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Theorem 1stpreima 29327
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 680 . . . . . . 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 3577 . . . . . . . 8 (𝐴𝐵 → ((1st𝑤) ∈ 𝐴 → (1st𝑤) ∈ 𝐵))
43pm4.71d 665 . . . . . . 7 (𝐴𝐵 → ((1st𝑤) ∈ 𝐴 ↔ ((1st𝑤) ∈ 𝐴 ∧ (1st𝑤) ∈ 𝐵)))
54anbi1d 740 . . . . . 6 (𝐴𝐵 → (((1st𝑤) ∈ 𝐴 ∧ (𝑤 ∈ (V × V) ∧ (2nd𝑤) ∈ 𝐶)) ↔ (((1st𝑤) ∈ 𝐴 ∧ (1st𝑤) ∈ 𝐵) ∧ (𝑤 ∈ (V × V) ∧ (2nd𝑤) ∈ 𝐶))))
6 an12 837 . . . . . . . 8 ((𝑤 ∈ (V × V) ∧ ((1st𝑤) ∈ 𝐵 ∧ (2nd𝑤) ∈ 𝐶)) ↔ ((1st𝑤) ∈ 𝐵 ∧ (𝑤 ∈ (V × V) ∧ (2nd𝑤) ∈ 𝐶)))
76anbi2i 729 . . . . . . 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 300 . . . . 5 (𝐴𝐵 → (((1st𝑤) ∈ 𝐴 ∧ (𝑤 ∈ (V × V) ∧ (2nd𝑤) ∈ 𝐶)) ↔ ((1st𝑤) ∈ 𝐴 ∧ (𝑤 ∈ (V × V) ∧ ((1st𝑤) ∈ 𝐵 ∧ (2nd𝑤) ∈ 𝐶)))))
10 elxp7 7146 . . . . . 6 (𝑤 ∈ (𝐵 × 𝐶) ↔ (𝑤 ∈ (V × V) ∧ ((1st𝑤) ∈ 𝐵 ∧ (2nd𝑤) ∈ 𝐶)))
1110anbi2i 729 . . . . 5 (((1st𝑤) ∈ 𝐴𝑤 ∈ (𝐵 × 𝐶)) ↔ ((1st𝑤) ∈ 𝐴 ∧ (𝑤 ∈ (V × V) ∧ ((1st𝑤) ∈ 𝐵 ∧ (2nd𝑤) ∈ 𝐶))))
129, 11syl6rbbr 279 . . . 4 (𝐴𝐵 → (((1st𝑤) ∈ 𝐴𝑤 ∈ (𝐵 × 𝐶)) ↔ ((1st𝑤) ∈ 𝐴 ∧ (𝑤 ∈ (V × V) ∧ (2nd𝑤) ∈ 𝐶))))
13 an12 837 . . . 4 (((1st𝑤) ∈ 𝐴 ∧ (𝑤 ∈ (V × V) ∧ (2nd𝑤) ∈ 𝐶)) ↔ (𝑤 ∈ (V × V) ∧ ((1st𝑤) ∈ 𝐴 ∧ (2nd𝑤) ∈ 𝐶)))
1412, 13syl6bb 276 . . 3 (𝐴𝐵 → (((1st𝑤) ∈ 𝐴𝑤 ∈ (𝐵 × 𝐶)) ↔ (𝑤 ∈ (V × V) ∧ ((1st𝑤) ∈ 𝐴 ∧ (2nd𝑤) ∈ 𝐶))))
15 cnvresima 5582 . . . . 5 ((1st ↾ (𝐵 × 𝐶)) “ 𝐴) = ((1st𝐴) ∩ (𝐵 × 𝐶))
1615eleq2i 2690 . . . 4 (𝑤 ∈ ((1st ↾ (𝐵 × 𝐶)) “ 𝐴) ↔ 𝑤 ∈ ((1st𝐴) ∩ (𝐵 × 𝐶)))
17 elin 3774 . . . 4 (𝑤 ∈ ((1st𝐴) ∩ (𝐵 × 𝐶)) ↔ (𝑤 ∈ (1st𝐴) ∧ 𝑤 ∈ (𝐵 × 𝐶)))
18 vex 3189 . . . . . 6 𝑤 ∈ V
19 fo1st 7133 . . . . . . 7 1st :V–onto→V
20 fofn 6074 . . . . . . 7 (1st :V–onto→V → 1st Fn V)
21 elpreima 6293 . . . . . . 7 (1st Fn V → (𝑤 ∈ (1st𝐴) ↔ (𝑤 ∈ V ∧ (1st𝑤) ∈ 𝐴)))
2219, 20, 21mp2b 10 . . . . . 6 (𝑤 ∈ (1st𝐴) ↔ (𝑤 ∈ V ∧ (1st𝑤) ∈ 𝐴))
2318, 22mpbiran 952 . . . . 5 (𝑤 ∈ (1st𝐴) ↔ (1st𝑤) ∈ 𝐴)
2423anbi1i 730 . . . 4 ((𝑤 ∈ (1st𝐴) ∧ 𝑤 ∈ (𝐵 × 𝐶)) ↔ ((1st𝑤) ∈ 𝐴𝑤 ∈ (𝐵 × 𝐶)))
2516, 17, 243bitri 286 . . 3 (𝑤 ∈ ((1st ↾ (𝐵 × 𝐶)) “ 𝐴) ↔ ((1st𝑤) ∈ 𝐴𝑤 ∈ (𝐵 × 𝐶)))
26 elxp7 7146 . . 3 (𝑤 ∈ (𝐴 × 𝐶) ↔ (𝑤 ∈ (V × V) ∧ ((1st𝑤) ∈ 𝐴 ∧ (2nd𝑤) ∈ 𝐶)))
2714, 25, 263bitr4g 303 . 2 (𝐴𝐵 → (𝑤 ∈ ((1st ↾ (𝐵 × 𝐶)) “ 𝐴) ↔ 𝑤 ∈ (𝐴 × 𝐶)))
2827eqrdv 2619 1 (𝐴𝐵 → ((1st ↾ (𝐵 × 𝐶)) “ 𝐴) = (𝐴 × 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 384   = wceq 1480  wcel 1987  Vcvv 3186  cin 3554  wss 3555   × cxp 5072  ccnv 5073  cres 5076  cima 5077   Fn wfn 5842  ontowfo 5845  cfv 5847  1st c1st 7111  2nd c2nd 7112
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pow 4803  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3418  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-mpt 4675  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-res 5086  df-ima 5087  df-iota 5810  df-fun 5849  df-fn 5850  df-f 5851  df-fo 5853  df-fv 5855  df-1st 7113  df-2nd 7114
This theorem is referenced by:  sxbrsigalem2  30129
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