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Theorem fv1stcnv 35777
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 4660 . . . . 5 (𝑌𝑉𝑌 ∈ {𝑌})
21anim2i 617 . . . 4 ((𝑋𝐴𝑌𝑉) → (𝑋𝐴𝑌 ∈ {𝑌}))
3 eqid 2737 . . . 4 𝑋 = 𝑋
42, 3jctir 520 . . 3 ((𝑋𝐴𝑌𝑉) → ((𝑋𝐴𝑌 ∈ {𝑌}) ∧ 𝑋 = 𝑋))
5 opex 5469 . . . . . . 7 𝑋, 𝑌⟩ ∈ V
6 brcnvg 5890 . . . . . . 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 6004 . . . . . 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 5721 . . . . . 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 8036 . . . . . 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 8125 . . . 4 (𝑌𝑉 → (1st ↾ (𝐴 × {𝑌})):(𝐴 × {𝑌})–1-1-onto𝐴)
19 f1ocnv 6860 . . . 4 ((1st ↾ (𝐴 × {𝑌})):(𝐴 × {𝑌})–1-1-onto𝐴(1st ↾ (𝐴 × {𝑌})):𝐴1-1-onto→(𝐴 × {𝑌}))
20 f1ofn 6849 . . . 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 6959 . . 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 1540  wcel 2108  Vcvv 3480  {csn 4626  cop 4632   class class class wbr 5143   × cxp 5683  ccnv 5684  cres 5687   Fn wfn 6556  1-1-ontowf1o 6560  cfv 6561  1st c1st 8012
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-csb 3900  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-iun 4993  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-1st 8014  df-2nd 8015
This theorem is referenced by: (None)
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