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Theorem 1stpreimas 28700
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 7075 . . . . . . . . 9 (𝑧 ∈ (V × V) ↔ 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
21biimpi 204 . . . . . . . 8 (𝑧 ∈ (V × V) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
32ad2antrl 759 . . . . . . 7 (((Rel 𝐴𝑋𝑉) ∧ (𝑧 ∈ (V × V) ∧ ((1st𝑧) ∈ {𝑋} ∧ (2nd𝑧) ∈ (𝐴 “ {𝑋})))) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
4 fvex 6098 . . . . . . . . . . . 12 (1st𝑧) ∈ V
54elsn 4139 . . . . . . . . . . 11 ((1st𝑧) ∈ {𝑋} ↔ (1st𝑧) = 𝑋)
65biimpi 204 . . . . . . . . . 10 ((1st𝑧) ∈ {𝑋} → (1st𝑧) = 𝑋)
76ad2antrl 759 . . . . . . . . 9 ((𝑧 ∈ (V × V) ∧ ((1st𝑧) ∈ {𝑋} ∧ (2nd𝑧) ∈ (𝐴 “ {𝑋}))) → (1st𝑧) = 𝑋)
87adantl 480 . . . . . . . 8 (((Rel 𝐴𝑋𝑉) ∧ (𝑧 ∈ (V × V) ∧ ((1st𝑧) ∈ {𝑋} ∧ (2nd𝑧) ∈ (𝐴 “ {𝑋})))) → (1st𝑧) = 𝑋)
98opeq1d 4340 . . . . . . 7 (((Rel 𝐴𝑋𝑉) ∧ (𝑧 ∈ (V × V) ∧ ((1st𝑧) ∈ {𝑋} ∧ (2nd𝑧) ∈ (𝐴 “ {𝑋})))) → ⟨(1st𝑧), (2nd𝑧)⟩ = ⟨𝑋, (2nd𝑧)⟩)
103, 9eqtrd 2643 . . . . . 6 (((Rel 𝐴𝑋𝑉) ∧ (𝑧 ∈ (V × V) ∧ ((1st𝑧) ∈ {𝑋} ∧ (2nd𝑧) ∈ (𝐴 “ {𝑋})))) → 𝑧 = ⟨𝑋, (2nd𝑧)⟩)
11 simplr 787 . . . . . . 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 5397 . . . . . . . 8 ((𝑋𝑉 ∧ (2nd𝑧) ∈ (𝐴 “ {𝑋})) → ((2nd𝑧) ∈ (𝐴 “ {𝑋}) ↔ ⟨𝑋, (2nd𝑧)⟩ ∈ 𝐴))
1413biimpa 499 . . . . . . 7 (((𝑋𝑉 ∧ (2nd𝑧) ∈ (𝐴 “ {𝑋})) ∧ (2nd𝑧) ∈ (𝐴 “ {𝑋})) → ⟨𝑋, (2nd𝑧)⟩ ∈ 𝐴)
1511, 12, 12, 14syl21anc 1316 . . . . . 6 (((Rel 𝐴𝑋𝑉) ∧ (𝑧 ∈ (V × V) ∧ ((1st𝑧) ∈ {𝑋} ∧ (2nd𝑧) ∈ (𝐴 “ {𝑋})))) → ⟨𝑋, (2nd𝑧)⟩ ∈ 𝐴)
1610, 15eqeltrd 2687 . . . . 5 (((Rel 𝐴𝑋𝑉) ∧ (𝑧 ∈ (V × V) ∧ ((1st𝑧) ∈ {𝑋} ∧ (2nd𝑧) ∈ (𝐴 “ {𝑋})))) → 𝑧𝐴)
17 fvres 6102 . . . . . . 7 (𝑧𝐴 → ((1st𝐴)‘𝑧) = (1st𝑧))
1816, 17syl 17 . . . . . 6 (((Rel 𝐴𝑋𝑉) ∧ (𝑧 ∈ (V × V) ∧ ((1st𝑧) ∈ {𝑋} ∧ (2nd𝑧) ∈ (𝐴 “ {𝑋})))) → ((1st𝐴)‘𝑧) = (1st𝑧))
1918, 8eqtrd 2643 . . . . 5 (((Rel 𝐴𝑋𝑉) ∧ (𝑧 ∈ (V × V) ∧ ((1st𝑧) ∈ {𝑋} ∧ (2nd𝑧) ∈ (𝐴 “ {𝑋})))) → ((1st𝐴)‘𝑧) = 𝑋)
2016, 19jca 552 . . . 4 (((Rel 𝐴𝑋𝑉) ∧ (𝑧 ∈ (V × V) ∧ ((1st𝑧) ∈ {𝑋} ∧ (2nd𝑧) ∈ (𝐴 “ {𝑋})))) → (𝑧𝐴 ∧ ((1st𝐴)‘𝑧) = 𝑋))
21 df-rel 5035 . . . . . . . . 9 (Rel 𝐴𝐴 ⊆ (V × V))
2221biimpi 204 . . . . . . . 8 (Rel 𝐴𝐴 ⊆ (V × V))
2322adantr 479 . . . . . . 7 ((Rel 𝐴𝑋𝑉) → 𝐴 ⊆ (V × V))
2423sselda 3567 . . . . . 6 (((Rel 𝐴𝑋𝑉) ∧ 𝑧𝐴) → 𝑧 ∈ (V × V))
2524adantrr 748 . . . . 5 (((Rel 𝐴𝑋𝑉) ∧ (𝑧𝐴 ∧ ((1st𝐴)‘𝑧) = 𝑋)) → 𝑧 ∈ (V × V))
2617ad2antrl 759 . . . . . . . 8 (((Rel 𝐴𝑋𝑉) ∧ (𝑧𝐴 ∧ ((1st𝐴)‘𝑧) = 𝑋)) → ((1st𝐴)‘𝑧) = (1st𝑧))
27 simprr 791 . . . . . . . 8 (((Rel 𝐴𝑋𝑉) ∧ (𝑧𝐴 ∧ ((1st𝐴)‘𝑧) = 𝑋)) → ((1st𝐴)‘𝑧) = 𝑋)
2826, 27eqtr3d 2645 . . . . . . 7 (((Rel 𝐴𝑋𝑉) ∧ (𝑧𝐴 ∧ ((1st𝐴)‘𝑧) = 𝑋)) → (1st𝑧) = 𝑋)
2928, 5sylibr 222 . . . . . 6 (((Rel 𝐴𝑋𝑉) ∧ (𝑧𝐴 ∧ ((1st𝐴)‘𝑧) = 𝑋)) → (1st𝑧) ∈ {𝑋})
3028, 29eqeltrrd 2688 . . . . . . . . 9 (((Rel 𝐴𝑋𝑉) ∧ (𝑧𝐴 ∧ ((1st𝐴)‘𝑧) = 𝑋)) → 𝑋 ∈ {𝑋})
31 simpr 475 . . . . . . . . . . 11 ((((Rel 𝐴𝑋𝑉) ∧ (𝑧𝐴 ∧ ((1st𝐴)‘𝑧) = 𝑋)) ∧ 𝑥 = 𝑋) → 𝑥 = 𝑋)
3231opeq1d 4340 . . . . . . . . . 10 ((((Rel 𝐴𝑋𝑉) ∧ (𝑧𝐴 ∧ ((1st𝐴)‘𝑧) = 𝑋)) ∧ 𝑥 = 𝑋) → ⟨𝑥, (2nd𝑧)⟩ = ⟨𝑋, (2nd𝑧)⟩)
3332eleq1d 2671 . . . . . . . . 9 ((((Rel 𝐴𝑋𝑉) ∧ (𝑧𝐴 ∧ ((1st𝐴)‘𝑧) = 𝑋)) ∧ 𝑥 = 𝑋) → (⟨𝑥, (2nd𝑧)⟩ ∈ 𝐴 ↔ ⟨𝑋, (2nd𝑧)⟩ ∈ 𝐴))
34 1st2nd 7083 . . . . . . . . . . . 12 ((Rel 𝐴𝑧𝐴) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
3534ad2ant2r 778 . . . . . . . . . . 11 (((Rel 𝐴𝑋𝑉) ∧ (𝑧𝐴 ∧ ((1st𝐴)‘𝑧) = 𝑋)) → 𝑧 = ⟨(1st𝑧), (2nd𝑧)⟩)
3628opeq1d 4340 . . . . . . . . . . 11 (((Rel 𝐴𝑋𝑉) ∧ (𝑧𝐴 ∧ ((1st𝐴)‘𝑧) = 𝑋)) → ⟨(1st𝑧), (2nd𝑧)⟩ = ⟨𝑋, (2nd𝑧)⟩)
3735, 36eqtrd 2643 . . . . . . . . . 10 (((Rel 𝐴𝑋𝑉) ∧ (𝑧𝐴 ∧ ((1st𝐴)‘𝑧) = 𝑋)) → 𝑧 = ⟨𝑋, (2nd𝑧)⟩)
38 simprl 789 . . . . . . . . . 10 (((Rel 𝐴𝑋𝑉) ∧ (𝑧𝐴 ∧ ((1st𝐴)‘𝑧) = 𝑋)) → 𝑧𝐴)
3937, 38eqeltrrd 2688 . . . . . . . . 9 (((Rel 𝐴𝑋𝑉) ∧ (𝑧𝐴 ∧ ((1st𝐴)‘𝑧) = 𝑋)) → ⟨𝑋, (2nd𝑧)⟩ ∈ 𝐴)
4030, 33, 39rspcedvd 3288 . . . . . . . 8 (((Rel 𝐴𝑋𝑉) ∧ (𝑧𝐴 ∧ ((1st𝐴)‘𝑧) = 𝑋)) → ∃𝑥 ∈ {𝑋}⟨𝑥, (2nd𝑧)⟩ ∈ 𝐴)
41 df-rex 2901 . . . . . . . 8 (∃𝑥 ∈ {𝑋}⟨𝑥, (2nd𝑧)⟩ ∈ 𝐴 ↔ ∃𝑥(𝑥 ∈ {𝑋} ∧ ⟨𝑥, (2nd𝑧)⟩ ∈ 𝐴))
4240, 41sylib 206 . . . . . . 7 (((Rel 𝐴𝑋𝑉) ∧ (𝑧𝐴 ∧ ((1st𝐴)‘𝑧) = 𝑋)) → ∃𝑥(𝑥 ∈ {𝑋} ∧ ⟨𝑥, (2nd𝑧)⟩ ∈ 𝐴))
43 fvex 6098 . . . . . . . 8 (2nd𝑧) ∈ V
4443elima3 5379 . . . . . . 7 ((2nd𝑧) ∈ (𝐴 “ {𝑋}) ↔ ∃𝑥(𝑥 ∈ {𝑋} ∧ ⟨𝑥, (2nd𝑧)⟩ ∈ 𝐴))
4542, 44sylibr 222 . . . . . 6 (((Rel 𝐴𝑋𝑉) ∧ (𝑧𝐴 ∧ ((1st𝐴)‘𝑧) = 𝑋)) → (2nd𝑧) ∈ (𝐴 “ {𝑋}))
4629, 45jca 552 . . . . 5 (((Rel 𝐴𝑋𝑉) ∧ (𝑧𝐴 ∧ ((1st𝐴)‘𝑧) = 𝑋)) → ((1st𝑧) ∈ {𝑋} ∧ (2nd𝑧) ∈ (𝐴 “ {𝑋})))
4725, 46jca 552 . . . 4 (((Rel 𝐴𝑋𝑉) ∧ (𝑧𝐴 ∧ ((1st𝐴)‘𝑧) = 𝑋)) → (𝑧 ∈ (V × V) ∧ ((1st𝑧) ∈ {𝑋} ∧ (2nd𝑧) ∈ (𝐴 “ {𝑋}))))
4820, 47impbida 872 . . 3 ((Rel 𝐴𝑋𝑉) → ((𝑧 ∈ (V × V) ∧ ((1st𝑧) ∈ {𝑋} ∧ (2nd𝑧) ∈ (𝐴 “ {𝑋}))) ↔ (𝑧𝐴 ∧ ((1st𝐴)‘𝑧) = 𝑋)))
49 elxp7 7070 . . . 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 7057 . . . . . . 7 1st :V–onto→V
52 fofn 6015 . . . . . . 7 (1st :V–onto→V → 1st Fn V)
5351, 52ax-mp 5 . . . . . 6 1st Fn V
54 ssv 3587 . . . . . 6 𝐴 ⊆ V
55 fnssres 5904 . . . . . 6 ((1st Fn V ∧ 𝐴 ⊆ V) → (1st𝐴) Fn 𝐴)
5653, 54, 55mp2an 703 . . . . 5 (1st𝐴) Fn 𝐴
57 fniniseg 6231 . . . . 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 299 . 2 ((Rel 𝐴𝑋𝑉) → (𝑧 ∈ ((1st𝐴) “ {𝑋}) ↔ 𝑧 ∈ ({𝑋} × (𝐴 “ {𝑋}))))
6160eqrdv 2607 1 ((Rel 𝐴𝑋𝑉) → ((1st𝐴) “ {𝑋}) = ({𝑋} × (𝐴 “ {𝑋})))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wa 382   = wceq 1474  wex 1694  wcel 1976  wrex 2896  Vcvv 3172  wss 3539  {csn 4124  cop 4130   × cxp 5026  ccnv 5027  cres 5030  cima 5031  Rel wrel 5033   Fn wfn 5785  ontowfo 5788  cfv 5790  1st c1st 7035  2nd c2nd 7036
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-8 1978  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4712  ax-pow 4764  ax-pr 4828  ax-un 6825
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-eu 2461  df-mo 2462  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ne 2781  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-uni 4367  df-br 4578  df-opab 4638  df-mpt 4639  df-id 4943  df-xp 5034  df-rel 5035  df-cnv 5036  df-co 5037  df-dm 5038  df-rn 5039  df-res 5040  df-ima 5041  df-iota 5754  df-fun 5792  df-fn 5793  df-f 5794  df-fo 5796  df-fv 5798  df-1st 7037  df-2nd 7038
This theorem is referenced by:  gsummpt2d  28946
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