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Theorem 1stpreima 30183
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 anass 461 . . . . . . 7 ((((1st𝑤) ∈ 𝐴 ∧ (1st𝑤) ∈ 𝐵) ∧ (𝑤 ∈ (V × V) ∧ (2nd𝑤) ∈ 𝐶)) ↔ ((1st𝑤) ∈ 𝐴 ∧ ((1st𝑤) ∈ 𝐵 ∧ (𝑤 ∈ (V × V) ∧ (2nd𝑤) ∈ 𝐶))))
21a1i 11 . . . . . 6 (𝐴𝐵 → ((((1st𝑤) ∈ 𝐴 ∧ (1st𝑤) ∈ 𝐵) ∧ (𝑤 ∈ (V × V) ∧ (2nd𝑤) ∈ 𝐶)) ↔ ((1st𝑤) ∈ 𝐴 ∧ ((1st𝑤) ∈ 𝐵 ∧ (𝑤 ∈ (V × V) ∧ (2nd𝑤) ∈ 𝐶)))))
3 ssel 3848 . . . . . . . 8 (𝐴𝐵 → ((1st𝑤) ∈ 𝐴 → (1st𝑤) ∈ 𝐵))
43pm4.71d 554 . . . . . . 7 (𝐴𝐵 → ((1st𝑤) ∈ 𝐴 ↔ ((1st𝑤) ∈ 𝐴 ∧ (1st𝑤) ∈ 𝐵)))
54anbi1d 620 . . . . . 6 (𝐴𝐵 → (((1st𝑤) ∈ 𝐴 ∧ (𝑤 ∈ (V × V) ∧ (2nd𝑤) ∈ 𝐶)) ↔ (((1st𝑤) ∈ 𝐴 ∧ (1st𝑤) ∈ 𝐵) ∧ (𝑤 ∈ (V × V) ∧ (2nd𝑤) ∈ 𝐶))))
6 an12 632 . . . . . . . 8 ((𝑤 ∈ (V × V) ∧ ((1st𝑤) ∈ 𝐵 ∧ (2nd𝑤) ∈ 𝐶)) ↔ ((1st𝑤) ∈ 𝐵 ∧ (𝑤 ∈ (V × V) ∧ (2nd𝑤) ∈ 𝐶)))
76anbi2i 613 . . . . . . 7 (((1st𝑤) ∈ 𝐴 ∧ (𝑤 ∈ (V × V) ∧ ((1st𝑤) ∈ 𝐵 ∧ (2nd𝑤) ∈ 𝐶))) ↔ ((1st𝑤) ∈ 𝐴 ∧ ((1st𝑤) ∈ 𝐵 ∧ (𝑤 ∈ (V × V) ∧ (2nd𝑤) ∈ 𝐶))))
87a1i 11 . . . . . 6 (𝐴𝐵 → (((1st𝑤) ∈ 𝐴 ∧ (𝑤 ∈ (V × V) ∧ ((1st𝑤) ∈ 𝐵 ∧ (2nd𝑤) ∈ 𝐶))) ↔ ((1st𝑤) ∈ 𝐴 ∧ ((1st𝑤) ∈ 𝐵 ∧ (𝑤 ∈ (V × V) ∧ (2nd𝑤) ∈ 𝐶)))))
92, 5, 83bitr4d 303 . . . . 5 (𝐴𝐵 → (((1st𝑤) ∈ 𝐴 ∧ (𝑤 ∈ (V × V) ∧ (2nd𝑤) ∈ 𝐶)) ↔ ((1st𝑤) ∈ 𝐴 ∧ (𝑤 ∈ (V × V) ∧ ((1st𝑤) ∈ 𝐵 ∧ (2nd𝑤) ∈ 𝐶)))))
10 elxp7 7529 . . . . . 6 (𝑤 ∈ (𝐵 × 𝐶) ↔ (𝑤 ∈ (V × V) ∧ ((1st𝑤) ∈ 𝐵 ∧ (2nd𝑤) ∈ 𝐶)))
1110anbi2i 613 . . . . 5 (((1st𝑤) ∈ 𝐴𝑤 ∈ (𝐵 × 𝐶)) ↔ ((1st𝑤) ∈ 𝐴 ∧ (𝑤 ∈ (V × V) ∧ ((1st𝑤) ∈ 𝐵 ∧ (2nd𝑤) ∈ 𝐶))))
129, 11syl6rbbr 282 . . . 4 (𝐴𝐵 → (((1st𝑤) ∈ 𝐴𝑤 ∈ (𝐵 × 𝐶)) ↔ ((1st𝑤) ∈ 𝐴 ∧ (𝑤 ∈ (V × V) ∧ (2nd𝑤) ∈ 𝐶))))
13 an12 632 . . . 4 (((1st𝑤) ∈ 𝐴 ∧ (𝑤 ∈ (V × V) ∧ (2nd𝑤) ∈ 𝐶)) ↔ (𝑤 ∈ (V × V) ∧ ((1st𝑤) ∈ 𝐴 ∧ (2nd𝑤) ∈ 𝐶)))
1412, 13syl6bb 279 . . 3 (𝐴𝐵 → (((1st𝑤) ∈ 𝐴𝑤 ∈ (𝐵 × 𝐶)) ↔ (𝑤 ∈ (V × V) ∧ ((1st𝑤) ∈ 𝐴 ∧ (2nd𝑤) ∈ 𝐶))))
15 cnvresima 5920 . . . . 5 ((1st ↾ (𝐵 × 𝐶)) “ 𝐴) = ((1st𝐴) ∩ (𝐵 × 𝐶))
1615eleq2i 2851 . . . 4 (𝑤 ∈ ((1st ↾ (𝐵 × 𝐶)) “ 𝐴) ↔ 𝑤 ∈ ((1st𝐴) ∩ (𝐵 × 𝐶)))
17 elin 4053 . . . 4 (𝑤 ∈ ((1st𝐴) ∩ (𝐵 × 𝐶)) ↔ (𝑤 ∈ (1st𝐴) ∧ 𝑤 ∈ (𝐵 × 𝐶)))
18 vex 3412 . . . . . 6 𝑤 ∈ V
19 fo1st 7514 . . . . . . 7 1st :V–onto→V
20 fofn 6415 . . . . . . 7 (1st :V–onto→V → 1st Fn V)
21 elpreima 6647 . . . . . . 7 (1st Fn V → (𝑤 ∈ (1st𝐴) ↔ (𝑤 ∈ V ∧ (1st𝑤) ∈ 𝐴)))
2219, 20, 21mp2b 10 . . . . . 6 (𝑤 ∈ (1st𝐴) ↔ (𝑤 ∈ V ∧ (1st𝑤) ∈ 𝐴))
2318, 22mpbiran 696 . . . . 5 (𝑤 ∈ (1st𝐴) ↔ (1st𝑤) ∈ 𝐴)
2423anbi1i 614 . . . 4 ((𝑤 ∈ (1st𝐴) ∧ 𝑤 ∈ (𝐵 × 𝐶)) ↔ ((1st𝑤) ∈ 𝐴𝑤 ∈ (𝐵 × 𝐶)))
2516, 17, 243bitri 289 . . 3 (𝑤 ∈ ((1st ↾ (𝐵 × 𝐶)) “ 𝐴) ↔ ((1st𝑤) ∈ 𝐴𝑤 ∈ (𝐵 × 𝐶)))
26 elxp7 7529 . . 3 (𝑤 ∈ (𝐴 × 𝐶) ↔ (𝑤 ∈ (V × V) ∧ ((1st𝑤) ∈ 𝐴 ∧ (2nd𝑤) ∈ 𝐶)))
2714, 25, 263bitr4g 306 . 2 (𝐴𝐵 → (𝑤 ∈ ((1st ↾ (𝐵 × 𝐶)) “ 𝐴) ↔ 𝑤 ∈ (𝐴 × 𝐶)))
2827eqrdv 2770 1 (𝐴𝐵 → ((1st ↾ (𝐵 × 𝐶)) “ 𝐴) = (𝐴 × 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 198  wa 387   = wceq 1507  wcel 2048  Vcvv 3409  cin 3824  wss 3825   × cxp 5398  ccnv 5399  cres 5402  cima 5403   Fn wfn 6177  ontowfo 6180  cfv 6182  1st c1st 7492  2nd c2nd 7493
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1964  ax-8 2050  ax-9 2057  ax-10 2077  ax-11 2091  ax-12 2104  ax-13 2299  ax-ext 2745  ax-sep 5054  ax-nul 5061  ax-pow 5113  ax-pr 5180  ax-un 7273
This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2014  df-mo 2544  df-eu 2580  df-clab 2754  df-cleq 2765  df-clel 2840  df-nfc 2912  df-ne 2962  df-ral 3087  df-rex 3088  df-rab 3091  df-v 3411  df-sbc 3678  df-dif 3828  df-un 3830  df-in 3832  df-ss 3839  df-nul 4174  df-if 4345  df-sn 4436  df-pr 4438  df-op 4442  df-uni 4707  df-br 4924  df-opab 4986  df-mpt 5003  df-id 5305  df-xp 5406  df-rel 5407  df-cnv 5408  df-co 5409  df-dm 5410  df-rn 5411  df-res 5412  df-ima 5413  df-iota 6146  df-fun 6184  df-fn 6185  df-f 6186  df-fo 6188  df-fv 6190  df-1st 7494  df-2nd 7495
This theorem is referenced by:  sxbrsigalem2  31146
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