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Theorem fv1stcnv 31374
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 4182 . . . . 5 (𝑌𝑉𝑌 ∈ {𝑌})
21anim2i 592 . . . 4 ((𝑋𝐴𝑌𝑉) → (𝑋𝐴𝑌 ∈ {𝑌}))
3 eqid 2626 . . . 4 𝑋 = 𝑋
42, 3jctil 559 . . 3 ((𝑋𝐴𝑌𝑉) → (𝑋 = 𝑋 ∧ (𝑋𝐴𝑌 ∈ {𝑌})))
5 opex 4898 . . . . . . 7 𝑋, 𝑌⟩ ∈ V
6 brcnvg 5268 . . . . . . 7 ((𝑋𝐴 ∧ ⟨𝑋, 𝑌⟩ ∈ V) → (𝑋(1st ↾ (𝐴 × {𝑌}))⟨𝑋, 𝑌⟩ ↔ ⟨𝑋, 𝑌⟩(1st ↾ (𝐴 × {𝑌}))𝑋))
75, 6mpan2 706 . . . . . 6 (𝑋𝐴 → (𝑋(1st ↾ (𝐴 × {𝑌}))⟨𝑋, 𝑌⟩ ↔ ⟨𝑋, 𝑌⟩(1st ↾ (𝐴 × {𝑌}))𝑋))
8 brresg 5368 . . . . . 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 481 . . . 4 ((𝑋𝐴𝑌𝑉) → (𝑋(1st ↾ (𝐴 × {𝑌}))⟨𝑋, 𝑌⟩ ↔ (⟨𝑋, 𝑌⟩1st 𝑋 ∧ ⟨𝑋, 𝑌⟩ ∈ (𝐴 × {𝑌}))))
11 opelxp 5111 . . . . . 6 (⟨𝑋, 𝑌⟩ ∈ (𝐴 × {𝑌}) ↔ (𝑋𝐴𝑌 ∈ {𝑌}))
1211anbi2i 729 . . . . 5 ((⟨𝑋, 𝑌⟩1st 𝑋 ∧ ⟨𝑋, 𝑌⟩ ∈ (𝐴 × {𝑌})) ↔ (⟨𝑋, 𝑌⟩1st 𝑋 ∧ (𝑋𝐴𝑌 ∈ {𝑌})))
13 br1steqg 31368 . . . . . . 7 ((𝑋𝐴𝑌𝑉𝑋𝐴) → (⟨𝑋, 𝑌⟩1st 𝑋𝑋 = 𝑋))
14133anidm13 1381 . . . . . 6 ((𝑋𝐴𝑌𝑉) → (⟨𝑋, 𝑌⟩1st 𝑋𝑋 = 𝑋))
1514anbi1d 740 . . . . 5 ((𝑋𝐴𝑌𝑉) → ((⟨𝑋, 𝑌⟩1st 𝑋 ∧ (𝑋𝐴𝑌 ∈ {𝑌})) ↔ (𝑋 = 𝑋 ∧ (𝑋𝐴𝑌 ∈ {𝑌}))))
1612, 15syl5bb 272 . . . 4 ((𝑋𝐴𝑌𝑉) → ((⟨𝑋, 𝑌⟩1st 𝑋 ∧ ⟨𝑋, 𝑌⟩ ∈ (𝐴 × {𝑌})) ↔ (𝑋 = 𝑋 ∧ (𝑋𝐴𝑌 ∈ {𝑌}))))
1710, 16bitrd 268 . . 3 ((𝑋𝐴𝑌𝑉) → (𝑋(1st ↾ (𝐴 × {𝑌}))⟨𝑋, 𝑌⟩ ↔ (𝑋 = 𝑋 ∧ (𝑋𝐴𝑌 ∈ {𝑌}))))
184, 17mpbird 247 . 2 ((𝑋𝐴𝑌𝑉) → 𝑋(1st ↾ (𝐴 × {𝑌}))⟨𝑋, 𝑌⟩)
19 1stconst 7211 . . . . 5 (𝑌𝑉 → (1st ↾ (𝐴 × {𝑌})):(𝐴 × {𝑌})–1-1-onto𝐴)
20 f1ocnv 6108 . . . . 5 ((1st ↾ (𝐴 × {𝑌})):(𝐴 × {𝑌})–1-1-onto𝐴(1st ↾ (𝐴 × {𝑌})):𝐴1-1-onto→(𝐴 × {𝑌}))
21 f1ofn 6097 . . . . 5 ((1st ↾ (𝐴 × {𝑌})):𝐴1-1-onto→(𝐴 × {𝑌}) → (1st ↾ (𝐴 × {𝑌})) Fn 𝐴)
2219, 20, 213syl 18 . . . 4 (𝑌𝑉(1st ↾ (𝐴 × {𝑌})) Fn 𝐴)
2322adantl 482 . . 3 ((𝑋𝐴𝑌𝑉) → (1st ↾ (𝐴 × {𝑌})) Fn 𝐴)
24 simpl 473 . . 3 ((𝑋𝐴𝑌𝑉) → 𝑋𝐴)
25 fnbrfvb 6194 . . 3 (((1st ↾ (𝐴 × {𝑌})) Fn 𝐴𝑋𝐴) → (((1st ↾ (𝐴 × {𝑌}))‘𝑋) = ⟨𝑋, 𝑌⟩ ↔ 𝑋(1st ↾ (𝐴 × {𝑌}))⟨𝑋, 𝑌⟩))
2623, 24, 25syl2anc 692 . 2 ((𝑋𝐴𝑌𝑉) → (((1st ↾ (𝐴 × {𝑌}))‘𝑋) = ⟨𝑋, 𝑌⟩ ↔ 𝑋(1st ↾ (𝐴 × {𝑌}))⟨𝑋, 𝑌⟩))
2718, 26mpbird 247 1 ((𝑋𝐴𝑌𝑉) → ((1st ↾ (𝐴 × {𝑌}))‘𝑋) = ⟨𝑋, 𝑌⟩)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1992  Vcvv 3191  {csn 4153  cop 4159   class class class wbr 4618   × cxp 5077  ccnv 5078  cres 5081   Fn wfn 5845  1-1-ontowf1o 5849  cfv 5850  1st c1st 7114
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pow 4808  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ne 2797  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-sbc 3423  df-csb 3520  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-iun 4492  df-br 4619  df-opab 4679  df-mpt 4680  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-res 5091  df-ima 5092  df-iota 5813  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-fv 5858  df-1st 7116  df-2nd 7117
This theorem is referenced by: (None)
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