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Theorem 2ndpreima 30202
 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 3852 . . . . . . . 8 (𝐴𝐶 → ((2nd𝑤) ∈ 𝐴 → (2nd𝑤) ∈ 𝐶))
21pm4.71rd 555 . . . . . . 7 (𝐴𝐶 → ((2nd𝑤) ∈ 𝐴 ↔ ((2nd𝑤) ∈ 𝐶 ∧ (2nd𝑤) ∈ 𝐴)))
32anbi2d 619 . . . . . 6 (𝐴𝐶 → (((𝑤 ∈ (V × V) ∧ (1st𝑤) ∈ 𝐵) ∧ (2nd𝑤) ∈ 𝐴) ↔ ((𝑤 ∈ (V × V) ∧ (1st𝑤) ∈ 𝐵) ∧ ((2nd𝑤) ∈ 𝐶 ∧ (2nd𝑤) ∈ 𝐴))))
4 anass 461 . . . . . . . 8 ((((𝑤 ∈ (V × V) ∧ (1st𝑤) ∈ 𝐵) ∧ (2nd𝑤) ∈ 𝐶) ∧ (2nd𝑤) ∈ 𝐴) ↔ ((𝑤 ∈ (V × V) ∧ (1st𝑤) ∈ 𝐵) ∧ ((2nd𝑤) ∈ 𝐶 ∧ (2nd𝑤) ∈ 𝐴)))
54bicomi 216 . . . . . . 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 461 . . . . . . . 8 (((𝑤 ∈ (V × V) ∧ (1st𝑤) ∈ 𝐵) ∧ (2nd𝑤) ∈ 𝐶) ↔ (𝑤 ∈ (V × V) ∧ ((1st𝑤) ∈ 𝐵 ∧ (2nd𝑤) ∈ 𝐶)))
87anbi1i 614 . . . . . . 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 297 . . . . 5 (𝐴𝐶 → (((𝑤 ∈ (V × V) ∧ (1st𝑤) ∈ 𝐵) ∧ (2nd𝑤) ∈ 𝐴) ↔ ((𝑤 ∈ (V × V) ∧ ((1st𝑤) ∈ 𝐵 ∧ (2nd𝑤) ∈ 𝐶)) ∧ (2nd𝑤) ∈ 𝐴)))
11 elxp7 7536 . . . . . 6 (𝑤 ∈ (𝐵 × 𝐶) ↔ (𝑤 ∈ (V × V) ∧ ((1st𝑤) ∈ 𝐵 ∧ (2nd𝑤) ∈ 𝐶)))
1211anbi1i 614 . . . . 5 ((𝑤 ∈ (𝐵 × 𝐶) ∧ (2nd𝑤) ∈ 𝐴) ↔ ((𝑤 ∈ (V × V) ∧ ((1st𝑤) ∈ 𝐵 ∧ (2nd𝑤) ∈ 𝐶)) ∧ (2nd𝑤) ∈ 𝐴))
1310, 12syl6rbbr 282 . . . 4 (𝐴𝐶 → ((𝑤 ∈ (𝐵 × 𝐶) ∧ (2nd𝑤) ∈ 𝐴) ↔ ((𝑤 ∈ (V × V) ∧ (1st𝑤) ∈ 𝐵) ∧ (2nd𝑤) ∈ 𝐴)))
14 ancom 453 . . . 4 ((𝑤 ∈ (𝐵 × 𝐶) ∧ (2nd𝑤) ∈ 𝐴) ↔ ((2nd𝑤) ∈ 𝐴𝑤 ∈ (𝐵 × 𝐶)))
15 anass 461 . . . 4 (((𝑤 ∈ (V × V) ∧ (1st𝑤) ∈ 𝐵) ∧ (2nd𝑤) ∈ 𝐴) ↔ (𝑤 ∈ (V × V) ∧ ((1st𝑤) ∈ 𝐵 ∧ (2nd𝑤) ∈ 𝐴)))
1613, 14, 153bitr3g 305 . . 3 (𝐴𝐶 → (((2nd𝑤) ∈ 𝐴𝑤 ∈ (𝐵 × 𝐶)) ↔ (𝑤 ∈ (V × V) ∧ ((1st𝑤) ∈ 𝐵 ∧ (2nd𝑤) ∈ 𝐴))))
17 cnvresima 5926 . . . . 5 ((2nd ↾ (𝐵 × 𝐶)) “ 𝐴) = ((2nd𝐴) ∩ (𝐵 × 𝐶))
1817eleq2i 2857 . . . 4 (𝑤 ∈ ((2nd ↾ (𝐵 × 𝐶)) “ 𝐴) ↔ 𝑤 ∈ ((2nd𝐴) ∩ (𝐵 × 𝐶)))
19 elin 4057 . . . 4 (𝑤 ∈ ((2nd𝐴) ∩ (𝐵 × 𝐶)) ↔ (𝑤 ∈ (2nd𝐴) ∧ 𝑤 ∈ (𝐵 × 𝐶)))
20 vex 3418 . . . . . 6 𝑤 ∈ V
21 fo2nd 7522 . . . . . . 7 2nd :V–onto→V
22 fofn 6421 . . . . . . 7 (2nd :V–onto→V → 2nd Fn V)
23 elpreima 6653 . . . . . . 7 (2nd Fn V → (𝑤 ∈ (2nd𝐴) ↔ (𝑤 ∈ V ∧ (2nd𝑤) ∈ 𝐴)))
2421, 22, 23mp2b 10 . . . . . 6 (𝑤 ∈ (2nd𝐴) ↔ (𝑤 ∈ V ∧ (2nd𝑤) ∈ 𝐴))
2520, 24mpbiran 696 . . . . 5 (𝑤 ∈ (2nd𝐴) ↔ (2nd𝑤) ∈ 𝐴)
2625anbi1i 614 . . . 4 ((𝑤 ∈ (2nd𝐴) ∧ 𝑤 ∈ (𝐵 × 𝐶)) ↔ ((2nd𝑤) ∈ 𝐴𝑤 ∈ (𝐵 × 𝐶)))
2718, 19, 263bitri 289 . . 3 (𝑤 ∈ ((2nd ↾ (𝐵 × 𝐶)) “ 𝐴) ↔ ((2nd𝑤) ∈ 𝐴𝑤 ∈ (𝐵 × 𝐶)))
28 elxp7 7536 . . 3 (𝑤 ∈ (𝐵 × 𝐴) ↔ (𝑤 ∈ (V × V) ∧ ((1st𝑤) ∈ 𝐵 ∧ (2nd𝑤) ∈ 𝐴)))
2916, 27, 283bitr4g 306 . 2 (𝐴𝐶 → (𝑤 ∈ ((2nd ↾ (𝐵 × 𝐶)) “ 𝐴) ↔ 𝑤 ∈ (𝐵 × 𝐴)))
3029eqrdv 2776 1 (𝐴𝐶 → ((2nd ↾ (𝐵 × 𝐶)) “ 𝐴) = (𝐵 × 𝐴))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 387   = wceq 1507   ∈ wcel 2050  Vcvv 3415   ∩ cin 3828   ⊆ wss 3829   × cxp 5405  ◡ccnv 5406   ↾ cres 5409   “ cima 5410   Fn wfn 6183  –onto→wfo 6186  ‘cfv 6188  1st c1st 7499  2nd c2nd 7500 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 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-sep 5060  ax-nul 5067  ax-pow 5119  ax-pr 5186  ax-un 7279 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 2016  df-mo 2547  df-eu 2584  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-rab 3097  df-v 3417  df-sbc 3682  df-dif 3832  df-un 3834  df-in 3836  df-ss 3843  df-nul 4179  df-if 4351  df-sn 4442  df-pr 4444  df-op 4448  df-uni 4713  df-br 4930  df-opab 4992  df-mpt 5009  df-id 5312  df-xp 5413  df-rel 5414  df-cnv 5415  df-co 5416  df-dm 5417  df-rn 5418  df-res 5419  df-ima 5420  df-iota 6152  df-fun 6190  df-fn 6191  df-f 6192  df-fo 6194  df-fv 6196  df-1st 7501  df-2nd 7502 This theorem is referenced by:  sxbrsigalem2  31195
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