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Theorem 2ndpreima 32717
Description: The preimage by 2nd is an 'horizontal band'. (Contributed by Thierry Arnoux, 13-Oct-2017.)
Assertion
Ref Expression
2ndpreima (𝐴𝐶 → ((2nd ↾ (𝐵 × 𝐶)) “ 𝐴) = (𝐵 × 𝐴))

Proof of Theorem 2ndpreima
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)))
21anbi1i 624 . . . . 5 ((𝑤 ∈ (𝐵 × 𝐶) ∧ (2nd𝑤) ∈ 𝐴) ↔ ((𝑤 ∈ (V × V) ∧ ((1st𝑤) ∈ 𝐵 ∧ (2nd𝑤) ∈ 𝐶)) ∧ (2nd𝑤) ∈ 𝐴))
3 ssel 3977 . . . . . . . 8 (𝐴𝐶 → ((2nd𝑤) ∈ 𝐴 → (2nd𝑤) ∈ 𝐶))
43pm4.71rd 562 . . . . . . 7 (𝐴𝐶 → ((2nd𝑤) ∈ 𝐴 ↔ ((2nd𝑤) ∈ 𝐶 ∧ (2nd𝑤) ∈ 𝐴)))
54anbi2d 630 . . . . . 6 (𝐴𝐶 → (((𝑤 ∈ (V × V) ∧ (1st𝑤) ∈ 𝐵) ∧ (2nd𝑤) ∈ 𝐴) ↔ ((𝑤 ∈ (V × V) ∧ (1st𝑤) ∈ 𝐵) ∧ ((2nd𝑤) ∈ 𝐶 ∧ (2nd𝑤) ∈ 𝐴))))
6 anass 468 . . . . . . . 8 ((((𝑤 ∈ (V × V) ∧ (1st𝑤) ∈ 𝐵) ∧ (2nd𝑤) ∈ 𝐶) ∧ (2nd𝑤) ∈ 𝐴) ↔ ((𝑤 ∈ (V × V) ∧ (1st𝑤) ∈ 𝐵) ∧ ((2nd𝑤) ∈ 𝐶 ∧ (2nd𝑤) ∈ 𝐴)))
76bicomi 224 . . . . . . 7 (((𝑤 ∈ (V × V) ∧ (1st𝑤) ∈ 𝐵) ∧ ((2nd𝑤) ∈ 𝐶 ∧ (2nd𝑤) ∈ 𝐴)) ↔ (((𝑤 ∈ (V × V) ∧ (1st𝑤) ∈ 𝐵) ∧ (2nd𝑤) ∈ 𝐶) ∧ (2nd𝑤) ∈ 𝐴))
87a1i 11 . . . . . 6 (𝐴𝐶 → (((𝑤 ∈ (V × V) ∧ (1st𝑤) ∈ 𝐵) ∧ ((2nd𝑤) ∈ 𝐶 ∧ (2nd𝑤) ∈ 𝐴)) ↔ (((𝑤 ∈ (V × V) ∧ (1st𝑤) ∈ 𝐵) ∧ (2nd𝑤) ∈ 𝐶) ∧ (2nd𝑤) ∈ 𝐴)))
9 anass 468 . . . . . . . 8 (((𝑤 ∈ (V × V) ∧ (1st𝑤) ∈ 𝐵) ∧ (2nd𝑤) ∈ 𝐶) ↔ (𝑤 ∈ (V × V) ∧ ((1st𝑤) ∈ 𝐵 ∧ (2nd𝑤) ∈ 𝐶)))
109anbi1i 624 . . . . . . 7 ((((𝑤 ∈ (V × V) ∧ (1st𝑤) ∈ 𝐵) ∧ (2nd𝑤) ∈ 𝐶) ∧ (2nd𝑤) ∈ 𝐴) ↔ ((𝑤 ∈ (V × V) ∧ ((1st𝑤) ∈ 𝐵 ∧ (2nd𝑤) ∈ 𝐶)) ∧ (2nd𝑤) ∈ 𝐴))
1110a1i 11 . . . . . 6 (𝐴𝐶 → ((((𝑤 ∈ (V × V) ∧ (1st𝑤) ∈ 𝐵) ∧ (2nd𝑤) ∈ 𝐶) ∧ (2nd𝑤) ∈ 𝐴) ↔ ((𝑤 ∈ (V × V) ∧ ((1st𝑤) ∈ 𝐵 ∧ (2nd𝑤) ∈ 𝐶)) ∧ (2nd𝑤) ∈ 𝐴)))
125, 8, 113bitrd 305 . . . . 5 (𝐴𝐶 → (((𝑤 ∈ (V × V) ∧ (1st𝑤) ∈ 𝐵) ∧ (2nd𝑤) ∈ 𝐴) ↔ ((𝑤 ∈ (V × V) ∧ ((1st𝑤) ∈ 𝐵 ∧ (2nd𝑤) ∈ 𝐶)) ∧ (2nd𝑤) ∈ 𝐴)))
132, 12bitr4id 290 . . . 4 (𝐴𝐶 → ((𝑤 ∈ (𝐵 × 𝐶) ∧ (2nd𝑤) ∈ 𝐴) ↔ ((𝑤 ∈ (V × V) ∧ (1st𝑤) ∈ 𝐵) ∧ (2nd𝑤) ∈ 𝐴)))
14 ancom 460 . . . 4 ((𝑤 ∈ (𝐵 × 𝐶) ∧ (2nd𝑤) ∈ 𝐴) ↔ ((2nd𝑤) ∈ 𝐴𝑤 ∈ (𝐵 × 𝐶)))
15 anass 468 . . . 4 (((𝑤 ∈ (V × V) ∧ (1st𝑤) ∈ 𝐵) ∧ (2nd𝑤) ∈ 𝐴) ↔ (𝑤 ∈ (V × V) ∧ ((1st𝑤) ∈ 𝐵 ∧ (2nd𝑤) ∈ 𝐴)))
1613, 14, 153bitr3g 313 . . 3 (𝐴𝐶 → (((2nd𝑤) ∈ 𝐴𝑤 ∈ (𝐵 × 𝐶)) ↔ (𝑤 ∈ (V × V) ∧ ((1st𝑤) ∈ 𝐵 ∧ (2nd𝑤) ∈ 𝐴))))
17 cnvresima 6250 . . . . 5 ((2nd ↾ (𝐵 × 𝐶)) “ 𝐴) = ((2nd𝐴) ∩ (𝐵 × 𝐶))
1817eleq2i 2833 . . . 4 (𝑤 ∈ ((2nd ↾ (𝐵 × 𝐶)) “ 𝐴) ↔ 𝑤 ∈ ((2nd𝐴) ∩ (𝐵 × 𝐶)))
19 elin 3967 . . . 4 (𝑤 ∈ ((2nd𝐴) ∩ (𝐵 × 𝐶)) ↔ (𝑤 ∈ (2nd𝐴) ∧ 𝑤 ∈ (𝐵 × 𝐶)))
20 vex 3484 . . . . . 6 𝑤 ∈ V
21 fo2nd 8035 . . . . . . 7 2nd :V–onto→V
22 fofn 6822 . . . . . . 7 (2nd :V–onto→V → 2nd Fn V)
23 elpreima 7078 . . . . . . 7 (2nd Fn V → (𝑤 ∈ (2nd𝐴) ↔ (𝑤 ∈ V ∧ (2nd𝑤) ∈ 𝐴)))
2421, 22, 23mp2b 10 . . . . . 6 (𝑤 ∈ (2nd𝐴) ↔ (𝑤 ∈ V ∧ (2nd𝑤) ∈ 𝐴))
2520, 24mpbiran 709 . . . . 5 (𝑤 ∈ (2nd𝐴) ↔ (2nd𝑤) ∈ 𝐴)
2625anbi1i 624 . . . 4 ((𝑤 ∈ (2nd𝐴) ∧ 𝑤 ∈ (𝐵 × 𝐶)) ↔ ((2nd𝑤) ∈ 𝐴𝑤 ∈ (𝐵 × 𝐶)))
2718, 19, 263bitri 297 . . 3 (𝑤 ∈ ((2nd ↾ (𝐵 × 𝐶)) “ 𝐴) ↔ ((2nd𝑤) ∈ 𝐴𝑤 ∈ (𝐵 × 𝐶)))
28 elxp7 8049 . . 3 (𝑤 ∈ (𝐵 × 𝐴) ↔ (𝑤 ∈ (V × V) ∧ ((1st𝑤) ∈ 𝐵 ∧ (2nd𝑤) ∈ 𝐴)))
2916, 27, 283bitr4g 314 . 2 (𝐴𝐶 → (𝑤 ∈ ((2nd ↾ (𝐵 × 𝐶)) “ 𝐴) ↔ 𝑤 ∈ (𝐵 × 𝐴)))
3029eqrdv 2735 1 (𝐴𝐶 → ((2nd ↾ (𝐵 × 𝐶)) “ 𝐴) = (𝐵 × 𝐴))
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
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