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Theorem 1stpreimas 29932
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 7406 . . . . . . . . 9 (𝑧 ∈ (V × V) ↔ 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
21biimpi 207 . . . . . . . 8 (𝑧 ∈ (V × V) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
32ad2antrl 719 . . . . . . 7 (((Rel 𝐴𝑋𝑉) ∧ (𝑧 ∈ (V × V) ∧ ((1st𝑧) ∈ {𝑋} ∧ (2nd𝑧) ∈ (𝐴 “ {𝑋})))) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
4 fvex 6388 . . . . . . . . . . . 12 (1st𝑧) ∈ V
54elsn 4349 . . . . . . . . . . 11 ((1st𝑧) ∈ {𝑋} ↔ (1st𝑧) = 𝑋)
65biimpi 207 . . . . . . . . . 10 ((1st𝑧) ∈ {𝑋} → (1st𝑧) = 𝑋)
76ad2antrl 719 . . . . . . . . 9 ((𝑧 ∈ (V × V) ∧ ((1st𝑧) ∈ {𝑋} ∧ (2nd𝑧) ∈ (𝐴 “ {𝑋}))) → (1st𝑧) = 𝑋)
87adantl 473 . . . . . . . 8 (((Rel 𝐴𝑋𝑉) ∧ (𝑧 ∈ (V × V) ∧ ((1st𝑧) ∈ {𝑋} ∧ (2nd𝑧) ∈ (𝐴 “ {𝑋})))) → (1st𝑧) = 𝑋)
98opeq1d 4565 . . . . . . 7 (((Rel 𝐴𝑋𝑉) ∧ (𝑧 ∈ (V × V) ∧ ((1st𝑧) ∈ {𝑋} ∧ (2nd𝑧) ∈ (𝐴 “ {𝑋})))) → ⟨(1st𝑧), (2nd𝑧)⟩ = ⟨𝑋, (2nd𝑧)⟩)
103, 9eqtrd 2799 . . . . . 6 (((Rel 𝐴𝑋𝑉) ∧ (𝑧 ∈ (V × V) ∧ ((1st𝑧) ∈ {𝑋} ∧ (2nd𝑧) ∈ (𝐴 “ {𝑋})))) → 𝑧 = ⟨𝑋, (2nd𝑧)⟩)
11 simplr 785 . . . . . . 7 (((Rel 𝐴𝑋𝑉) ∧ (𝑧 ∈ (V × V) ∧ ((1st𝑧) ∈ {𝑋} ∧ (2nd𝑧) ∈ (𝐴 “ {𝑋})))) → 𝑋𝑉)
12 simprrr 800 . . . . . . 7 (((Rel 𝐴𝑋𝑉) ∧ (𝑧 ∈ (V × V) ∧ ((1st𝑧) ∈ {𝑋} ∧ (2nd𝑧) ∈ (𝐴 “ {𝑋})))) → (2nd𝑧) ∈ (𝐴 “ {𝑋}))
13 elimasng 5673 . . . . . . . 8 ((𝑋𝑉 ∧ (2nd𝑧) ∈ (𝐴 “ {𝑋})) → ((2nd𝑧) ∈ (𝐴 “ {𝑋}) ↔ ⟨𝑋, (2nd𝑧)⟩ ∈ 𝐴))
1413biimpa 468 . . . . . . 7 (((𝑋𝑉 ∧ (2nd𝑧) ∈ (𝐴 “ {𝑋})) ∧ (2nd𝑧) ∈ (𝐴 “ {𝑋})) → ⟨𝑋, (2nd𝑧)⟩ ∈ 𝐴)
1511, 12, 12, 14syl21anc 866 . . . . . 6 (((Rel 𝐴𝑋𝑉) ∧ (𝑧 ∈ (V × V) ∧ ((1st𝑧) ∈ {𝑋} ∧ (2nd𝑧) ∈ (𝐴 “ {𝑋})))) → ⟨𝑋, (2nd𝑧)⟩ ∈ 𝐴)
1610, 15eqeltrd 2844 . . . . 5 (((Rel 𝐴𝑋𝑉) ∧ (𝑧 ∈ (V × V) ∧ ((1st𝑧) ∈ {𝑋} ∧ (2nd𝑧) ∈ (𝐴 “ {𝑋})))) → 𝑧𝐴)
17 fvres 6394 . . . . . . 7 (𝑧𝐴 → ((1st𝐴)‘𝑧) = (1st𝑧))
1816, 17syl 17 . . . . . 6 (((Rel 𝐴𝑋𝑉) ∧ (𝑧 ∈ (V × V) ∧ ((1st𝑧) ∈ {𝑋} ∧ (2nd𝑧) ∈ (𝐴 “ {𝑋})))) → ((1st𝐴)‘𝑧) = (1st𝑧))
1918, 8eqtrd 2799 . . . . 5 (((Rel 𝐴𝑋𝑉) ∧ (𝑧 ∈ (V × V) ∧ ((1st𝑧) ∈ {𝑋} ∧ (2nd𝑧) ∈ (𝐴 “ {𝑋})))) → ((1st𝐴)‘𝑧) = 𝑋)
2016, 19jca 507 . . . 4 (((Rel 𝐴𝑋𝑉) ∧ (𝑧 ∈ (V × V) ∧ ((1st𝑧) ∈ {𝑋} ∧ (2nd𝑧) ∈ (𝐴 “ {𝑋})))) → (𝑧𝐴 ∧ ((1st𝐴)‘𝑧) = 𝑋))
21 df-rel 5284 . . . . . . . . 9 (Rel 𝐴𝐴 ⊆ (V × V))
2221biimpi 207 . . . . . . . 8 (Rel 𝐴𝐴 ⊆ (V × V))
2322adantr 472 . . . . . . 7 ((Rel 𝐴𝑋𝑉) → 𝐴 ⊆ (V × V))
2423sselda 3761 . . . . . 6 (((Rel 𝐴𝑋𝑉) ∧ 𝑧𝐴) → 𝑧 ∈ (V × V))
2524adantrr 708 . . . . 5 (((Rel 𝐴𝑋𝑉) ∧ (𝑧𝐴 ∧ ((1st𝐴)‘𝑧) = 𝑋)) → 𝑧 ∈ (V × V))
2617ad2antrl 719 . . . . . . . 8 (((Rel 𝐴𝑋𝑉) ∧ (𝑧𝐴 ∧ ((1st𝐴)‘𝑧) = 𝑋)) → ((1st𝐴)‘𝑧) = (1st𝑧))
27 simprr 789 . . . . . . . 8 (((Rel 𝐴𝑋𝑉) ∧ (𝑧𝐴 ∧ ((1st𝐴)‘𝑧) = 𝑋)) → ((1st𝐴)‘𝑧) = 𝑋)
2826, 27eqtr3d 2801 . . . . . . 7 (((Rel 𝐴𝑋𝑉) ∧ (𝑧𝐴 ∧ ((1st𝐴)‘𝑧) = 𝑋)) → (1st𝑧) = 𝑋)
2928, 5sylibr 225 . . . . . 6 (((Rel 𝐴𝑋𝑉) ∧ (𝑧𝐴 ∧ ((1st𝐴)‘𝑧) = 𝑋)) → (1st𝑧) ∈ {𝑋})
3028, 29eqeltrrd 2845 . . . . . . . . 9 (((Rel 𝐴𝑋𝑉) ∧ (𝑧𝐴 ∧ ((1st𝐴)‘𝑧) = 𝑋)) → 𝑋 ∈ {𝑋})
31 simpr 477 . . . . . . . . . . 11 ((((Rel 𝐴𝑋𝑉) ∧ (𝑧𝐴 ∧ ((1st𝐴)‘𝑧) = 𝑋)) ∧ 𝑥 = 𝑋) → 𝑥 = 𝑋)
3231opeq1d 4565 . . . . . . . . . 10 ((((Rel 𝐴𝑋𝑉) ∧ (𝑧𝐴 ∧ ((1st𝐴)‘𝑧) = 𝑋)) ∧ 𝑥 = 𝑋) → ⟨𝑥, (2nd𝑧)⟩ = ⟨𝑋, (2nd𝑧)⟩)
3332eleq1d 2829 . . . . . . . . 9 ((((Rel 𝐴𝑋𝑉) ∧ (𝑧𝐴 ∧ ((1st𝐴)‘𝑧) = 𝑋)) ∧ 𝑥 = 𝑋) → (⟨𝑥, (2nd𝑧)⟩ ∈ 𝐴 ↔ ⟨𝑋, (2nd𝑧)⟩ ∈ 𝐴))
34 1st2nd 7414 . . . . . . . . . . . 12 ((Rel 𝐴𝑧𝐴) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
3534ad2ant2r 753 . . . . . . . . . . 11 (((Rel 𝐴𝑋𝑉) ∧ (𝑧𝐴 ∧ ((1st𝐴)‘𝑧) = 𝑋)) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
3628opeq1d 4565 . . . . . . . . . . 11 (((Rel 𝐴𝑋𝑉) ∧ (𝑧𝐴 ∧ ((1st𝐴)‘𝑧) = 𝑋)) → ⟨(1st𝑧), (2nd𝑧)⟩ = ⟨𝑋, (2nd𝑧)⟩)
3735, 36eqtrd 2799 . . . . . . . . . 10 (((Rel 𝐴𝑋𝑉) ∧ (𝑧𝐴 ∧ ((1st𝐴)‘𝑧) = 𝑋)) → 𝑧 = ⟨𝑋, (2nd𝑧)⟩)
38 simprl 787 . . . . . . . . . 10 (((Rel 𝐴𝑋𝑉) ∧ (𝑧𝐴 ∧ ((1st𝐴)‘𝑧) = 𝑋)) → 𝑧𝐴)
3937, 38eqeltrrd 2845 . . . . . . . . 9 (((Rel 𝐴𝑋𝑉) ∧ (𝑧𝐴 ∧ ((1st𝐴)‘𝑧) = 𝑋)) → ⟨𝑋, (2nd𝑧)⟩ ∈ 𝐴)
4030, 33, 39rspcedvd 3468 . . . . . . . 8 (((Rel 𝐴𝑋𝑉) ∧ (𝑧𝐴 ∧ ((1st𝐴)‘𝑧) = 𝑋)) → ∃𝑥 ∈ {𝑋}⟨𝑥, (2nd𝑧)⟩ ∈ 𝐴)
41 df-rex 3061 . . . . . . . 8 (∃𝑥 ∈ {𝑋}⟨𝑥, (2nd𝑧)⟩ ∈ 𝐴 ↔ ∃𝑥(𝑥 ∈ {𝑋} ∧ ⟨𝑥, (2nd𝑧)⟩ ∈ 𝐴))
4240, 41sylib 209 . . . . . . 7 (((Rel 𝐴𝑋𝑉) ∧ (𝑧𝐴 ∧ ((1st𝐴)‘𝑧) = 𝑋)) → ∃𝑥(𝑥 ∈ {𝑋} ∧ ⟨𝑥, (2nd𝑧)⟩ ∈ 𝐴))
43 fvex 6388 . . . . . . . 8 (2nd𝑧) ∈ V
4443elima3 5655 . . . . . . 7 ((2nd𝑧) ∈ (𝐴 “ {𝑋}) ↔ ∃𝑥(𝑥 ∈ {𝑋} ∧ ⟨𝑥, (2nd𝑧)⟩ ∈ 𝐴))
4542, 44sylibr 225 . . . . . 6 (((Rel 𝐴𝑋𝑉) ∧ (𝑧𝐴 ∧ ((1st𝐴)‘𝑧) = 𝑋)) → (2nd𝑧) ∈ (𝐴 “ {𝑋}))
4629, 45jca 507 . . . . 5 (((Rel 𝐴𝑋𝑉) ∧ (𝑧𝐴 ∧ ((1st𝐴)‘𝑧) = 𝑋)) → ((1st𝑧) ∈ {𝑋} ∧ (2nd𝑧) ∈ (𝐴 “ {𝑋})))
4725, 46jca 507 . . . 4 (((Rel 𝐴𝑋𝑉) ∧ (𝑧𝐴 ∧ ((1st𝐴)‘𝑧) = 𝑋)) → (𝑧 ∈ (V × V) ∧ ((1st𝑧) ∈ {𝑋} ∧ (2nd𝑧) ∈ (𝐴 “ {𝑋}))))
4820, 47impbida 835 . . 3 ((Rel 𝐴𝑋𝑉) → ((𝑧 ∈ (V × V) ∧ ((1st𝑧) ∈ {𝑋} ∧ (2nd𝑧) ∈ (𝐴 “ {𝑋}))) ↔ (𝑧𝐴 ∧ ((1st𝐴)‘𝑧) = 𝑋)))
49 elxp7 7401 . . . 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 7386 . . . . . . 7 1st :V–onto→V
52 fofn 6300 . . . . . . 7 (1st :V–onto→V → 1st Fn V)
5351, 52ax-mp 5 . . . . . 6 1st Fn V
54 ssv 3785 . . . . . 6 𝐴 ⊆ V
55 fnssres 6182 . . . . . 6 ((1st Fn V ∧ 𝐴 ⊆ V) → (1st𝐴) Fn 𝐴)
5653, 54, 55mp2an 683 . . . . 5 (1st𝐴) Fn 𝐴
57 fniniseg 6528 . . . . 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 303 . 2 ((Rel 𝐴𝑋𝑉) → (𝑧 ∈ ((1st𝐴) “ {𝑋}) ↔ 𝑧 ∈ ({𝑋} × (𝐴 “ {𝑋}))))
6160eqrdv 2763 1 ((Rel 𝐴𝑋𝑉) → ((1st𝐴) “ {𝑋}) = ({𝑋} × (𝐴 “ {𝑋})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 197  wa 384   = wceq 1652  wex 1874  wcel 2155  wrex 3056  Vcvv 3350  wss 3732  {csn 4334  cop 4340   × cxp 5275  ccnv 5276  cres 5279  cima 5280  Rel wrel 5282   Fn wfn 6063  ontowfo 6066  cfv 6068  1st c1st 7364  2nd c2nd 7365
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-rab 3064  df-v 3352  df-sbc 3597  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-nul 4080  df-if 4244  df-sn 4335  df-pr 4337  df-op 4341  df-uni 4595  df-br 4810  df-opab 4872  df-mpt 4889  df-id 5185  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-fo 6074  df-fv 6076  df-1st 7366  df-2nd 7367
This theorem is referenced by:  gsummpt2d  30228
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