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Theorem fv1stcnv 31664
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 4204 . . . . 5 (𝑌𝑉𝑌 ∈ {𝑌})
21anim2i 593 . . . 4 ((𝑋𝐴𝑌𝑉) → (𝑋𝐴𝑌 ∈ {𝑌}))
3 eqid 2621 . . . 4 𝑋 = 𝑋
42, 3jctil 560 . . 3 ((𝑋𝐴𝑌𝑉) → (𝑋 = 𝑋 ∧ (𝑋𝐴𝑌 ∈ {𝑌})))
5 opex 4930 . . . . . . 7 𝑋, 𝑌⟩ ∈ V
6 brcnvg 5301 . . . . . . 7 ((𝑋𝐴 ∧ ⟨𝑋, 𝑌⟩ ∈ V) → (𝑋(1st ↾ (𝐴 × {𝑌}))⟨𝑋, 𝑌⟩ ↔ ⟨𝑋, 𝑌⟩(1st ↾ (𝐴 × {𝑌}))𝑋))
75, 6mpan2 707 . . . . . 6 (𝑋𝐴 → (𝑋(1st ↾ (𝐴 × {𝑌}))⟨𝑋, 𝑌⟩ ↔ ⟨𝑋, 𝑌⟩(1st ↾ (𝐴 × {𝑌}))𝑋))
8 brresg 5403 . . . . . 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 5144 . . . . . 6 (⟨𝑋, 𝑌⟩ ∈ (𝐴 × {𝑌}) ↔ (𝑋𝐴𝑌 ∈ {𝑌}))
1211anbi2i 730 . . . . 5 ((⟨𝑋, 𝑌⟩1st 𝑋 ∧ ⟨𝑋, 𝑌⟩ ∈ (𝐴 × {𝑌})) ↔ (⟨𝑋, 𝑌⟩1st 𝑋 ∧ (𝑋𝐴𝑌 ∈ {𝑌})))
13 br1steqg 31658 . . . . . . 7 ((𝑋𝐴𝑌𝑉𝑋𝐴) → (⟨𝑋, 𝑌⟩1st 𝑋𝑋 = 𝑋))
14133anidm13 1383 . . . . . 6 ((𝑋𝐴𝑌𝑉) → (⟨𝑋, 𝑌⟩1st 𝑋𝑋 = 𝑋))
1514anbi1d 741 . . . . 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 7262 . . . . 5 (𝑌𝑉 → (1st ↾ (𝐴 × {𝑌})):(𝐴 × {𝑌})–1-1-onto𝐴)
20 f1ocnv 6147 . . . . 5 ((1st ↾ (𝐴 × {𝑌})):(𝐴 × {𝑌})–1-1-onto𝐴(1st ↾ (𝐴 × {𝑌})):𝐴1-1-onto→(𝐴 × {𝑌}))
21 f1ofn 6136 . . . . 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 6234 . . 3 (((1st ↾ (𝐴 × {𝑌})) Fn 𝐴𝑋𝐴) → (((1st ↾ (𝐴 × {𝑌}))‘𝑋) = ⟨𝑋, 𝑌⟩ ↔ 𝑋(1st ↾ (𝐴 × {𝑌}))⟨𝑋, 𝑌⟩))
2623, 24, 25syl2anc 693 . 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 1482  wcel 1989  Vcvv 3198  {csn 4175  cop 4181   class class class wbr 4651   × cxp 5110  ccnv 5111  cres 5114   Fn wfn 5881  1-1-ontowf1o 5885  cfv 5886  1st c1st 7163
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-1st 7165  df-2nd 7166
This theorem is referenced by: (None)
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