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Theorem 2ndpreima 29328
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 ssel 3577 . . . . . . . 8 (𝐴𝐶 → ((2nd𝑤) ∈ 𝐴 → (2nd𝑤) ∈ 𝐶))
21pm4.71rd 666 . . . . . . 7 (𝐴𝐶 → ((2nd𝑤) ∈ 𝐴 ↔ ((2nd𝑤) ∈ 𝐶 ∧ (2nd𝑤) ∈ 𝐴)))
32anbi2d 739 . . . . . 6 (𝐴𝐶 → (((𝑤 ∈ (V × V) ∧ (1st𝑤) ∈ 𝐵) ∧ (2nd𝑤) ∈ 𝐴) ↔ ((𝑤 ∈ (V × V) ∧ (1st𝑤) ∈ 𝐵) ∧ ((2nd𝑤) ∈ 𝐶 ∧ (2nd𝑤) ∈ 𝐴))))
4 anass 680 . . . . . . . 8 ((((𝑤 ∈ (V × V) ∧ (1st𝑤) ∈ 𝐵) ∧ (2nd𝑤) ∈ 𝐶) ∧ (2nd𝑤) ∈ 𝐴) ↔ ((𝑤 ∈ (V × V) ∧ (1st𝑤) ∈ 𝐵) ∧ ((2nd𝑤) ∈ 𝐶 ∧ (2nd𝑤) ∈ 𝐴)))
54bicomi 214 . . . . . . 7 (((𝑤 ∈ (V × V) ∧ (1st𝑤) ∈ 𝐵) ∧ ((2nd𝑤) ∈ 𝐶 ∧ (2nd𝑤) ∈ 𝐴)) ↔ (((𝑤 ∈ (V × V) ∧ (1st𝑤) ∈ 𝐵) ∧ (2nd𝑤) ∈ 𝐶) ∧ (2nd𝑤) ∈ 𝐴))
65a1i 11 . . . . . 6 (𝐴𝐶 → (((𝑤 ∈ (V × V) ∧ (1st𝑤) ∈ 𝐵) ∧ ((2nd𝑤) ∈ 𝐶 ∧ (2nd𝑤) ∈ 𝐴)) ↔ (((𝑤 ∈ (V × V) ∧ (1st𝑤) ∈ 𝐵) ∧ (2nd𝑤) ∈ 𝐶) ∧ (2nd𝑤) ∈ 𝐴)))
7 anass 680 . . . . . . . 8 (((𝑤 ∈ (V × V) ∧ (1st𝑤) ∈ 𝐵) ∧ (2nd𝑤) ∈ 𝐶) ↔ (𝑤 ∈ (V × V) ∧ ((1st𝑤) ∈ 𝐵 ∧ (2nd𝑤) ∈ 𝐶)))
87anbi1i 730 . . . . . . 7 ((((𝑤 ∈ (V × V) ∧ (1st𝑤) ∈ 𝐵) ∧ (2nd𝑤) ∈ 𝐶) ∧ (2nd𝑤) ∈ 𝐴) ↔ ((𝑤 ∈ (V × V) ∧ ((1st𝑤) ∈ 𝐵 ∧ (2nd𝑤) ∈ 𝐶)) ∧ (2nd𝑤) ∈ 𝐴))
98a1i 11 . . . . . 6 (𝐴𝐶 → ((((𝑤 ∈ (V × V) ∧ (1st𝑤) ∈ 𝐵) ∧ (2nd𝑤) ∈ 𝐶) ∧ (2nd𝑤) ∈ 𝐴) ↔ ((𝑤 ∈ (V × V) ∧ ((1st𝑤) ∈ 𝐵 ∧ (2nd𝑤) ∈ 𝐶)) ∧ (2nd𝑤) ∈ 𝐴)))
103, 6, 93bitrd 294 . . . . 5 (𝐴𝐶 → (((𝑤 ∈ (V × V) ∧ (1st𝑤) ∈ 𝐵) ∧ (2nd𝑤) ∈ 𝐴) ↔ ((𝑤 ∈ (V × V) ∧ ((1st𝑤) ∈ 𝐵 ∧ (2nd𝑤) ∈ 𝐶)) ∧ (2nd𝑤) ∈ 𝐴)))
11 elxp7 7146 . . . . . 6 (𝑤 ∈ (𝐵 × 𝐶) ↔ (𝑤 ∈ (V × V) ∧ ((1st𝑤) ∈ 𝐵 ∧ (2nd𝑤) ∈ 𝐶)))
1211anbi1i 730 . . . . 5 ((𝑤 ∈ (𝐵 × 𝐶) ∧ (2nd𝑤) ∈ 𝐴) ↔ ((𝑤 ∈ (V × V) ∧ ((1st𝑤) ∈ 𝐵 ∧ (2nd𝑤) ∈ 𝐶)) ∧ (2nd𝑤) ∈ 𝐴))
1310, 12syl6rbbr 279 . . . 4 (𝐴𝐶 → ((𝑤 ∈ (𝐵 × 𝐶) ∧ (2nd𝑤) ∈ 𝐴) ↔ ((𝑤 ∈ (V × V) ∧ (1st𝑤) ∈ 𝐵) ∧ (2nd𝑤) ∈ 𝐴)))
14 ancom 466 . . . 4 ((𝑤 ∈ (𝐵 × 𝐶) ∧ (2nd𝑤) ∈ 𝐴) ↔ ((2nd𝑤) ∈ 𝐴𝑤 ∈ (𝐵 × 𝐶)))
15 anass 680 . . . 4 (((𝑤 ∈ (V × V) ∧ (1st𝑤) ∈ 𝐵) ∧ (2nd𝑤) ∈ 𝐴) ↔ (𝑤 ∈ (V × V) ∧ ((1st𝑤) ∈ 𝐵 ∧ (2nd𝑤) ∈ 𝐴)))
1613, 14, 153bitr3g 302 . . 3 (𝐴𝐶 → (((2nd𝑤) ∈ 𝐴𝑤 ∈ (𝐵 × 𝐶)) ↔ (𝑤 ∈ (V × V) ∧ ((1st𝑤) ∈ 𝐵 ∧ (2nd𝑤) ∈ 𝐴))))
17 cnvresima 5582 . . . . 5 ((2nd ↾ (𝐵 × 𝐶)) “ 𝐴) = ((2nd𝐴) ∩ (𝐵 × 𝐶))
1817eleq2i 2690 . . . 4 (𝑤 ∈ ((2nd ↾ (𝐵 × 𝐶)) “ 𝐴) ↔ 𝑤 ∈ ((2nd𝐴) ∩ (𝐵 × 𝐶)))
19 elin 3774 . . . 4 (𝑤 ∈ ((2nd𝐴) ∩ (𝐵 × 𝐶)) ↔ (𝑤 ∈ (2nd𝐴) ∧ 𝑤 ∈ (𝐵 × 𝐶)))
20 vex 3189 . . . . . 6 𝑤 ∈ V
21 fo2nd 7134 . . . . . . 7 2nd :V–onto→V
22 fofn 6074 . . . . . . 7 (2nd :V–onto→V → 2nd Fn V)
23 elpreima 6293 . . . . . . 7 (2nd Fn V → (𝑤 ∈ (2nd𝐴) ↔ (𝑤 ∈ V ∧ (2nd𝑤) ∈ 𝐴)))
2421, 22, 23mp2b 10 . . . . . 6 (𝑤 ∈ (2nd𝐴) ↔ (𝑤 ∈ V ∧ (2nd𝑤) ∈ 𝐴))
2520, 24mpbiran 952 . . . . 5 (𝑤 ∈ (2nd𝐴) ↔ (2nd𝑤) ∈ 𝐴)
2625anbi1i 730 . . . 4 ((𝑤 ∈ (2nd𝐴) ∧ 𝑤 ∈ (𝐵 × 𝐶)) ↔ ((2nd𝑤) ∈ 𝐴𝑤 ∈ (𝐵 × 𝐶)))
2718, 19, 263bitri 286 . . 3 (𝑤 ∈ ((2nd ↾ (𝐵 × 𝐶)) “ 𝐴) ↔ ((2nd𝑤) ∈ 𝐴𝑤 ∈ (𝐵 × 𝐶)))
28 elxp7 7146 . . 3 (𝑤 ∈ (𝐵 × 𝐴) ↔ (𝑤 ∈ (V × V) ∧ ((1st𝑤) ∈ 𝐵 ∧ (2nd𝑤) ∈ 𝐴)))
2916, 27, 283bitr4g 303 . 2 (𝐴𝐶 → (𝑤 ∈ ((2nd ↾ (𝐵 × 𝐶)) “ 𝐴) ↔ 𝑤 ∈ (𝐵 × 𝐴)))
3029eqrdv 2619 1 (𝐴𝐶 → ((2nd ↾ (𝐵 × 𝐶)) “ 𝐴) = (𝐵 × 𝐴))
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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