𝑋(1st ↾ (𝐴 × {𝑌}))⟨𝑋, 𝑌⟩ ↔ ⟨𝑋, 𝑌⟩(1st ↾ (𝐴 × {𝑌}))𝑋))
75, 6mpan2 691 . . . . . 6 (𝑋𝐴 → (𝑋(1st ↾ (𝐴 × {𝑌}))⟨𝑋, 𝑌⟩ ↔ ⟨𝑋, 𝑌⟩(1st ↾ (𝐴 × {𝑌}))𝑋))
8 brres 6007 . . . . . 6 (𝑋𝐴 → (⟨𝑋, 𝑌⟩(1st ↾ (𝐴 × {𝑌}))𝑋 ↔ (⟨𝑋, 𝑌⟩ ∈ (𝐴 × {𝑌}) ∧ ⟨𝑋, 𝑌⟩1st 𝑋)))
97, 8bitrd 279 . . . . 5 (𝑋𝐴 → (𝑋(1st ↾ (𝐴 × {𝑌}))⟨𝑋, 𝑌⟩ ↔ (⟨𝑋, 𝑌⟩ ∈ (𝐴 × {𝑌}) ∧ ⟨𝑋, 𝑌⟩1st 𝑋)))
109adantr 480 . . . 4 ((𝑋𝐴𝑌𝑉) → (𝑋(1st ↾ (𝐴 × {𝑌}))⟨𝑋, 𝑌⟩ ↔ (⟨𝑋, 𝑌⟩ ∈ (𝐴 × {𝑌}) ∧ ⟨𝑋, 𝑌⟩1st 𝑋)))
11 opelxp 5725 . . . . . 6 (⟨𝑋, 𝑌⟩ ∈ (𝐴 × {𝑌}) ↔ (𝑋𝐴𝑌 ∈ {𝑌}))
1211anbi1i 624 . . . . 5 ((⟨𝑋, 𝑌⟩ ∈ (𝐴 × {𝑌}) ∧ ⟨𝑋, 𝑌⟩1st 𝑋) ↔ ((𝑋𝐴𝑌 ∈ {𝑌}) ∧ ⟨𝑋, 𝑌⟩1st 𝑋))
13 br1steqg 8035 . . . . . 6 ((𝑋𝐴𝑌𝑉) → (⟨𝑋, 𝑌⟩1st 𝑋𝑋 = 𝑋))
1413anbi2d 630 . . . . 5 ((𝑋𝐴𝑌𝑉) → (((𝑋𝐴𝑌 ∈ {𝑌}) ∧ ⟨𝑋, 𝑌⟩1st 𝑋) ↔ ((𝑋𝐴𝑌 ∈ {𝑌}) ∧ 𝑋 = 𝑋)))
1512, 14bitrid 283 . . . 4 ((𝑋𝐴𝑌𝑉) → ((⟨𝑋, 𝑌⟩ ∈ (𝐴 × {𝑌}) ∧ ⟨𝑋, 𝑌⟩1st 𝑋) ↔ ((𝑋𝐴𝑌 ∈ {𝑌}) ∧ 𝑋 = 𝑋)))
1610, 15bitrd 279 . . 3 ((𝑋𝐴𝑌𝑉) → (𝑋(1st ↾ (𝐴 × {𝑌}))⟨𝑋, 𝑌⟩ ↔ ((𝑋𝐴𝑌 ∈ {𝑌}) ∧ 𝑋 = 𝑋)))
174, 16mpbird 257 . 2 ((𝑋𝐴𝑌𝑉) → 𝑋(1st ↾ (𝐴 × {𝑌}))⟨𝑋, 𝑌⟩)
18 1stconst 8124 . . . 4 (𝑌𝑉 → (1st ↾ (𝐴 × {𝑌})):(𝐴 × {𝑌})–1-1-onto𝐴)
19 f1ocnv 6861 . . . 4 ((1st ↾ (𝐴 × {𝑌})):(𝐴 × {𝑌})–1-1-onto𝐴(1st ↾ (𝐴 × {𝑌})):𝐴1-1-onto→(𝐴 × {𝑌}))
20 f1ofn 6850 . . . 4 ((1st ↾ (𝐴 × {𝑌})):𝐴1-1-onto→(𝐴 × {𝑌}) → (1st ↾ (𝐴 × {𝑌})) Fn 𝐴)
2118, 19, 203syl 18 . . 3 (𝑌𝑉(1st ↾ (𝐴 × {𝑌})) Fn 𝐴)
22 simpl 482 . . 3 ((𝑋𝐴𝑌𝑉) → 𝑋𝐴)
23 fnbrfvb 6960 . . 3 (((1st ↾ (𝐴 × {𝑌})) Fn 𝐴𝑋𝐴) → (((1st ↾ (𝐴 × {𝑌}))‘𝑋) = ⟨𝑋, 𝑌⟩ ↔ 𝑋(1st ↾ (𝐴 × {𝑌}))⟨𝑋, 𝑌⟩))
2421, 22, 23syl2an2 686 . 2 ((𝑋𝐴𝑌𝑉) → (((1st ↾ (𝐴 × {𝑌}))‘𝑋) = ⟨𝑋, 𝑌⟩ ↔ 𝑋(1st ↾ (𝐴 × {𝑌}))⟨𝑋, 𝑌⟩))
2517, 24mpbird 257 1 ((𝑋𝐴𝑌𝑉) → ((1st ↾ (𝐴 × {𝑌}))‘𝑋) = ⟨𝑋, 𝑌⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  Vcvv 3478  {csn 4631  cop 4637   class class class wbr 5148   × cxp 5687  ccnv 5688  cres 5691   Fn wfn 6558  1-1-ontowf1o 6562  cfv 6563  1st c1st 8011
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-1st 8013  df-2nd 8014
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator