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Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  1stpreimas Structured version   Visualization version   GIF version

Theorem 1stpreimas 31922
Description: The preimage of a singleton. (Contributed by Thierry Arnoux, 27-Apr-2020.)
Assertion
Ref Expression
1stpreimas ((Rel š“ āˆ§ š‘‹ āˆˆ š‘‰) ā†’ (ā—”(1st ā†¾ š“) ā€œ {š‘‹}) = ({š‘‹} Ɨ (š“ ā€œ {š‘‹})))

Proof of Theorem 1stpreimas
Dummy variables š‘„ š‘§ are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 1st2ndb 8014 . . . . . . . . 9 (š‘§ āˆˆ (V Ɨ V) ā†” š‘§ = āŸØ(1st ā€˜š‘§), (2nd ā€˜š‘§)āŸ©)
21biimpi 215 . . . . . . . 8 (š‘§ āˆˆ (V Ɨ V) ā†’ š‘§ = āŸØ(1st ā€˜š‘§), (2nd ā€˜š‘§)āŸ©)
32ad2antrl 726 . . . . . . 7 (((Rel š“ āˆ§ š‘‹ āˆˆ š‘‰) āˆ§ (š‘§ āˆˆ (V Ɨ V) āˆ§ ((1st ā€˜š‘§) āˆˆ {š‘‹} āˆ§ (2nd ā€˜š‘§) āˆˆ (š“ ā€œ {š‘‹})))) ā†’ š‘§ = āŸØ(1st ā€˜š‘§), (2nd ā€˜š‘§)āŸ©)
4 fvex 6904 . . . . . . . . . . . 12 (1st ā€˜š‘§) āˆˆ V
54elsn 4643 . . . . . . . . . . 11 ((1st ā€˜š‘§) āˆˆ {š‘‹} ā†” (1st ā€˜š‘§) = š‘‹)
65biimpi 215 . . . . . . . . . 10 ((1st ā€˜š‘§) āˆˆ {š‘‹} ā†’ (1st ā€˜š‘§) = š‘‹)
76ad2antrl 726 . . . . . . . . 9 ((š‘§ āˆˆ (V Ɨ V) āˆ§ ((1st ā€˜š‘§) āˆˆ {š‘‹} āˆ§ (2nd ā€˜š‘§) āˆˆ (š“ ā€œ {š‘‹}))) ā†’ (1st ā€˜š‘§) = š‘‹)
87adantl 482 . . . . . . . 8 (((Rel š“ āˆ§ š‘‹ āˆˆ š‘‰) āˆ§ (š‘§ āˆˆ (V Ɨ V) āˆ§ ((1st ā€˜š‘§) āˆˆ {š‘‹} āˆ§ (2nd ā€˜š‘§) āˆˆ (š“ ā€œ {š‘‹})))) ā†’ (1st ā€˜š‘§) = š‘‹)
98opeq1d 4879 . . . . . . 7 (((Rel š“ āˆ§ š‘‹ āˆˆ š‘‰) āˆ§ (š‘§ āˆˆ (V Ɨ V) āˆ§ ((1st ā€˜š‘§) āˆˆ {š‘‹} āˆ§ (2nd ā€˜š‘§) āˆˆ (š“ ā€œ {š‘‹})))) ā†’ āŸØ(1st ā€˜š‘§), (2nd ā€˜š‘§)āŸ© = āŸØš‘‹, (2nd ā€˜š‘§)āŸ©)
103, 9eqtrd 2772 . . . . . 6 (((Rel š“ āˆ§ š‘‹ āˆˆ š‘‰) āˆ§ (š‘§ āˆˆ (V Ɨ V) āˆ§ ((1st ā€˜š‘§) āˆˆ {š‘‹} āˆ§ (2nd ā€˜š‘§) āˆˆ (š“ ā€œ {š‘‹})))) ā†’ š‘§ = āŸØš‘‹, (2nd ā€˜š‘§)āŸ©)
11 simplr 767 . . . . . . 7 (((Rel š“ āˆ§ š‘‹ āˆˆ š‘‰) āˆ§ (š‘§ āˆˆ (V Ɨ V) āˆ§ ((1st ā€˜š‘§) āˆˆ {š‘‹} āˆ§ (2nd ā€˜š‘§) āˆˆ (š“ ā€œ {š‘‹})))) ā†’ š‘‹ āˆˆ š‘‰)
12 simprrr 780 . . . . . . 7 (((Rel š“ āˆ§ š‘‹ āˆˆ š‘‰) āˆ§ (š‘§ āˆˆ (V Ɨ V) āˆ§ ((1st ā€˜š‘§) āˆˆ {š‘‹} āˆ§ (2nd ā€˜š‘§) āˆˆ (š“ ā€œ {š‘‹})))) ā†’ (2nd ā€˜š‘§) āˆˆ (š“ ā€œ {š‘‹}))
13 elimasng 6087 . . . . . . . 8 ((š‘‹ āˆˆ š‘‰ āˆ§ (2nd ā€˜š‘§) āˆˆ (š“ ā€œ {š‘‹})) ā†’ ((2nd ā€˜š‘§) āˆˆ (š“ ā€œ {š‘‹}) ā†” āŸØš‘‹, (2nd ā€˜š‘§)āŸ© āˆˆ š“))
1413biimpa 477 . . . . . . 7 (((š‘‹ āˆˆ š‘‰ āˆ§ (2nd ā€˜š‘§) āˆˆ (š“ ā€œ {š‘‹})) āˆ§ (2nd ā€˜š‘§) āˆˆ (š“ ā€œ {š‘‹})) ā†’ āŸØš‘‹, (2nd ā€˜š‘§)āŸ© āˆˆ š“)
1511, 12, 12, 14syl21anc 836 . . . . . 6 (((Rel š“ āˆ§ š‘‹ āˆˆ š‘‰) āˆ§ (š‘§ āˆˆ (V Ɨ V) āˆ§ ((1st ā€˜š‘§) āˆˆ {š‘‹} āˆ§ (2nd ā€˜š‘§) āˆˆ (š“ ā€œ {š‘‹})))) ā†’ āŸØš‘‹, (2nd ā€˜š‘§)āŸ© āˆˆ š“)
1610, 15eqeltrd 2833 . . . . 5 (((Rel š“ āˆ§ š‘‹ āˆˆ š‘‰) āˆ§ (š‘§ āˆˆ (V Ɨ V) āˆ§ ((1st ā€˜š‘§) āˆˆ {š‘‹} āˆ§ (2nd ā€˜š‘§) āˆˆ (š“ ā€œ {š‘‹})))) ā†’ š‘§ āˆˆ š“)
17 fvres 6910 . . . . . . 7 (š‘§ āˆˆ š“ ā†’ ((1st ā†¾ š“)ā€˜š‘§) = (1st ā€˜š‘§))
1816, 17syl 17 . . . . . 6 (((Rel š“ āˆ§ š‘‹ āˆˆ š‘‰) āˆ§ (š‘§ āˆˆ (V Ɨ V) āˆ§ ((1st ā€˜š‘§) āˆˆ {š‘‹} āˆ§ (2nd ā€˜š‘§) āˆˆ (š“ ā€œ {š‘‹})))) ā†’ ((1st ā†¾ š“)ā€˜š‘§) = (1st ā€˜š‘§))
1918, 8eqtrd 2772 . . . . 5 (((Rel š“ āˆ§ š‘‹ āˆˆ š‘‰) āˆ§ (š‘§ āˆˆ (V Ɨ V) āˆ§ ((1st ā€˜š‘§) āˆˆ {š‘‹} āˆ§ (2nd ā€˜š‘§) āˆˆ (š“ ā€œ {š‘‹})))) ā†’ ((1st ā†¾ š“)ā€˜š‘§) = š‘‹)
2016, 19jca 512 . . . 4 (((Rel š“ āˆ§ š‘‹ āˆˆ š‘‰) āˆ§ (š‘§ āˆˆ (V Ɨ V) āˆ§ ((1st ā€˜š‘§) āˆˆ {š‘‹} āˆ§ (2nd ā€˜š‘§) āˆˆ (š“ ā€œ {š‘‹})))) ā†’ (š‘§ āˆˆ š“ āˆ§ ((1st ā†¾ š“)ā€˜š‘§) = š‘‹))
21 df-rel 5683 . . . . . . . . 9 (Rel š“ ā†” š“ āŠ† (V Ɨ V))
2221biimpi 215 . . . . . . . 8 (Rel š“ ā†’ š“ āŠ† (V Ɨ V))
2322adantr 481 . . . . . . 7 ((Rel š“ āˆ§ š‘‹ āˆˆ š‘‰) ā†’ š“ āŠ† (V Ɨ V))
2423sselda 3982 . . . . . 6 (((Rel š“ āˆ§ š‘‹ āˆˆ š‘‰) āˆ§ š‘§ āˆˆ š“) ā†’ š‘§ āˆˆ (V Ɨ V))
2524adantrr 715 . . . . 5 (((Rel š“ āˆ§ š‘‹ āˆˆ š‘‰) āˆ§ (š‘§ āˆˆ š“ āˆ§ ((1st ā†¾ š“)ā€˜š‘§) = š‘‹)) ā†’ š‘§ āˆˆ (V Ɨ V))
2617ad2antrl 726 . . . . . . . 8 (((Rel š“ āˆ§ š‘‹ āˆˆ š‘‰) āˆ§ (š‘§ āˆˆ š“ āˆ§ ((1st ā†¾ š“)ā€˜š‘§) = š‘‹)) ā†’ ((1st ā†¾ š“)ā€˜š‘§) = (1st ā€˜š‘§))
27 simprr 771 . . . . . . . 8 (((Rel š“ āˆ§ š‘‹ āˆˆ š‘‰) āˆ§ (š‘§ āˆˆ š“ āˆ§ ((1st ā†¾ š“)ā€˜š‘§) = š‘‹)) ā†’ ((1st ā†¾ š“)ā€˜š‘§) = š‘‹)
2826, 27eqtr3d 2774 . . . . . . 7 (((Rel š“ āˆ§ š‘‹ āˆˆ š‘‰) āˆ§ (š‘§ āˆˆ š“ āˆ§ ((1st ā†¾ š“)ā€˜š‘§) = š‘‹)) ā†’ (1st ā€˜š‘§) = š‘‹)
2928, 5sylibr 233 . . . . . 6 (((Rel š“ āˆ§ š‘‹ āˆˆ š‘‰) āˆ§ (š‘§ āˆˆ š“ āˆ§ ((1st ā†¾ š“)ā€˜š‘§) = š‘‹)) ā†’ (1st ā€˜š‘§) āˆˆ {š‘‹})
3028, 29eqeltrrd 2834 . . . . . . . . 9 (((Rel š“ āˆ§ š‘‹ āˆˆ š‘‰) āˆ§ (š‘§ āˆˆ š“ āˆ§ ((1st ā†¾ š“)ā€˜š‘§) = š‘‹)) ā†’ š‘‹ āˆˆ {š‘‹})
31 simpr 485 . . . . . . . . . . 11 ((((Rel š“ āˆ§ š‘‹ āˆˆ š‘‰) āˆ§ (š‘§ āˆˆ š“ āˆ§ ((1st ā†¾ š“)ā€˜š‘§) = š‘‹)) āˆ§ š‘„ = š‘‹) ā†’ š‘„ = š‘‹)
3231opeq1d 4879 . . . . . . . . . 10 ((((Rel š“ āˆ§ š‘‹ āˆˆ š‘‰) āˆ§ (š‘§ āˆˆ š“ āˆ§ ((1st ā†¾ š“)ā€˜š‘§) = š‘‹)) āˆ§ š‘„ = š‘‹) ā†’ āŸØš‘„, (2nd ā€˜š‘§)āŸ© = āŸØš‘‹, (2nd ā€˜š‘§)āŸ©)
3332eleq1d 2818 . . . . . . . . 9 ((((Rel š“ āˆ§ š‘‹ āˆˆ š‘‰) āˆ§ (š‘§ āˆˆ š“ āˆ§ ((1st ā†¾ š“)ā€˜š‘§) = š‘‹)) āˆ§ š‘„ = š‘‹) ā†’ (āŸØš‘„, (2nd ā€˜š‘§)āŸ© āˆˆ š“ ā†” āŸØš‘‹, (2nd ā€˜š‘§)āŸ© āˆˆ š“))
34 1st2nd 8024 . . . . . . . . . . . 12 ((Rel š“ āˆ§ š‘§ āˆˆ š“) ā†’ š‘§ = āŸØ(1st ā€˜š‘§), (2nd ā€˜š‘§)āŸ©)
3534ad2ant2r 745 . . . . . . . . . . 11 (((Rel š“ āˆ§ š‘‹ āˆˆ š‘‰) āˆ§ (š‘§ āˆˆ š“ āˆ§ ((1st ā†¾ š“)ā€˜š‘§) = š‘‹)) ā†’ š‘§ = āŸØ(1st ā€˜š‘§), (2nd ā€˜š‘§)āŸ©)
3628opeq1d 4879 . . . . . . . . . . 11 (((Rel š“ āˆ§ š‘‹ āˆˆ š‘‰) āˆ§ (š‘§ āˆˆ š“ āˆ§ ((1st ā†¾ š“)ā€˜š‘§) = š‘‹)) ā†’ āŸØ(1st ā€˜š‘§), (2nd ā€˜š‘§)āŸ© = āŸØš‘‹, (2nd ā€˜š‘§)āŸ©)
3735, 36eqtrd 2772 . . . . . . . . . 10 (((Rel š“ āˆ§ š‘‹ āˆˆ š‘‰) āˆ§ (š‘§ āˆˆ š“ āˆ§ ((1st ā†¾ š“)ā€˜š‘§) = š‘‹)) ā†’ š‘§ = āŸØš‘‹, (2nd ā€˜š‘§)āŸ©)
38 simprl 769 . . . . . . . . . 10 (((Rel š“ āˆ§ š‘‹ āˆˆ š‘‰) āˆ§ (š‘§ āˆˆ š“ āˆ§ ((1st ā†¾ š“)ā€˜š‘§) = š‘‹)) ā†’ š‘§ āˆˆ š“)
3937, 38eqeltrrd 2834 . . . . . . . . 9 (((Rel š“ āˆ§ š‘‹ āˆˆ š‘‰) āˆ§ (š‘§ āˆˆ š“ āˆ§ ((1st ā†¾ š“)ā€˜š‘§) = š‘‹)) ā†’ āŸØš‘‹, (2nd ā€˜š‘§)āŸ© āˆˆ š“)
4030, 33, 39rspcedvd 3614 . . . . . . . 8 (((Rel š“ āˆ§ š‘‹ āˆˆ š‘‰) āˆ§ (š‘§ āˆˆ š“ āˆ§ ((1st ā†¾ š“)ā€˜š‘§) = š‘‹)) ā†’ āˆƒš‘„ āˆˆ {š‘‹}āŸØš‘„, (2nd ā€˜š‘§)āŸ© āˆˆ š“)
41 df-rex 3071 . . . . . . . 8 (āˆƒš‘„ āˆˆ {š‘‹}āŸØš‘„, (2nd ā€˜š‘§)āŸ© āˆˆ š“ ā†” āˆƒš‘„(š‘„ āˆˆ {š‘‹} āˆ§ āŸØš‘„, (2nd ā€˜š‘§)āŸ© āˆˆ š“))
4240, 41sylib 217 . . . . . . 7 (((Rel š“ āˆ§ š‘‹ āˆˆ š‘‰) āˆ§ (š‘§ āˆˆ š“ āˆ§ ((1st ā†¾ š“)ā€˜š‘§) = š‘‹)) ā†’ āˆƒš‘„(š‘„ āˆˆ {š‘‹} āˆ§ āŸØš‘„, (2nd ā€˜š‘§)āŸ© āˆˆ š“))
43 fvex 6904 . . . . . . . 8 (2nd ā€˜š‘§) āˆˆ V
4443elima3 6066 . . . . . . 7 ((2nd ā€˜š‘§) āˆˆ (š“ ā€œ {š‘‹}) ā†” āˆƒš‘„(š‘„ āˆˆ {š‘‹} āˆ§ āŸØš‘„, (2nd ā€˜š‘§)āŸ© āˆˆ š“))
4542, 44sylibr 233 . . . . . 6 (((Rel š“ āˆ§ š‘‹ āˆˆ š‘‰) āˆ§ (š‘§ āˆˆ š“ āˆ§ ((1st ā†¾ š“)ā€˜š‘§) = š‘‹)) ā†’ (2nd ā€˜š‘§) āˆˆ (š“ ā€œ {š‘‹}))
4629, 45jca 512 . . . . 5 (((Rel š“ āˆ§ š‘‹ āˆˆ š‘‰) āˆ§ (š‘§ āˆˆ š“ āˆ§ ((1st ā†¾ š“)ā€˜š‘§) = š‘‹)) ā†’ ((1st ā€˜š‘§) āˆˆ {š‘‹} āˆ§ (2nd ā€˜š‘§) āˆˆ (š“ ā€œ {š‘‹})))
4725, 46jca 512 . . . 4 (((Rel š“ āˆ§ š‘‹ āˆˆ š‘‰) āˆ§ (š‘§ āˆˆ š“ āˆ§ ((1st ā†¾ š“)ā€˜š‘§) = š‘‹)) ā†’ (š‘§ āˆˆ (V Ɨ V) āˆ§ ((1st ā€˜š‘§) āˆˆ {š‘‹} āˆ§ (2nd ā€˜š‘§) āˆˆ (š“ ā€œ {š‘‹}))))
4820, 47impbida 799 . . 3 ((Rel š“ āˆ§ š‘‹ āˆˆ š‘‰) ā†’ ((š‘§ āˆˆ (V Ɨ V) āˆ§ ((1st ā€˜š‘§) āˆˆ {š‘‹} āˆ§ (2nd ā€˜š‘§) āˆˆ (š“ ā€œ {š‘‹}))) ā†” (š‘§ āˆˆ š“ āˆ§ ((1st ā†¾ š“)ā€˜š‘§) = š‘‹)))
49 elxp7 8009 . . . 4 (š‘§ āˆˆ ({š‘‹} Ɨ (š“ ā€œ {š‘‹})) ā†” (š‘§ āˆˆ (V Ɨ V) āˆ§ ((1st ā€˜š‘§) āˆˆ {š‘‹} āˆ§ (2nd ā€˜š‘§) āˆˆ (š“ ā€œ {š‘‹}))))
5049a1i 11 . . 3 ((Rel š“ āˆ§ š‘‹ āˆˆ š‘‰) ā†’ (š‘§ āˆˆ ({š‘‹} Ɨ (š“ ā€œ {š‘‹})) ā†” (š‘§ āˆˆ (V Ɨ V) āˆ§ ((1st ā€˜š‘§) āˆˆ {š‘‹} āˆ§ (2nd ā€˜š‘§) āˆˆ (š“ ā€œ {š‘‹})))))
51 fo1st 7994 . . . . . . 7 1st :Vā€“ontoā†’V
52 fofn 6807 . . . . . . 7 (1st :Vā€“ontoā†’V ā†’ 1st Fn V)
5351, 52ax-mp 5 . . . . . 6 1st Fn V
54 ssv 4006 . . . . . 6 š“ āŠ† V
55 fnssres 6673 . . . . . 6 ((1st Fn V āˆ§ š“ āŠ† V) ā†’ (1st ā†¾ š“) Fn š“)
5653, 54, 55mp2an 690 . . . . 5 (1st ā†¾ š“) Fn š“
57 fniniseg 7061 . . . . 5 ((1st ā†¾ š“) Fn š“ ā†’ (š‘§ āˆˆ (ā—”(1st ā†¾ š“) ā€œ {š‘‹}) ā†” (š‘§ āˆˆ š“ āˆ§ ((1st ā†¾ š“)ā€˜š‘§) = š‘‹)))
5856, 57ax-mp 5 . . . 4 (š‘§ āˆˆ (ā—”(1st ā†¾ š“) ā€œ {š‘‹}) ā†” (š‘§ āˆˆ š“ āˆ§ ((1st ā†¾ š“)ā€˜š‘§) = š‘‹))
5958a1i 11 . . 3 ((Rel š“ āˆ§ š‘‹ āˆˆ š‘‰) ā†’ (š‘§ āˆˆ (ā—”(1st ā†¾ š“) ā€œ {š‘‹}) ā†” (š‘§ āˆˆ š“ āˆ§ ((1st ā†¾ š“)ā€˜š‘§) = š‘‹)))
6048, 50, 593bitr4rd 311 . 2 ((Rel š“ āˆ§ š‘‹ āˆˆ š‘‰) ā†’ (š‘§ āˆˆ (ā—”(1st ā†¾ š“) ā€œ {š‘‹}) ā†” š‘§ āˆˆ ({š‘‹} Ɨ (š“ ā€œ {š‘‹}))))
6160eqrdv 2730 1 ((Rel š“ āˆ§ š‘‹ āˆˆ š‘‰) ā†’ (ā—”(1st ā†¾ š“) ā€œ {š‘‹}) = ({š‘‹} Ɨ (š“ ā€œ {š‘‹})))
Colors of variables: wff setvar class
Syntax hints:   ā†’ wi 4   ā†” wb 205   āˆ§ wa 396   = wceq 1541  āˆƒwex 1781   āˆˆ wcel 2106  āˆƒwrex 3070  Vcvv 3474   āŠ† wss 3948  {csn 4628  āŸØcop 4634   Ɨ cxp 5674  ā—”ccnv 5675   ā†¾ cres 5678   ā€œ cima 5679  Rel wrel 5681   Fn wfn 6538  ā€“ontoā†’wfo 6541  ā€˜cfv 6543  1st c1st 7972  2nd c2nd 7973
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 2703  ax-sep 5299  ax-nul 5306  ax-pr 5427  ax-un 7724
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 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-fo 6549  df-fv 6551  df-1st 7974  df-2nd 7975
This theorem is referenced by:  gsummpt2d  32196
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