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Theorem 1stpreimas 29611
Description: The preimage of a singleton. (Contributed by Thierry Arnoux, 27-Apr-2020.)
Assertion
Ref Expression
1stpreimas ((Rel 𝐴𝑋𝑉) → ((1st𝐴) “ {𝑋}) = ({𝑋} × (𝐴 “ {𝑋})))

Proof of Theorem 1stpreimas
Dummy variables 𝑥 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 1st2ndb 7250 . . . . . . . . 9 (𝑧 ∈ (V × V) ↔ 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
21biimpi 206 . . . . . . . 8 (𝑧 ∈ (V × V) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
32ad2antrl 764 . . . . . . 7 (((Rel 𝐴𝑋𝑉) ∧ (𝑧 ∈ (V × V) ∧ ((1st𝑧) ∈ {𝑋} ∧ (2nd𝑧) ∈ (𝐴 “ {𝑋})))) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
4 fvex 6239 . . . . . . . . . . . 12 (1st𝑧) ∈ V
54elsn 4225 . . . . . . . . . . 11 ((1st𝑧) ∈ {𝑋} ↔ (1st𝑧) = 𝑋)
65biimpi 206 . . . . . . . . . 10 ((1st𝑧) ∈ {𝑋} → (1st𝑧) = 𝑋)
76ad2antrl 764 . . . . . . . . 9 ((𝑧 ∈ (V × V) ∧ ((1st𝑧) ∈ {𝑋} ∧ (2nd𝑧) ∈ (𝐴 “ {𝑋}))) → (1st𝑧) = 𝑋)
87adantl 481 . . . . . . . 8 (((Rel 𝐴𝑋𝑉) ∧ (𝑧 ∈ (V × V) ∧ ((1st𝑧) ∈ {𝑋} ∧ (2nd𝑧) ∈ (𝐴 “ {𝑋})))) → (1st𝑧) = 𝑋)
98opeq1d 4439 . . . . . . 7 (((Rel 𝐴𝑋𝑉) ∧ (𝑧 ∈ (V × V) ∧ ((1st𝑧) ∈ {𝑋} ∧ (2nd𝑧) ∈ (𝐴 “ {𝑋})))) → ⟨(1st𝑧), (2nd𝑧)⟩ = ⟨𝑋, (2nd𝑧)⟩)
103, 9eqtrd 2685 . . . . . 6 (((Rel 𝐴𝑋𝑉) ∧ (𝑧 ∈ (V × V) ∧ ((1st𝑧) ∈ {𝑋} ∧ (2nd𝑧) ∈ (𝐴 “ {𝑋})))) → 𝑧 = ⟨𝑋, (2nd𝑧)⟩)
11 simplr 807 . . . . . . 7 (((Rel 𝐴𝑋𝑉) ∧ (𝑧 ∈ (V × V) ∧ ((1st𝑧) ∈ {𝑋} ∧ (2nd𝑧) ∈ (𝐴 “ {𝑋})))) → 𝑋𝑉)
12 simprrr 822 . . . . . . 7 (((Rel 𝐴𝑋𝑉) ∧ (𝑧 ∈ (V × V) ∧ ((1st𝑧) ∈ {𝑋} ∧ (2nd𝑧) ∈ (𝐴 “ {𝑋})))) → (2nd𝑧) ∈ (𝐴 “ {𝑋}))
13 elimasng 5526 . . . . . . . 8 ((𝑋𝑉 ∧ (2nd𝑧) ∈ (𝐴 “ {𝑋})) → ((2nd𝑧) ∈ (𝐴 “ {𝑋}) ↔ ⟨𝑋, (2nd𝑧)⟩ ∈ 𝐴))
1413biimpa 500 . . . . . . 7 (((𝑋𝑉 ∧ (2nd𝑧) ∈ (𝐴 “ {𝑋})) ∧ (2nd𝑧) ∈ (𝐴 “ {𝑋})) → ⟨𝑋, (2nd𝑧)⟩ ∈ 𝐴)
1511, 12, 12, 14syl21anc 1365 . . . . . 6 (((Rel 𝐴𝑋𝑉) ∧ (𝑧 ∈ (V × V) ∧ ((1st𝑧) ∈ {𝑋} ∧ (2nd𝑧) ∈ (𝐴 “ {𝑋})))) → ⟨𝑋, (2nd𝑧)⟩ ∈ 𝐴)
1610, 15eqeltrd 2730 . . . . 5 (((Rel 𝐴𝑋𝑉) ∧ (𝑧 ∈ (V × V) ∧ ((1st𝑧) ∈ {𝑋} ∧ (2nd𝑧) ∈ (𝐴 “ {𝑋})))) → 𝑧𝐴)
17 fvres 6245 . . . . . . 7 (𝑧𝐴 → ((1st𝐴)‘𝑧) = (1st𝑧))
1816, 17syl 17 . . . . . 6 (((Rel 𝐴𝑋𝑉) ∧ (𝑧 ∈ (V × V) ∧ ((1st𝑧) ∈ {𝑋} ∧ (2nd𝑧) ∈ (𝐴 “ {𝑋})))) → ((1st𝐴)‘𝑧) = (1st𝑧))
1918, 8eqtrd 2685 . . . . 5 (((Rel 𝐴𝑋𝑉) ∧ (𝑧 ∈ (V × V) ∧ ((1st𝑧) ∈ {𝑋} ∧ (2nd𝑧) ∈ (𝐴 “ {𝑋})))) → ((1st𝐴)‘𝑧) = 𝑋)
2016, 19jca 553 . . . 4 (((Rel 𝐴𝑋𝑉) ∧ (𝑧 ∈ (V × V) ∧ ((1st𝑧) ∈ {𝑋} ∧ (2nd𝑧) ∈ (𝐴 “ {𝑋})))) → (𝑧𝐴 ∧ ((1st𝐴)‘𝑧) = 𝑋))
21 df-rel 5150 . . . . . . . . 9 (Rel 𝐴𝐴 ⊆ (V × V))
2221biimpi 206 . . . . . . . 8 (Rel 𝐴𝐴 ⊆ (V × V))
2322adantr 480 . . . . . . 7 ((Rel 𝐴𝑋𝑉) → 𝐴 ⊆ (V × V))
2423sselda 3636 . . . . . 6 (((Rel 𝐴𝑋𝑉) ∧ 𝑧𝐴) → 𝑧 ∈ (V × V))
2524adantrr 753 . . . . 5 (((Rel 𝐴𝑋𝑉) ∧ (𝑧𝐴 ∧ ((1st𝐴)‘𝑧) = 𝑋)) → 𝑧 ∈ (V × V))
2617ad2antrl 764 . . . . . . . 8 (((Rel 𝐴𝑋𝑉) ∧ (𝑧𝐴 ∧ ((1st𝐴)‘𝑧) = 𝑋)) → ((1st𝐴)‘𝑧) = (1st𝑧))
27 simprr 811 . . . . . . . 8 (((Rel 𝐴𝑋𝑉) ∧ (𝑧𝐴 ∧ ((1st𝐴)‘𝑧) = 𝑋)) → ((1st𝐴)‘𝑧) = 𝑋)
2826, 27eqtr3d 2687 . . . . . . 7 (((Rel 𝐴𝑋𝑉) ∧ (𝑧𝐴 ∧ ((1st𝐴)‘𝑧) = 𝑋)) → (1st𝑧) = 𝑋)
2928, 5sylibr 224 . . . . . 6 (((Rel 𝐴𝑋𝑉) ∧ (𝑧𝐴 ∧ ((1st𝐴)‘𝑧) = 𝑋)) → (1st𝑧) ∈ {𝑋})
3028, 29eqeltrrd 2731 . . . . . . . . 9 (((Rel 𝐴𝑋𝑉) ∧ (𝑧𝐴 ∧ ((1st𝐴)‘𝑧) = 𝑋)) → 𝑋 ∈ {𝑋})
31 simpr 476 . . . . . . . . . . 11 ((((Rel 𝐴𝑋𝑉) ∧ (𝑧𝐴 ∧ ((1st𝐴)‘𝑧) = 𝑋)) ∧ 𝑥 = 𝑋) → 𝑥 = 𝑋)
3231opeq1d 4439 . . . . . . . . . 10 ((((Rel 𝐴𝑋𝑉) ∧ (𝑧𝐴 ∧ ((1st𝐴)‘𝑧) = 𝑋)) ∧ 𝑥 = 𝑋) → ⟨𝑥, (2nd𝑧)⟩ = ⟨𝑋, (2nd𝑧)⟩)
3332eleq1d 2715 . . . . . . . . 9 ((((Rel 𝐴𝑋𝑉) ∧ (𝑧𝐴 ∧ ((1st𝐴)‘𝑧) = 𝑋)) ∧ 𝑥 = 𝑋) → (⟨𝑥, (2nd𝑧)⟩ ∈ 𝐴 ↔ ⟨𝑋, (2nd𝑧)⟩ ∈ 𝐴))
34 1st2nd 7258 . . . . . . . . . . . 12 ((Rel 𝐴𝑧𝐴) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
3534ad2ant2r 798 . . . . . . . . . . 11 (((Rel 𝐴𝑋𝑉) ∧ (𝑧𝐴 ∧ ((1st𝐴)‘𝑧) = 𝑋)) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
3628opeq1d 4439 . . . . . . . . . . 11 (((Rel 𝐴𝑋𝑉) ∧ (𝑧𝐴 ∧ ((1st𝐴)‘𝑧) = 𝑋)) → ⟨(1st𝑧), (2nd𝑧)⟩ = ⟨𝑋, (2nd𝑧)⟩)
3735, 36eqtrd 2685 . . . . . . . . . 10 (((Rel 𝐴𝑋𝑉) ∧ (𝑧𝐴 ∧ ((1st𝐴)‘𝑧) = 𝑋)) → 𝑧 = ⟨𝑋, (2nd𝑧)⟩)
38 simprl 809 . . . . . . . . . 10 (((Rel 𝐴𝑋𝑉) ∧ (𝑧𝐴 ∧ ((1st𝐴)‘𝑧) = 𝑋)) → 𝑧𝐴)
3937, 38eqeltrrd 2731 . . . . . . . . 9 (((Rel 𝐴𝑋𝑉) ∧ (𝑧𝐴 ∧ ((1st𝐴)‘𝑧) = 𝑋)) → ⟨𝑋, (2nd𝑧)⟩ ∈ 𝐴)
4030, 33, 39rspcedvd 3348 . . . . . . . 8 (((Rel 𝐴𝑋𝑉) ∧ (𝑧𝐴 ∧ ((1st𝐴)‘𝑧) = 𝑋)) → ∃𝑥 ∈ {𝑋}⟨𝑥, (2nd𝑧)⟩ ∈ 𝐴)
41 df-rex 2947 . . . . . . . 8 (∃𝑥 ∈ {𝑋}⟨𝑥, (2nd𝑧)⟩ ∈ 𝐴 ↔ ∃𝑥(𝑥 ∈ {𝑋} ∧ ⟨𝑥, (2nd𝑧)⟩ ∈ 𝐴))
4240, 41sylib 208 . . . . . . 7 (((Rel 𝐴𝑋𝑉) ∧ (𝑧𝐴 ∧ ((1st𝐴)‘𝑧) = 𝑋)) → ∃𝑥(𝑥 ∈ {𝑋} ∧ ⟨𝑥, (2nd𝑧)⟩ ∈ 𝐴))
43 fvex 6239 . . . . . . . 8 (2nd𝑧) ∈ V
4443elima3 5508 . . . . . . 7 ((2nd𝑧) ∈ (𝐴 “ {𝑋}) ↔ ∃𝑥(𝑥 ∈ {𝑋} ∧ ⟨𝑥, (2nd𝑧)⟩ ∈ 𝐴))
4542, 44sylibr 224 . . . . . 6 (((Rel 𝐴𝑋𝑉) ∧ (𝑧𝐴 ∧ ((1st𝐴)‘𝑧) = 𝑋)) → (2nd𝑧) ∈ (𝐴 “ {𝑋}))
4629, 45jca 553 . . . . 5 (((Rel 𝐴𝑋𝑉) ∧ (𝑧𝐴 ∧ ((1st𝐴)‘𝑧) = 𝑋)) → ((1st𝑧) ∈ {𝑋} ∧ (2nd𝑧) ∈ (𝐴 “ {𝑋})))
4725, 46jca 553 . . . 4 (((Rel 𝐴𝑋𝑉) ∧ (𝑧𝐴 ∧ ((1st𝐴)‘𝑧) = 𝑋)) → (𝑧 ∈ (V × V) ∧ ((1st𝑧) ∈ {𝑋} ∧ (2nd𝑧) ∈ (𝐴 “ {𝑋}))))
4820, 47impbida 895 . . 3 ((Rel 𝐴𝑋𝑉) → ((𝑧 ∈ (V × V) ∧ ((1st𝑧) ∈ {𝑋} ∧ (2nd𝑧) ∈ (𝐴 “ {𝑋}))) ↔ (𝑧𝐴 ∧ ((1st𝐴)‘𝑧) = 𝑋)))
49 elxp7 7245 . . . 4 (𝑧 ∈ ({𝑋} × (𝐴 “ {𝑋})) ↔ (𝑧 ∈ (V × V) ∧ ((1st𝑧) ∈ {𝑋} ∧ (2nd𝑧) ∈ (𝐴 “ {𝑋}))))
5049a1i 11 . . 3 ((Rel 𝐴𝑋𝑉) → (𝑧 ∈ ({𝑋} × (𝐴 “ {𝑋})) ↔ (𝑧 ∈ (V × V) ∧ ((1st𝑧) ∈ {𝑋} ∧ (2nd𝑧) ∈ (𝐴 “ {𝑋})))))
51 fo1st 7230 . . . . . . 7 1st :V–onto→V
52 fofn 6155 . . . . . . 7 (1st :V–onto→V → 1st Fn V)
5351, 52ax-mp 5 . . . . . 6 1st Fn V
54 ssv 3658 . . . . . 6 𝐴 ⊆ V
55 fnssres 6042 . . . . . 6 ((1st Fn V ∧ 𝐴 ⊆ V) → (1st𝐴) Fn 𝐴)
5653, 54, 55mp2an 708 . . . . 5 (1st𝐴) Fn 𝐴
57 fniniseg 6378 . . . . 5 ((1st𝐴) Fn 𝐴 → (𝑧 ∈ ((1st𝐴) “ {𝑋}) ↔ (𝑧𝐴 ∧ ((1st𝐴)‘𝑧) = 𝑋)))
5856, 57ax-mp 5 . . . 4 (𝑧 ∈ ((1st𝐴) “ {𝑋}) ↔ (𝑧𝐴 ∧ ((1st𝐴)‘𝑧) = 𝑋))
5958a1i 11 . . 3 ((Rel 𝐴𝑋𝑉) → (𝑧 ∈ ((1st𝐴) “ {𝑋}) ↔ (𝑧𝐴 ∧ ((1st𝐴)‘𝑧) = 𝑋)))
6048, 50, 593bitr4rd 301 . 2 ((Rel 𝐴𝑋𝑉) → (𝑧 ∈ ((1st𝐴) “ {𝑋}) ↔ 𝑧 ∈ ({𝑋} × (𝐴 “ {𝑋}))))
6160eqrdv 2649 1 ((Rel 𝐴𝑋𝑉) → ((1st𝐴) “ {𝑋}) = ({𝑋} × (𝐴 “ {𝑋})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383   = wceq 1523  wex 1744  wcel 2030  wrex 2942  Vcvv 3231  wss 3607  {csn 4210  cop 4216   × cxp 5141  ccnv 5142  cres 5145  cima 5146  Rel wrel 5148   Fn wfn 5921  ontowfo 5924  cfv 5926  1st c1st 7208  2nd c2nd 7209
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-fo 5932  df-fv 5934  df-1st 7210  df-2nd 7211
This theorem is referenced by:  gsummpt2d  29909
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