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Theorem 1stpreimas 31462
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 7953 . . . . . . . . 9 (𝑧 ∈ (V × V) ↔ 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
21biimpi 215 . . . . . . . 8 (𝑧 ∈ (V × V) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
32ad2antrl 726 . . . . . . 7 (((Rel 𝐴𝑋𝑉) ∧ (𝑧 ∈ (V × V) ∧ ((1st𝑧) ∈ {𝑋} ∧ (2nd𝑧) ∈ (𝐴 “ {𝑋})))) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
4 fvex 6852 . . . . . . . . . . . 12 (1st𝑧) ∈ V
54elsn 4599 . . . . . . . . . . 11 ((1st𝑧) ∈ {𝑋} ↔ (1st𝑧) = 𝑋)
65biimpi 215 . . . . . . . . . 10 ((1st𝑧) ∈ {𝑋} → (1st𝑧) = 𝑋)
76ad2antrl 726 . . . . . . . . 9 ((𝑧 ∈ (V × V) ∧ ((1st𝑧) ∈ {𝑋} ∧ (2nd𝑧) ∈ (𝐴 “ {𝑋}))) → (1st𝑧) = 𝑋)
87adantl 482 . . . . . . . 8 (((Rel 𝐴𝑋𝑉) ∧ (𝑧 ∈ (V × V) ∧ ((1st𝑧) ∈ {𝑋} ∧ (2nd𝑧) ∈ (𝐴 “ {𝑋})))) → (1st𝑧) = 𝑋)
98opeq1d 4834 . . . . . . 7 (((Rel 𝐴𝑋𝑉) ∧ (𝑧 ∈ (V × V) ∧ ((1st𝑧) ∈ {𝑋} ∧ (2nd𝑧) ∈ (𝐴 “ {𝑋})))) → ⟨(1st𝑧), (2nd𝑧)⟩ = ⟨𝑋, (2nd𝑧)⟩)
103, 9eqtrd 2776 . . . . . 6 (((Rel 𝐴𝑋𝑉) ∧ (𝑧 ∈ (V × V) ∧ ((1st𝑧) ∈ {𝑋} ∧ (2nd𝑧) ∈ (𝐴 “ {𝑋})))) → 𝑧 = ⟨𝑋, (2nd𝑧)⟩)
11 simplr 767 . . . . . . 7 (((Rel 𝐴𝑋𝑉) ∧ (𝑧 ∈ (V × V) ∧ ((1st𝑧) ∈ {𝑋} ∧ (2nd𝑧) ∈ (𝐴 “ {𝑋})))) → 𝑋𝑉)
12 simprrr 780 . . . . . . 7 (((Rel 𝐴𝑋𝑉) ∧ (𝑧 ∈ (V × V) ∧ ((1st𝑧) ∈ {𝑋} ∧ (2nd𝑧) ∈ (𝐴 “ {𝑋})))) → (2nd𝑧) ∈ (𝐴 “ {𝑋}))
13 elimasng 6038 . . . . . . . 8 ((𝑋𝑉 ∧ (2nd𝑧) ∈ (𝐴 “ {𝑋})) → ((2nd𝑧) ∈ (𝐴 “ {𝑋}) ↔ ⟨𝑋, (2nd𝑧)⟩ ∈ 𝐴))
1413biimpa 477 . . . . . . 7 (((𝑋𝑉 ∧ (2nd𝑧) ∈ (𝐴 “ {𝑋})) ∧ (2nd𝑧) ∈ (𝐴 “ {𝑋})) → ⟨𝑋, (2nd𝑧)⟩ ∈ 𝐴)
1511, 12, 12, 14syl21anc 836 . . . . . 6 (((Rel 𝐴𝑋𝑉) ∧ (𝑧 ∈ (V × V) ∧ ((1st𝑧) ∈ {𝑋} ∧ (2nd𝑧) ∈ (𝐴 “ {𝑋})))) → ⟨𝑋, (2nd𝑧)⟩ ∈ 𝐴)
1610, 15eqeltrd 2838 . . . . 5 (((Rel 𝐴𝑋𝑉) ∧ (𝑧 ∈ (V × V) ∧ ((1st𝑧) ∈ {𝑋} ∧ (2nd𝑧) ∈ (𝐴 “ {𝑋})))) → 𝑧𝐴)
17 fvres 6858 . . . . . . 7 (𝑧𝐴 → ((1st𝐴)‘𝑧) = (1st𝑧))
1816, 17syl 17 . . . . . 6 (((Rel 𝐴𝑋𝑉) ∧ (𝑧 ∈ (V × V) ∧ ((1st𝑧) ∈ {𝑋} ∧ (2nd𝑧) ∈ (𝐴 “ {𝑋})))) → ((1st𝐴)‘𝑧) = (1st𝑧))
1918, 8eqtrd 2776 . . . . 5 (((Rel 𝐴𝑋𝑉) ∧ (𝑧 ∈ (V × V) ∧ ((1st𝑧) ∈ {𝑋} ∧ (2nd𝑧) ∈ (𝐴 “ {𝑋})))) → ((1st𝐴)‘𝑧) = 𝑋)
2016, 19jca 512 . . . 4 (((Rel 𝐴𝑋𝑉) ∧ (𝑧 ∈ (V × V) ∧ ((1st𝑧) ∈ {𝑋} ∧ (2nd𝑧) ∈ (𝐴 “ {𝑋})))) → (𝑧𝐴 ∧ ((1st𝐴)‘𝑧) = 𝑋))
21 df-rel 5638 . . . . . . . . 9 (Rel 𝐴𝐴 ⊆ (V × V))
2221biimpi 215 . . . . . . . 8 (Rel 𝐴𝐴 ⊆ (V × V))
2322adantr 481 . . . . . . 7 ((Rel 𝐴𝑋𝑉) → 𝐴 ⊆ (V × V))
2423sselda 3942 . . . . . 6 (((Rel 𝐴𝑋𝑉) ∧ 𝑧𝐴) → 𝑧 ∈ (V × V))
2524adantrr 715 . . . . 5 (((Rel 𝐴𝑋𝑉) ∧ (𝑧𝐴 ∧ ((1st𝐴)‘𝑧) = 𝑋)) → 𝑧 ∈ (V × V))
2617ad2antrl 726 . . . . . . . 8 (((Rel 𝐴𝑋𝑉) ∧ (𝑧𝐴 ∧ ((1st𝐴)‘𝑧) = 𝑋)) → ((1st𝐴)‘𝑧) = (1st𝑧))
27 simprr 771 . . . . . . . 8 (((Rel 𝐴𝑋𝑉) ∧ (𝑧𝐴 ∧ ((1st𝐴)‘𝑧) = 𝑋)) → ((1st𝐴)‘𝑧) = 𝑋)
2826, 27eqtr3d 2778 . . . . . . 7 (((Rel 𝐴𝑋𝑉) ∧ (𝑧𝐴 ∧ ((1st𝐴)‘𝑧) = 𝑋)) → (1st𝑧) = 𝑋)
2928, 5sylibr 233 . . . . . 6 (((Rel 𝐴𝑋𝑉) ∧ (𝑧𝐴 ∧ ((1st𝐴)‘𝑧) = 𝑋)) → (1st𝑧) ∈ {𝑋})
3028, 29eqeltrrd 2839 . . . . . . . . 9 (((Rel 𝐴𝑋𝑉) ∧ (𝑧𝐴 ∧ ((1st𝐴)‘𝑧) = 𝑋)) → 𝑋 ∈ {𝑋})
31 simpr 485 . . . . . . . . . . 11 ((((Rel 𝐴𝑋𝑉) ∧ (𝑧𝐴 ∧ ((1st𝐴)‘𝑧) = 𝑋)) ∧ 𝑥 = 𝑋) → 𝑥 = 𝑋)
3231opeq1d 4834 . . . . . . . . . 10 ((((Rel 𝐴𝑋𝑉) ∧ (𝑧𝐴 ∧ ((1st𝐴)‘𝑧) = 𝑋)) ∧ 𝑥 = 𝑋) → ⟨𝑥, (2nd𝑧)⟩ = ⟨𝑋, (2nd𝑧)⟩)
3332eleq1d 2822 . . . . . . . . 9 ((((Rel 𝐴𝑋𝑉) ∧ (𝑧𝐴 ∧ ((1st𝐴)‘𝑧) = 𝑋)) ∧ 𝑥 = 𝑋) → (⟨𝑥, (2nd𝑧)⟩ ∈ 𝐴 ↔ ⟨𝑋, (2nd𝑧)⟩ ∈ 𝐴))
34 1st2nd 7963 . . . . . . . . . . . 12 ((Rel 𝐴𝑧𝐴) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
3534ad2ant2r 745 . . . . . . . . . . 11 (((Rel 𝐴𝑋𝑉) ∧ (𝑧𝐴 ∧ ((1st𝐴)‘𝑧) = 𝑋)) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
3628opeq1d 4834 . . . . . . . . . . 11 (((Rel 𝐴𝑋𝑉) ∧ (𝑧𝐴 ∧ ((1st𝐴)‘𝑧) = 𝑋)) → ⟨(1st𝑧), (2nd𝑧)⟩ = ⟨𝑋, (2nd𝑧)⟩)
3735, 36eqtrd 2776 . . . . . . . . . 10 (((Rel 𝐴𝑋𝑉) ∧ (𝑧𝐴 ∧ ((1st𝐴)‘𝑧) = 𝑋)) → 𝑧 = ⟨𝑋, (2nd𝑧)⟩)
38 simprl 769 . . . . . . . . . 10 (((Rel 𝐴𝑋𝑉) ∧ (𝑧𝐴 ∧ ((1st𝐴)‘𝑧) = 𝑋)) → 𝑧𝐴)
3937, 38eqeltrrd 2839 . . . . . . . . 9 (((Rel 𝐴𝑋𝑉) ∧ (𝑧𝐴 ∧ ((1st𝐴)‘𝑧) = 𝑋)) → ⟨𝑋, (2nd𝑧)⟩ ∈ 𝐴)
4030, 33, 39rspcedvd 3581 . . . . . . . 8 (((Rel 𝐴𝑋𝑉) ∧ (𝑧𝐴 ∧ ((1st𝐴)‘𝑧) = 𝑋)) → ∃𝑥 ∈ {𝑋}⟨𝑥, (2nd𝑧)⟩ ∈ 𝐴)
41 df-rex 3072 . . . . . . . 8 (∃𝑥 ∈ {𝑋}⟨𝑥, (2nd𝑧)⟩ ∈ 𝐴 ↔ ∃𝑥(𝑥 ∈ {𝑋} ∧ ⟨𝑥, (2nd𝑧)⟩ ∈ 𝐴))
4240, 41sylib 217 . . . . . . 7 (((Rel 𝐴𝑋𝑉) ∧ (𝑧𝐴 ∧ ((1st𝐴)‘𝑧) = 𝑋)) → ∃𝑥(𝑥 ∈ {𝑋} ∧ ⟨𝑥, (2nd𝑧)⟩ ∈ 𝐴))
43 fvex 6852 . . . . . . . 8 (2nd𝑧) ∈ V
4443elima3 6018 . . . . . . 7 ((2nd𝑧) ∈ (𝐴 “ {𝑋}) ↔ ∃𝑥(𝑥 ∈ {𝑋} ∧ ⟨𝑥, (2nd𝑧)⟩ ∈ 𝐴))
4542, 44sylibr 233 . . . . . 6 (((Rel 𝐴𝑋𝑉) ∧ (𝑧𝐴 ∧ ((1st𝐴)‘𝑧) = 𝑋)) → (2nd𝑧) ∈ (𝐴 “ {𝑋}))
4629, 45jca 512 . . . . 5 (((Rel 𝐴𝑋𝑉) ∧ (𝑧𝐴 ∧ ((1st𝐴)‘𝑧) = 𝑋)) → ((1st𝑧) ∈ {𝑋} ∧ (2nd𝑧) ∈ (𝐴 “ {𝑋})))
4725, 46jca 512 . . . 4 (((Rel 𝐴𝑋𝑉) ∧ (𝑧𝐴 ∧ ((1st𝐴)‘𝑧) = 𝑋)) → (𝑧 ∈ (V × V) ∧ ((1st𝑧) ∈ {𝑋} ∧ (2nd𝑧) ∈ (𝐴 “ {𝑋}))))
4820, 47impbida 799 . . 3 ((Rel 𝐴𝑋𝑉) → ((𝑧 ∈ (V × V) ∧ ((1st𝑧) ∈ {𝑋} ∧ (2nd𝑧) ∈ (𝐴 “ {𝑋}))) ↔ (𝑧𝐴 ∧ ((1st𝐴)‘𝑧) = 𝑋)))
49 elxp7 7948 . . . 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 7933 . . . . . . 7 1st :V–onto→V
52 fofn 6755 . . . . . . 7 (1st :V–onto→V → 1st Fn V)
5351, 52ax-mp 5 . . . . . 6 1st Fn V
54 ssv 3966 . . . . . 6 𝐴 ⊆ V
55 fnssres 6621 . . . . . 6 ((1st Fn V ∧ 𝐴 ⊆ V) → (1st𝐴) Fn 𝐴)
5653, 54, 55mp2an 690 . . . . 5 (1st𝐴) Fn 𝐴
57 fniniseg 7007 . . . . 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 311 . 2 ((Rel 𝐴𝑋𝑉) → (𝑧 ∈ ((1st𝐴) “ {𝑋}) ↔ 𝑧 ∈ ({𝑋} × (𝐴 “ {𝑋}))))
6160eqrdv 2734 1 ((Rel 𝐴𝑋𝑉) → ((1st𝐴) “ {𝑋}) = ({𝑋} × (𝐴 “ {𝑋})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396   = wceq 1541  wex 1781  wcel 2106  wrex 3071  Vcvv 3443  wss 3908  {csn 4584  cop 4590   × cxp 5629  ccnv 5630  cres 5633  cima 5634  Rel wrel 5636   Fn wfn 6488  ontowfo 6491  cfv 6493  1st c1st 7911  2nd c2nd 7912
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5254  ax-nul 5261  ax-pr 5382  ax-un 7664
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4864  df-br 5104  df-opab 5166  df-mpt 5187  df-id 5529  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6445  df-fun 6495  df-fn 6496  df-f 6497  df-fo 6499  df-fv 6501  df-1st 7913  df-2nd 7914
This theorem is referenced by:  gsummpt2d  31733
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