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Theorem fv1stcnv 33751
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 4595 . . . . 5 (𝑌𝑉𝑌 ∈ {𝑌})
21anim2i 617 . . . 4 ((𝑋𝐴𝑌𝑉) → (𝑋𝐴𝑌 ∈ {𝑌}))
3 eqid 2738 . . . 4 𝑋 = 𝑋
42, 3jctir 521 . . 3 ((𝑋𝐴𝑌𝑉) → ((𝑋𝐴𝑌 ∈ {𝑌}) ∧ 𝑋 = 𝑋))
5 opex 5379 . . . . . . 7 𝑋, 𝑌⟩ ∈ V
6 brcnvg 5788 . . . . . . 7 ((𝑋𝐴 ∧ ⟨𝑋, 𝑌⟩ ∈ V) → (𝑋(1st ↾ (𝐴 × {𝑌}))⟨𝑋, 𝑌⟩ ↔ ⟨𝑋, 𝑌⟩(1st ↾ (𝐴 × {𝑌}))𝑋))
75, 6mpan2 688 . . . . . 6 (𝑋𝐴 → (𝑋(1st ↾ (𝐴 × {𝑌}))⟨𝑋, 𝑌⟩ ↔ ⟨𝑋, 𝑌⟩(1st ↾ (𝐴 × {𝑌}))𝑋))
8 brres 5898 . . . . . 6 (𝑋𝐴 → (⟨𝑋, 𝑌⟩(1st ↾ (𝐴 × {𝑌}))𝑋 ↔ (⟨𝑋, 𝑌⟩ ∈ (𝐴 × {𝑌}) ∧ ⟨𝑋, 𝑌⟩1st 𝑋)))
97, 8bitrd 278 . . . . 5 (𝑋𝐴 → (𝑋(1st ↾ (𝐴 × {𝑌}))⟨𝑋, 𝑌⟩ ↔ (⟨𝑋, 𝑌⟩ ∈ (𝐴 × {𝑌}) ∧ ⟨𝑋, 𝑌⟩1st 𝑋)))
109adantr 481 . . . 4 ((𝑋𝐴𝑌𝑉) → (𝑋(1st ↾ (𝐴 × {𝑌}))⟨𝑋, 𝑌⟩ ↔ (⟨𝑋, 𝑌⟩ ∈ (𝐴 × {𝑌}) ∧ ⟨𝑋, 𝑌⟩1st 𝑋)))
11 opelxp 5625 . . . . . 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 7853 . . . . . 6 ((𝑋𝐴𝑌𝑉) → (⟨𝑋, 𝑌⟩1st 𝑋𝑋 = 𝑋))
1413anbi2d 629 . . . . 5 ((𝑋𝐴𝑌𝑉) → (((𝑋𝐴𝑌 ∈ {𝑌}) ∧ ⟨𝑋, 𝑌⟩1st 𝑋) ↔ ((𝑋𝐴𝑌 ∈ {𝑌}) ∧ 𝑋 = 𝑋)))
1512, 14syl5bb 283 . . . 4 ((𝑋𝐴𝑌𝑉) → ((⟨𝑋, 𝑌⟩ ∈ (𝐴 × {𝑌}) ∧ ⟨𝑋, 𝑌⟩1st 𝑋) ↔ ((𝑋𝐴𝑌 ∈ {𝑌}) ∧ 𝑋 = 𝑋)))
1610, 15bitrd 278 . . 3 ((𝑋𝐴𝑌𝑉) → (𝑋(1st ↾ (𝐴 × {𝑌}))⟨𝑋, 𝑌⟩ ↔ ((𝑋𝐴𝑌 ∈ {𝑌}) ∧ 𝑋 = 𝑋)))
174, 16mpbird 256 . 2 ((𝑋𝐴𝑌𝑉) → 𝑋(1st ↾ (𝐴 × {𝑌}))⟨𝑋, 𝑌⟩)
18 1stconst 7940 . . . 4 (𝑌𝑉 → (1st ↾ (𝐴 × {𝑌})):(𝐴 × {𝑌})–1-1-onto𝐴)
19 f1ocnv 6728 . . . 4 ((1st ↾ (𝐴 × {𝑌})):(𝐴 × {𝑌})–1-1-onto𝐴(1st ↾ (𝐴 × {𝑌})):𝐴1-1-onto→(𝐴 × {𝑌}))
20 f1ofn 6717 . . . 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 483 . . 3 ((𝑋𝐴𝑌𝑉) → 𝑋𝐴)
23 fnbrfvb 6822 . . 3 (((1st ↾ (𝐴 × {𝑌})) Fn 𝐴𝑋𝐴) → (((1st ↾ (𝐴 × {𝑌}))‘𝑋) = ⟨𝑋, 𝑌⟩ ↔ 𝑋(1st ↾ (𝐴 × {𝑌}))⟨𝑋, 𝑌⟩))
2421, 22, 23syl2an2 683 . 2 ((𝑋𝐴𝑌𝑉) → (((1st ↾ (𝐴 × {𝑌}))‘𝑋) = ⟨𝑋, 𝑌⟩ ↔ 𝑋(1st ↾ (𝐴 × {𝑌}))⟨𝑋, 𝑌⟩))
2517, 24mpbird 256 1 ((𝑋𝐴𝑌𝑉) → ((1st ↾ (𝐴 × {𝑌}))‘𝑋) = ⟨𝑋, 𝑌⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1539  wcel 2106  Vcvv 3432  {csn 4561  cop 4567   class class class wbr 5074   × cxp 5587  ccnv 5588  cres 5591   Fn wfn 6428  1-1-ontowf1o 6432  cfv 6433  1st c1st 7829
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-mo 2540  df-eu 2569  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-sbc 3717  df-csb 3833  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-mpt 5158  df-id 5489  df-xp 5595  df-rel 5596  df-cnv 5597  df-co 5598  df-dm 5599  df-rn 5600  df-res 5601  df-ima 5602  df-iota 6391  df-fun 6435  df-fn 6436  df-f 6437  df-f1 6438  df-fo 6439  df-f1o 6440  df-fv 6441  df-1st 7831  df-2nd 7832
This theorem is referenced by: (None)
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