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Theorem 2ndpreima 29470
 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 3595 . . . . . . . 8 (𝐴𝐶 → ((2nd𝑤) ∈ 𝐴 → (2nd𝑤) ∈ 𝐶))
21pm4.71rd 667 . . . . . . 7 (𝐴𝐶 → ((2nd𝑤) ∈ 𝐴 ↔ ((2nd𝑤) ∈ 𝐶 ∧ (2nd𝑤) ∈ 𝐴)))
32anbi2d 740 . . . . . 6 (𝐴𝐶 → (((𝑤 ∈ (V × V) ∧ (1st𝑤) ∈ 𝐵) ∧ (2nd𝑤) ∈ 𝐴) ↔ ((𝑤 ∈ (V × V) ∧ (1st𝑤) ∈ 𝐵) ∧ ((2nd𝑤) ∈ 𝐶 ∧ (2nd𝑤) ∈ 𝐴))))
4 anass 681 . . . . . . . 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 681 . . . . . . . 8 (((𝑤 ∈ (V × V) ∧ (1st𝑤) ∈ 𝐵) ∧ (2nd𝑤) ∈ 𝐶) ↔ (𝑤 ∈ (V × V) ∧ ((1st𝑤) ∈ 𝐵 ∧ (2nd𝑤) ∈ 𝐶)))
87anbi1i 731 . . . . . . 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 7198 . . . . . 6 (𝑤 ∈ (𝐵 × 𝐶) ↔ (𝑤 ∈ (V × V) ∧ ((1st𝑤) ∈ 𝐵 ∧ (2nd𝑤) ∈ 𝐶)))
1211anbi1i 731 . . . . 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 681 . . . 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 5621 . . . . 5 ((2nd ↾ (𝐵 × 𝐶)) “ 𝐴) = ((2nd𝐴) ∩ (𝐵 × 𝐶))
1817eleq2i 2692 . . . 4 (𝑤 ∈ ((2nd ↾ (𝐵 × 𝐶)) “ 𝐴) ↔ 𝑤 ∈ ((2nd𝐴) ∩ (𝐵 × 𝐶)))
19 elin 3794 . . . 4 (𝑤 ∈ ((2nd𝐴) ∩ (𝐵 × 𝐶)) ↔ (𝑤 ∈ (2nd𝐴) ∧ 𝑤 ∈ (𝐵 × 𝐶)))
20 vex 3201 . . . . . 6 𝑤 ∈ V
21 fo2nd 7186 . . . . . . 7 2nd :V–onto→V
22 fofn 6115 . . . . . . 7 (2nd :V–onto→V → 2nd Fn V)
23 elpreima 6335 . . . . . . 7 (2nd Fn V → (𝑤 ∈ (2nd𝐴) ↔ (𝑤 ∈ V ∧ (2nd𝑤) ∈ 𝐴)))
2421, 22, 23mp2b 10 . . . . . 6 (𝑤 ∈ (2nd𝐴) ↔ (𝑤 ∈ V ∧ (2nd𝑤) ∈ 𝐴))
2520, 24mpbiran 953 . . . . 5 (𝑤 ∈ (2nd𝐴) ↔ (2nd𝑤) ∈ 𝐴)
2625anbi1i 731 . . . 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 7198 . . 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 1482   ∈ wcel 1989  Vcvv 3198   ∩ cin 3571   ⊆ wss 3572   × cxp 5110  ◡ccnv 5111   ↾ cres 5114   “ cima 5115   Fn wfn 5881  –onto→wfo 5884  ‘cfv 5886  1st c1st 7163  2nd c2nd 7164 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-fo 5892  df-fv 5894  df-1st 7165  df-2nd 7166 This theorem is referenced by:  sxbrsigalem2  30333
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