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Theorem 1stpreima 32722
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 8048 . . . . . 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 3989 . . . . . . . 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 6252 . . . . 5 ((1st ↾ (𝐵 × 𝐶)) “ 𝐴) = ((1st𝐴) ∩ (𝐵 × 𝐶))
1615eleq2i 2831 . . . 4 (𝑤 ∈ ((1st ↾ (𝐵 × 𝐶)) “ 𝐴) ↔ 𝑤 ∈ ((1st𝐴) ∩ (𝐵 × 𝐶)))
17 elin 3979 . . . 4 (𝑤 ∈ ((1st𝐴) ∩ (𝐵 × 𝐶)) ↔ (𝑤 ∈ (1st𝐴) ∧ 𝑤 ∈ (𝐵 × 𝐶)))
18 vex 3482 . . . . . 6 𝑤 ∈ V
19 fo1st 8033 . . . . . . 7 1st :V–onto→V
20 fofn 6823 . . . . . . 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 8048 . . 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 2733 1 (𝐴𝐵 → ((1st ↾ (𝐵 × 𝐶)) “ 𝐴) = (𝐴 × 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wa 395   = wceq 1537  wcel 2106  Vcvv 3478  cin 3962  wss 3963   × cxp 5687  ccnv 5688  cres 5691  cima 5692   Fn wfn 6558  ontowfo 6561  cfv 6563  1st c1st 8011  2nd c2nd 8012
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-fo 6569  df-fv 6571  df-1st 8013  df-2nd 8014
This theorem is referenced by:  sxbrsigalem2  34268
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