Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  1stpreima Structured version   Visualization version   GIF version

Theorem 1stpreima 31667
Description: The preimage by 1st is a 'vertical band'. (Contributed by Thierry Arnoux, 13-Oct-2017.)
Assertion
Ref Expression
1stpreima (š“ āŠ† šµ ā†’ (ā—”(1st ā†¾ (šµ Ɨ š¶)) ā€œ š“) = (š“ Ɨ š¶))

Proof of Theorem 1stpreima
Dummy variable š‘¤ is distinct from all other variables.
StepHypRef Expression
1 elxp7 7957 . . . . . 6 (š‘¤ āˆˆ (šµ Ɨ š¶) ā†” (š‘¤ āˆˆ (V Ɨ V) āˆ§ ((1st ā€˜š‘¤) āˆˆ šµ āˆ§ (2nd ā€˜š‘¤) āˆˆ š¶)))
21anbi2i 624 . . . . 5 (((1st ā€˜š‘¤) āˆˆ š“ āˆ§ š‘¤ āˆˆ (šµ Ɨ š¶)) ā†” ((1st ā€˜š‘¤) āˆˆ š“ āˆ§ (š‘¤ āˆˆ (V Ɨ V) āˆ§ ((1st ā€˜š‘¤) āˆˆ šµ āˆ§ (2nd ā€˜š‘¤) āˆˆ š¶))))
3 anass 470 . . . . . . 7 ((((1st ā€˜š‘¤) āˆˆ š“ āˆ§ (1st ā€˜š‘¤) āˆˆ šµ) āˆ§ (š‘¤ āˆˆ (V Ɨ V) āˆ§ (2nd ā€˜š‘¤) āˆˆ š¶)) ā†” ((1st ā€˜š‘¤) āˆˆ š“ āˆ§ ((1st ā€˜š‘¤) āˆˆ šµ āˆ§ (š‘¤ āˆˆ (V Ɨ V) āˆ§ (2nd ā€˜š‘¤) āˆˆ š¶))))
43a1i 11 . . . . . 6 (š“ āŠ† šµ ā†’ ((((1st ā€˜š‘¤) āˆˆ š“ āˆ§ (1st ā€˜š‘¤) āˆˆ šµ) āˆ§ (š‘¤ āˆˆ (V Ɨ V) āˆ§ (2nd ā€˜š‘¤) āˆˆ š¶)) ā†” ((1st ā€˜š‘¤) āˆˆ š“ āˆ§ ((1st ā€˜š‘¤) āˆˆ šµ āˆ§ (š‘¤ āˆˆ (V Ɨ V) āˆ§ (2nd ā€˜š‘¤) āˆˆ š¶)))))
5 ssel 3938 . . . . . . . 8 (š“ āŠ† šµ ā†’ ((1st ā€˜š‘¤) āˆˆ š“ ā†’ (1st ā€˜š‘¤) āˆˆ šµ))
65pm4.71d 563 . . . . . . 7 (š“ āŠ† šµ ā†’ ((1st ā€˜š‘¤) āˆˆ š“ ā†” ((1st ā€˜š‘¤) āˆˆ š“ āˆ§ (1st ā€˜š‘¤) āˆˆ šµ)))
76anbi1d 631 . . . . . 6 (š“ āŠ† šµ ā†’ (((1st ā€˜š‘¤) āˆˆ š“ āˆ§ (š‘¤ āˆˆ (V Ɨ V) āˆ§ (2nd ā€˜š‘¤) āˆˆ š¶)) ā†” (((1st ā€˜š‘¤) āˆˆ š“ āˆ§ (1st ā€˜š‘¤) āˆˆ šµ) āˆ§ (š‘¤ āˆˆ (V Ɨ V) āˆ§ (2nd ā€˜š‘¤) āˆˆ š¶))))
8 an12 644 . . . . . . . 8 ((š‘¤ āˆˆ (V Ɨ V) āˆ§ ((1st ā€˜š‘¤) āˆˆ šµ āˆ§ (2nd ā€˜š‘¤) āˆˆ š¶)) ā†” ((1st ā€˜š‘¤) āˆˆ šµ āˆ§ (š‘¤ āˆˆ (V Ɨ V) āˆ§ (2nd ā€˜š‘¤) āˆˆ š¶)))
98anbi2i 624 . . . . . . 7 (((1st ā€˜š‘¤) āˆˆ š“ āˆ§ (š‘¤ āˆˆ (V Ɨ V) āˆ§ ((1st ā€˜š‘¤) āˆˆ šµ āˆ§ (2nd ā€˜š‘¤) āˆˆ š¶))) ā†” ((1st ā€˜š‘¤) āˆˆ š“ āˆ§ ((1st ā€˜š‘¤) āˆˆ šµ āˆ§ (š‘¤ āˆˆ (V Ɨ V) āˆ§ (2nd ā€˜š‘¤) āˆˆ š¶))))
109a1i 11 . . . . . 6 (š“ āŠ† šµ ā†’ (((1st ā€˜š‘¤) āˆˆ š“ āˆ§ (š‘¤ āˆˆ (V Ɨ V) āˆ§ ((1st ā€˜š‘¤) āˆˆ šµ āˆ§ (2nd ā€˜š‘¤) āˆˆ š¶))) ā†” ((1st ā€˜š‘¤) āˆˆ š“ āˆ§ ((1st ā€˜š‘¤) āˆˆ šµ āˆ§ (š‘¤ āˆˆ (V Ɨ V) āˆ§ (2nd ā€˜š‘¤) āˆˆ š¶)))))
114, 7, 103bitr4d 311 . . . . 5 (š“ āŠ† šµ ā†’ (((1st ā€˜š‘¤) āˆˆ š“ āˆ§ (š‘¤ āˆˆ (V Ɨ V) āˆ§ (2nd ā€˜š‘¤) āˆˆ š¶)) ā†” ((1st ā€˜š‘¤) āˆˆ š“ āˆ§ (š‘¤ āˆˆ (V Ɨ V) āˆ§ ((1st ā€˜š‘¤) āˆˆ šµ āˆ§ (2nd ā€˜š‘¤) āˆˆ š¶)))))
122, 11bitr4id 290 . . . 4 (š“ āŠ† šµ ā†’ (((1st ā€˜š‘¤) āˆˆ š“ āˆ§ š‘¤ āˆˆ (šµ Ɨ š¶)) ā†” ((1st ā€˜š‘¤) āˆˆ š“ āˆ§ (š‘¤ āˆˆ (V Ɨ V) āˆ§ (2nd ā€˜š‘¤) āˆˆ š¶))))
13 an12 644 . . . 4 (((1st ā€˜š‘¤) āˆˆ š“ āˆ§ (š‘¤ āˆˆ (V Ɨ V) āˆ§ (2nd ā€˜š‘¤) āˆˆ š¶)) ā†” (š‘¤ āˆˆ (V Ɨ V) āˆ§ ((1st ā€˜š‘¤) āˆˆ š“ āˆ§ (2nd ā€˜š‘¤) āˆˆ š¶)))
1412, 13bitrdi 287 . . 3 (š“ āŠ† šµ ā†’ (((1st ā€˜š‘¤) āˆˆ š“ āˆ§ š‘¤ āˆˆ (šµ Ɨ š¶)) ā†” (š‘¤ āˆˆ (V Ɨ V) āˆ§ ((1st ā€˜š‘¤) āˆˆ š“ āˆ§ (2nd ā€˜š‘¤) āˆˆ š¶))))
15 cnvresima 6183 . . . . 5 (ā—”(1st ā†¾ (šµ Ɨ š¶)) ā€œ š“) = ((ā—”1st ā€œ š“) āˆ© (šµ Ɨ š¶))
1615eleq2i 2826 . . . 4 (š‘¤ āˆˆ (ā—”(1st ā†¾ (šµ Ɨ š¶)) ā€œ š“) ā†” š‘¤ āˆˆ ((ā—”1st ā€œ š“) āˆ© (šµ Ɨ š¶)))
17 elin 3927 . . . 4 (š‘¤ āˆˆ ((ā—”1st ā€œ š“) āˆ© (šµ Ɨ š¶)) ā†” (š‘¤ āˆˆ (ā—”1st ā€œ š“) āˆ§ š‘¤ āˆˆ (šµ Ɨ š¶)))
18 vex 3448 . . . . . 6 š‘¤ āˆˆ V
19 fo1st 7942 . . . . . . 7 1st :Vā€“ontoā†’V
20 fofn 6759 . . . . . . 7 (1st :Vā€“ontoā†’V ā†’ 1st Fn V)
21 elpreima 7009 . . . . . . 7 (1st Fn V ā†’ (š‘¤ āˆˆ (ā—”1st ā€œ š“) ā†” (š‘¤ āˆˆ V āˆ§ (1st ā€˜š‘¤) āˆˆ š“)))
2219, 20, 21mp2b 10 . . . . . 6 (š‘¤ āˆˆ (ā—”1st ā€œ š“) ā†” (š‘¤ āˆˆ V āˆ§ (1st ā€˜š‘¤) āˆˆ š“))
2318, 22mpbiran 708 . . . . 5 (š‘¤ āˆˆ (ā—”1st ā€œ š“) ā†” (1st ā€˜š‘¤) āˆˆ š“)
2423anbi1i 625 . . . 4 ((š‘¤ āˆˆ (ā—”1st ā€œ š“) āˆ§ š‘¤ āˆˆ (šµ Ɨ š¶)) ā†” ((1st ā€˜š‘¤) āˆˆ š“ āˆ§ š‘¤ āˆˆ (šµ Ɨ š¶)))
2516, 17, 243bitri 297 . . 3 (š‘¤ āˆˆ (ā—”(1st ā†¾ (šµ Ɨ š¶)) ā€œ š“) ā†” ((1st ā€˜š‘¤) āˆˆ š“ āˆ§ š‘¤ āˆˆ (šµ Ɨ š¶)))
26 elxp7 7957 . . 3 (š‘¤ āˆˆ (š“ Ɨ š¶) ā†” (š‘¤ āˆˆ (V Ɨ V) āˆ§ ((1st ā€˜š‘¤) āˆˆ š“ āˆ§ (2nd ā€˜š‘¤) āˆˆ š¶)))
2714, 25, 263bitr4g 314 . 2 (š“ āŠ† šµ ā†’ (š‘¤ āˆˆ (ā—”(1st ā†¾ (šµ Ɨ š¶)) ā€œ š“) ā†” š‘¤ āˆˆ (š“ Ɨ š¶)))
2827eqrdv 2731 1 (š“ āŠ† šµ ā†’ (ā—”(1st ā†¾ (šµ Ɨ š¶)) ā€œ š“) = (š“ Ɨ š¶))
Colors of variables: wff setvar class
Syntax hints:   ā†’ wi 4   ā†” wb 205   āˆ§ wa 397   = wceq 1542   āˆˆ wcel 2107  Vcvv 3444   āˆ© cin 3910   āŠ† wss 3911   Ɨ cxp 5632  ā—”ccnv 5633   ā†¾ cres 5636   ā€œ cima 5637   Fn wfn 6492  ā€“ontoā†’wfo 6495  ā€˜cfv 6497  1st c1st 7920  2nd c2nd 7921
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385  ax-un 7673
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-mpt 5190  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-iota 6449  df-fun 6499  df-fn 6500  df-f 6501  df-fo 6503  df-fv 6505  df-1st 7922  df-2nd 7923
This theorem is referenced by:  sxbrsigalem2  32943
  Copyright terms: Public domain W3C validator