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 32716
Description: The preimage by 1st is a 'vertical band'. (Contributed by Thierry Arnoux, 13-Oct-2017.)
Assertion
Ref Expression
1stpreima (𝐴𝐵 → ((1st ↾ (𝐵 × 𝐶)) “ 𝐴) = (𝐴 × 𝐶))

Proof of Theorem 1stpreima
Dummy variable 𝑤 is distinct from all other variables.
StepHypRef Expression
1 elxp7 8049 . . . . . 6 (𝑤 ∈ (𝐵 × 𝐶) ↔ (𝑤 ∈ (V × V) ∧ ((1st𝑤) ∈ 𝐵 ∧ (2nd𝑤) ∈ 𝐶)))
21anbi2i 623 . . . . 5 (((1st𝑤) ∈ 𝐴𝑤 ∈ (𝐵 × 𝐶)) ↔ ((1st𝑤) ∈ 𝐴 ∧ (𝑤 ∈ (V × V) ∧ ((1st𝑤) ∈ 𝐵 ∧ (2nd𝑤) ∈ 𝐶))))
3 anass 468 . . . . . . 7 ((((1st𝑤) ∈ 𝐴 ∧ (1st𝑤) ∈ 𝐵) ∧ (𝑤 ∈ (V × V) ∧ (2nd𝑤) ∈ 𝐶)) ↔ ((1st𝑤) ∈ 𝐴 ∧ ((1st𝑤) ∈ 𝐵 ∧ (𝑤 ∈ (V × V) ∧ (2nd𝑤) ∈ 𝐶))))
43a1i 11 . . . . . 6 (𝐴𝐵 → ((((1st𝑤) ∈ 𝐴 ∧ (1st𝑤) ∈ 𝐵) ∧ (𝑤 ∈ (V × V) ∧ (2nd𝑤) ∈ 𝐶)) ↔ ((1st𝑤) ∈ 𝐴 ∧ ((1st𝑤) ∈ 𝐵 ∧ (𝑤 ∈ (V × V) ∧ (2nd𝑤) ∈ 𝐶)))))
5 ssel 3977 . . . . . . . 8 (𝐴𝐵 → ((1st𝑤) ∈ 𝐴 → (1st𝑤) ∈ 𝐵))
65pm4.71d 561 . . . . . . 7 (𝐴𝐵 → ((1st𝑤) ∈ 𝐴 ↔ ((1st𝑤) ∈ 𝐴 ∧ (1st𝑤) ∈ 𝐵)))
76anbi1d 631 . . . . . 6 (𝐴𝐵 → (((1st𝑤) ∈ 𝐴 ∧ (𝑤 ∈ (V × V) ∧ (2nd𝑤) ∈ 𝐶)) ↔ (((1st𝑤) ∈ 𝐴 ∧ (1st𝑤) ∈ 𝐵) ∧ (𝑤 ∈ (V × V) ∧ (2nd𝑤) ∈ 𝐶))))
8 an12 645 . . . . . . . 8 ((𝑤 ∈ (V × V) ∧ ((1st𝑤) ∈ 𝐵 ∧ (2nd𝑤) ∈ 𝐶)) ↔ ((1st𝑤) ∈ 𝐵 ∧ (𝑤 ∈ (V × V) ∧ (2nd𝑤) ∈ 𝐶)))
98anbi2i 623 . . . . . . 7 (((1st𝑤) ∈ 𝐴 ∧ (𝑤 ∈ (V × V) ∧ ((1st𝑤) ∈ 𝐵 ∧ (2nd𝑤) ∈ 𝐶))) ↔ ((1st𝑤) ∈ 𝐴 ∧ ((1st𝑤) ∈ 𝐵 ∧ (𝑤 ∈ (V × V) ∧ (2nd𝑤) ∈ 𝐶))))
109a1i 11 . . . . . 6 (𝐴𝐵 → (((1st𝑤) ∈ 𝐴 ∧ (𝑤 ∈ (V × V) ∧ ((1st𝑤) ∈ 𝐵 ∧ (2nd𝑤) ∈ 𝐶))) ↔ ((1st𝑤) ∈ 𝐴 ∧ ((1st𝑤) ∈ 𝐵 ∧ (𝑤 ∈ (V × V) ∧ (2nd𝑤) ∈ 𝐶)))))
114, 7, 103bitr4d 311 . . . . 5 (𝐴𝐵 → (((1st𝑤) ∈ 𝐴 ∧ (𝑤 ∈ (V × V) ∧ (2nd𝑤) ∈ 𝐶)) ↔ ((1st𝑤) ∈ 𝐴 ∧ (𝑤 ∈ (V × V) ∧ ((1st𝑤) ∈ 𝐵 ∧ (2nd𝑤) ∈ 𝐶)))))
122, 11bitr4id 290 . . . 4 (𝐴𝐵 → (((1st𝑤) ∈ 𝐴𝑤 ∈ (𝐵 × 𝐶)) ↔ ((1st𝑤) ∈ 𝐴 ∧ (𝑤 ∈ (V × V) ∧ (2nd𝑤) ∈ 𝐶))))
13 an12 645 . . . 4 (((1st𝑤) ∈ 𝐴 ∧ (𝑤 ∈ (V × V) ∧ (2nd𝑤) ∈ 𝐶)) ↔ (𝑤 ∈ (V × V) ∧ ((1st𝑤) ∈ 𝐴 ∧ (2nd𝑤) ∈ 𝐶)))
1412, 13bitrdi 287 . . 3 (𝐴𝐵 → (((1st𝑤) ∈ 𝐴𝑤 ∈ (𝐵 × 𝐶)) ↔ (𝑤 ∈ (V × V) ∧ ((1st𝑤) ∈ 𝐴 ∧ (2nd𝑤) ∈ 𝐶))))
15 cnvresima 6250 . . . . 5 ((1st ↾ (𝐵 × 𝐶)) “ 𝐴) = ((1st𝐴) ∩ (𝐵 × 𝐶))
1615eleq2i 2833 . . . 4 (𝑤 ∈ ((1st ↾ (𝐵 × 𝐶)) “ 𝐴) ↔ 𝑤 ∈ ((1st𝐴) ∩ (𝐵 × 𝐶)))
17 elin 3967 . . . 4 (𝑤 ∈ ((1st𝐴) ∩ (𝐵 × 𝐶)) ↔ (𝑤 ∈ (1st𝐴) ∧ 𝑤 ∈ (𝐵 × 𝐶)))
18 vex 3484 . . . . . 6 𝑤 ∈ V
19 fo1st 8034 . . . . . . 7 1st :V–onto→V
20 fofn 6822 . . . . . . 7 (1st :V–onto→V → 1st Fn V)
21 elpreima 7078 . . . . . . 7 (1st Fn V → (𝑤 ∈ (1st𝐴) ↔ (𝑤 ∈ V ∧ (1st𝑤) ∈ 𝐴)))
2219, 20, 21mp2b 10 . . . . . 6 (𝑤 ∈ (1st𝐴) ↔ (𝑤 ∈ V ∧ (1st𝑤) ∈ 𝐴))
2318, 22mpbiran 709 . . . . 5 (𝑤 ∈ (1st𝐴) ↔ (1st𝑤) ∈ 𝐴)
2423anbi1i 624 . . . 4 ((𝑤 ∈ (1st𝐴) ∧ 𝑤 ∈ (𝐵 × 𝐶)) ↔ ((1st𝑤) ∈ 𝐴𝑤 ∈ (𝐵 × 𝐶)))
2516, 17, 243bitri 297 . . 3 (𝑤 ∈ ((1st ↾ (𝐵 × 𝐶)) “ 𝐴) ↔ ((1st𝑤) ∈ 𝐴𝑤 ∈ (𝐵 × 𝐶)))
26 elxp7 8049 . . 3 (𝑤 ∈ (𝐴 × 𝐶) ↔ (𝑤 ∈ (V × V) ∧ ((1st𝑤) ∈ 𝐴 ∧ (2nd𝑤) ∈ 𝐶)))
2714, 25, 263bitr4g 314 . 2 (𝐴𝐵 → (𝑤 ∈ ((1st ↾ (𝐵 × 𝐶)) “ 𝐴) ↔ 𝑤 ∈ (𝐴 × 𝐶)))
2827eqrdv 2735 1 (𝐴𝐵 → ((1st ↾ (𝐵 × 𝐶)) “ 𝐴) = (𝐴 × 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1540  wcel 2108  Vcvv 3480  cin 3950  wss 3951   × cxp 5683  ccnv 5684  cres 5687  cima 5688   Fn wfn 6556  ontowfo 6559  cfv 6561  1st c1st 8012  2nd c2nd 8013
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432  ax-un 7755
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fo 6567  df-fv 6569  df-1st 8014  df-2nd 8015
This theorem is referenced by:  sxbrsigalem2  34288
  Copyright terms: Public domain W3C validator