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Theorem 2ndpreima 28868
 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 3562 . . . . . . . 8 (𝐴𝐶 → ((2nd𝑤) ∈ 𝐴 → (2nd𝑤) ∈ 𝐶))
21pm4.71rd 665 . . . . . . 7 (𝐴𝐶 → ((2nd𝑤) ∈ 𝐴 ↔ ((2nd𝑤) ∈ 𝐶 ∧ (2nd𝑤) ∈ 𝐴)))
32anbi2d 736 . . . . . 6 (𝐴𝐶 → (((𝑤 ∈ (V × V) ∧ (1st𝑤) ∈ 𝐵) ∧ (2nd𝑤) ∈ 𝐴) ↔ ((𝑤 ∈ (V × V) ∧ (1st𝑤) ∈ 𝐵) ∧ ((2nd𝑤) ∈ 𝐶 ∧ (2nd𝑤) ∈ 𝐴))))
4 anass 679 . . . . . . . 8 ((((𝑤 ∈ (V × V) ∧ (1st𝑤) ∈ 𝐵) ∧ (2nd𝑤) ∈ 𝐶) ∧ (2nd𝑤) ∈ 𝐴) ↔ ((𝑤 ∈ (V × V) ∧ (1st𝑤) ∈ 𝐵) ∧ ((2nd𝑤) ∈ 𝐶 ∧ (2nd𝑤) ∈ 𝐴)))
54bicomi 213 . . . . . . 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 679 . . . . . . . 8 (((𝑤 ∈ (V × V) ∧ (1st𝑤) ∈ 𝐵) ∧ (2nd𝑤) ∈ 𝐶) ↔ (𝑤 ∈ (V × V) ∧ ((1st𝑤) ∈ 𝐵 ∧ (2nd𝑤) ∈ 𝐶)))
87anbi1i 727 . . . . . . 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 293 . . . . 5 (𝐴𝐶 → (((𝑤 ∈ (V × V) ∧ (1st𝑤) ∈ 𝐵) ∧ (2nd𝑤) ∈ 𝐴) ↔ ((𝑤 ∈ (V × V) ∧ ((1st𝑤) ∈ 𝐵 ∧ (2nd𝑤) ∈ 𝐶)) ∧ (2nd𝑤) ∈ 𝐴)))
11 elxp7 7092 . . . . . 6 (𝑤 ∈ (𝐵 × 𝐶) ↔ (𝑤 ∈ (V × V) ∧ ((1st𝑤) ∈ 𝐵 ∧ (2nd𝑤) ∈ 𝐶)))
1211anbi1i 727 . . . . 5 ((𝑤 ∈ (𝐵 × 𝐶) ∧ (2nd𝑤) ∈ 𝐴) ↔ ((𝑤 ∈ (V × V) ∧ ((1st𝑤) ∈ 𝐵 ∧ (2nd𝑤) ∈ 𝐶)) ∧ (2nd𝑤) ∈ 𝐴))
1310, 12syl6rbbr 278 . . . 4 (𝐴𝐶 → ((𝑤 ∈ (𝐵 × 𝐶) ∧ (2nd𝑤) ∈ 𝐴) ↔ ((𝑤 ∈ (V × V) ∧ (1st𝑤) ∈ 𝐵) ∧ (2nd𝑤) ∈ 𝐴)))
14 ancom 465 . . . 4 ((𝑤 ∈ (𝐵 × 𝐶) ∧ (2nd𝑤) ∈ 𝐴) ↔ ((2nd𝑤) ∈ 𝐴𝑤 ∈ (𝐵 × 𝐶)))
15 anass 679 . . . 4 (((𝑤 ∈ (V × V) ∧ (1st𝑤) ∈ 𝐵) ∧ (2nd𝑤) ∈ 𝐴) ↔ (𝑤 ∈ (V × V) ∧ ((1st𝑤) ∈ 𝐵 ∧ (2nd𝑤) ∈ 𝐴)))
1613, 14, 153bitr3g 301 . . 3 (𝐴𝐶 → (((2nd𝑤) ∈ 𝐴𝑤 ∈ (𝐵 × 𝐶)) ↔ (𝑤 ∈ (V × V) ∧ ((1st𝑤) ∈ 𝐵 ∧ (2nd𝑤) ∈ 𝐴))))
17 cnvresima 5541 . . . . 5 ((2nd ↾ (𝐵 × 𝐶)) “ 𝐴) = ((2nd𝐴) ∩ (𝐵 × 𝐶))
1817eleq2i 2680 . . . 4 (𝑤 ∈ ((2nd ↾ (𝐵 × 𝐶)) “ 𝐴) ↔ 𝑤 ∈ ((2nd𝐴) ∩ (𝐵 × 𝐶)))
19 elin 3758 . . . 4 (𝑤 ∈ ((2nd𝐴) ∩ (𝐵 × 𝐶)) ↔ (𝑤 ∈ (2nd𝐴) ∧ 𝑤 ∈ (𝐵 × 𝐶)))
20 vex 3176 . . . . . 6 𝑤 ∈ V
21 fo2nd 7080 . . . . . . 7 2nd :V–onto→V
22 fofn 6030 . . . . . . 7 (2nd :V–onto→V → 2nd Fn V)
23 elpreima 6245 . . . . . . 7 (2nd Fn V → (𝑤 ∈ (2nd𝐴) ↔ (𝑤 ∈ V ∧ (2nd𝑤) ∈ 𝐴)))
2421, 22, 23mp2b 10 . . . . . 6 (𝑤 ∈ (2nd𝐴) ↔ (𝑤 ∈ V ∧ (2nd𝑤) ∈ 𝐴))
2520, 24mpbiran 955 . . . . 5 (𝑤 ∈ (2nd𝐴) ↔ (2nd𝑤) ∈ 𝐴)
2625anbi1i 727 . . . 4 ((𝑤 ∈ (2nd𝐴) ∧ 𝑤 ∈ (𝐵 × 𝐶)) ↔ ((2nd𝑤) ∈ 𝐴𝑤 ∈ (𝐵 × 𝐶)))
2718, 19, 263bitri 285 . . 3 (𝑤 ∈ ((2nd ↾ (𝐵 × 𝐶)) “ 𝐴) ↔ ((2nd𝑤) ∈ 𝐴𝑤 ∈ (𝐵 × 𝐶)))
28 elxp7 7092 . . 3 (𝑤 ∈ (𝐵 × 𝐴) ↔ (𝑤 ∈ (V × V) ∧ ((1st𝑤) ∈ 𝐵 ∧ (2nd𝑤) ∈ 𝐴)))
2916, 27, 283bitr4g 302 . 2 (𝐴𝐶 → (𝑤 ∈ ((2nd ↾ (𝐵 × 𝐶)) “ 𝐴) ↔ 𝑤 ∈ (𝐵 × 𝐴)))
3029eqrdv 2608 1 (𝐴𝐶 → ((2nd ↾ (𝐵 × 𝐶)) “ 𝐴) = (𝐵 × 𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 195   ∧ wa 383   = wceq 1475   ∈ wcel 1977  Vcvv 3173   ∩ cin 3539   ⊆ wss 3540   × cxp 5036  ◡ccnv 5037   ↾ cres 5040   “ cima 5041   Fn wfn 5799  –onto→wfo 5802  ‘cfv 5804  1st c1st 7057  2nd c2nd 7058 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-fo 5810  df-fv 5812  df-1st 7059  df-2nd 7060 This theorem is referenced by:  sxbrsigalem2  29675
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