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Theorem fv1stcnv 32010
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 4343 . . . . 5 (𝑌𝑉𝑌 ∈ {𝑌})
21anim2i 595 . . . 4 ((𝑋𝐴𝑌𝑉) → (𝑋𝐴𝑌 ∈ {𝑌}))
3 eqid 2770 . . . 4 𝑋 = 𝑋
42, 3jctil 503 . . 3 ((𝑋𝐴𝑌𝑉) → (𝑋 = 𝑋 ∧ (𝑋𝐴𝑌 ∈ {𝑌})))
5 opex 5060 . . . . . . 7 𝑋, 𝑌⟩ ∈ V
6 brcnvg 5441 . . . . . . 7 ((𝑋𝐴 ∧ ⟨𝑋, 𝑌⟩ ∈ V) → (𝑋(1st ↾ (𝐴 × {𝑌}))⟨𝑋, 𝑌⟩ ↔ ⟨𝑋, 𝑌⟩(1st ↾ (𝐴 × {𝑌}))𝑋))
75, 6mpan2 663 . . . . . 6 (𝑋𝐴 → (𝑋(1st ↾ (𝐴 × {𝑌}))⟨𝑋, 𝑌⟩ ↔ ⟨𝑋, 𝑌⟩(1st ↾ (𝐴 × {𝑌}))𝑋))
8 brresg 5541 . . . . . 6 (𝑋𝐴 → (⟨𝑋, 𝑌⟩(1st ↾ (𝐴 × {𝑌}))𝑋 ↔ (⟨𝑋, 𝑌⟩1st 𝑋 ∧ ⟨𝑋, 𝑌⟩ ∈ (𝐴 × {𝑌}))))
97, 8bitrd 268 . . . . 5 (𝑋𝐴 → (𝑋(1st ↾ (𝐴 × {𝑌}))⟨𝑋, 𝑌⟩ ↔ (⟨𝑋, 𝑌⟩1st 𝑋 ∧ ⟨𝑋, 𝑌⟩ ∈ (𝐴 × {𝑌}))))
109adantr 466 . . . 4 ((𝑋𝐴𝑌𝑉) → (𝑋(1st ↾ (𝐴 × {𝑌}))⟨𝑋, 𝑌⟩ ↔ (⟨𝑋, 𝑌⟩1st 𝑋 ∧ ⟨𝑋, 𝑌⟩ ∈ (𝐴 × {𝑌}))))
11 opelxp 5286 . . . . . 6 (⟨𝑋, 𝑌⟩ ∈ (𝐴 × {𝑌}) ↔ (𝑋𝐴𝑌 ∈ {𝑌}))
1211anbi2i 601 . . . . 5 ((⟨𝑋, 𝑌⟩1st 𝑋 ∧ ⟨𝑋, 𝑌⟩ ∈ (𝐴 × {𝑌})) ↔ (⟨𝑋, 𝑌⟩1st 𝑋 ∧ (𝑋𝐴𝑌 ∈ {𝑌})))
13 br1steqg 7336 . . . . . 6 ((𝑋𝐴𝑌𝑉) → (⟨𝑋, 𝑌⟩1st 𝑋𝑋 = 𝑋))
1413anbi1d 607 . . . . 5 ((𝑋𝐴𝑌𝑉) → ((⟨𝑋, 𝑌⟩1st 𝑋 ∧ (𝑋𝐴𝑌 ∈ {𝑌})) ↔ (𝑋 = 𝑋 ∧ (𝑋𝐴𝑌 ∈ {𝑌}))))
1512, 14syl5bb 272 . . . 4 ((𝑋𝐴𝑌𝑉) → ((⟨𝑋, 𝑌⟩1st 𝑋 ∧ ⟨𝑋, 𝑌⟩ ∈ (𝐴 × {𝑌})) ↔ (𝑋 = 𝑋 ∧ (𝑋𝐴𝑌 ∈ {𝑌}))))
1610, 15bitrd 268 . . 3 ((𝑋𝐴𝑌𝑉) → (𝑋(1st ↾ (𝐴 × {𝑌}))⟨𝑋, 𝑌⟩ ↔ (𝑋 = 𝑋 ∧ (𝑋𝐴𝑌 ∈ {𝑌}))))
174, 16mpbird 247 . 2 ((𝑋𝐴𝑌𝑉) → 𝑋(1st ↾ (𝐴 × {𝑌}))⟨𝑋, 𝑌⟩)
18 1stconst 7415 . . . 4 (𝑌𝑉 → (1st ↾ (𝐴 × {𝑌})):(𝐴 × {𝑌})–1-1-onto𝐴)
19 f1ocnv 6290 . . . 4 ((1st ↾ (𝐴 × {𝑌})):(𝐴 × {𝑌})–1-1-onto𝐴(1st ↾ (𝐴 × {𝑌})):𝐴1-1-onto→(𝐴 × {𝑌}))
20 f1ofn 6279 . . . 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 468 . . 3 ((𝑋𝐴𝑌𝑉) → 𝑋𝐴)
23 fnbrfvb 6377 . . 3 (((1st ↾ (𝐴 × {𝑌})) Fn 𝐴𝑋𝐴) → (((1st ↾ (𝐴 × {𝑌}))‘𝑋) = ⟨𝑋, 𝑌⟩ ↔ 𝑋(1st ↾ (𝐴 × {𝑌}))⟨𝑋, 𝑌⟩))
2421, 22, 23syl2an2 658 . 2 ((𝑋𝐴𝑌𝑉) → (((1st ↾ (𝐴 × {𝑌}))‘𝑋) = ⟨𝑋, 𝑌⟩ ↔ 𝑋(1st ↾ (𝐴 × {𝑌}))⟨𝑋, 𝑌⟩))
2517, 24mpbird 247 1 ((𝑋𝐴𝑌𝑉) → ((1st ↾ (𝐴 × {𝑌}))‘𝑋) = ⟨𝑋, 𝑌⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 382   = wceq 1630  wcel 2144  Vcvv 3349  {csn 4314  cop 4320   class class class wbr 4784   × cxp 5247  ccnv 5248  cres 5251   Fn wfn 6026  1-1-ontowf1o 6030  cfv 6031  1st c1st 7312
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-1st 7314  df-2nd 7315
This theorem is referenced by: (None)
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