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Theorem xppreima 30322
Description: The preimage of a Cartesian product is the intersection of the preimages of each component function. (Contributed by Thierry Arnoux, 6-Jun-2017.)
Assertion
Ref Expression
xppreima ((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) → (𝐹 “ (𝑌 × 𝑍)) = (((1st𝐹) “ 𝑌) ∩ ((2nd𝐹) “ 𝑍)))

Proof of Theorem xppreima
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 funfn 6378 . . . . 5 (Fun 𝐹𝐹 Fn dom 𝐹)
2 fncnvima2 6823 . . . . 5 (𝐹 Fn dom 𝐹 → (𝐹 “ (𝑌 × 𝑍)) = {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) ∈ (𝑌 × 𝑍)})
31, 2sylbi 218 . . . 4 (Fun 𝐹 → (𝐹 “ (𝑌 × 𝑍)) = {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) ∈ (𝑌 × 𝑍)})
43adantr 481 . . 3 ((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) → (𝐹 “ (𝑌 × 𝑍)) = {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) ∈ (𝑌 × 𝑍)})
5 fvco 6752 . . . . . . . . . 10 ((Fun 𝐹𝑥 ∈ dom 𝐹) → ((1st𝐹)‘𝑥) = (1st ‘(𝐹𝑥)))
6 fvco 6752 . . . . . . . . . 10 ((Fun 𝐹𝑥 ∈ dom 𝐹) → ((2nd𝐹)‘𝑥) = (2nd ‘(𝐹𝑥)))
75, 6opeq12d 4803 . . . . . . . . 9 ((Fun 𝐹𝑥 ∈ dom 𝐹) → ⟨((1st𝐹)‘𝑥), ((2nd𝐹)‘𝑥)⟩ = ⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩)
87eqeq2d 2829 . . . . . . . 8 ((Fun 𝐹𝑥 ∈ dom 𝐹) → ((𝐹𝑥) = ⟨((1st𝐹)‘𝑥), ((2nd𝐹)‘𝑥)⟩ ↔ (𝐹𝑥) = ⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩))
95eleq1d 2894 . . . . . . . . 9 ((Fun 𝐹𝑥 ∈ dom 𝐹) → (((1st𝐹)‘𝑥) ∈ 𝑌 ↔ (1st ‘(𝐹𝑥)) ∈ 𝑌))
106eleq1d 2894 . . . . . . . . 9 ((Fun 𝐹𝑥 ∈ dom 𝐹) → (((2nd𝐹)‘𝑥) ∈ 𝑍 ↔ (2nd ‘(𝐹𝑥)) ∈ 𝑍))
119, 10anbi12d 630 . . . . . . . 8 ((Fun 𝐹𝑥 ∈ dom 𝐹) → ((((1st𝐹)‘𝑥) ∈ 𝑌 ∧ ((2nd𝐹)‘𝑥) ∈ 𝑍) ↔ ((1st ‘(𝐹𝑥)) ∈ 𝑌 ∧ (2nd ‘(𝐹𝑥)) ∈ 𝑍)))
128, 11anbi12d 630 . . . . . . 7 ((Fun 𝐹𝑥 ∈ dom 𝐹) → (((𝐹𝑥) = ⟨((1st𝐹)‘𝑥), ((2nd𝐹)‘𝑥)⟩ ∧ (((1st𝐹)‘𝑥) ∈ 𝑌 ∧ ((2nd𝐹)‘𝑥) ∈ 𝑍)) ↔ ((𝐹𝑥) = ⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩ ∧ ((1st ‘(𝐹𝑥)) ∈ 𝑌 ∧ (2nd ‘(𝐹𝑥)) ∈ 𝑍))))
13 elxp6 7712 . . . . . . 7 ((𝐹𝑥) ∈ (𝑌 × 𝑍) ↔ ((𝐹𝑥) = ⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩ ∧ ((1st ‘(𝐹𝑥)) ∈ 𝑌 ∧ (2nd ‘(𝐹𝑥)) ∈ 𝑍)))
1412, 13syl6rbbr 291 . . . . . 6 ((Fun 𝐹𝑥 ∈ dom 𝐹) → ((𝐹𝑥) ∈ (𝑌 × 𝑍) ↔ ((𝐹𝑥) = ⟨((1st𝐹)‘𝑥), ((2nd𝐹)‘𝑥)⟩ ∧ (((1st𝐹)‘𝑥) ∈ 𝑌 ∧ ((2nd𝐹)‘𝑥) ∈ 𝑍))))
1514adantlr 711 . . . . 5 (((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) ∧ 𝑥 ∈ dom 𝐹) → ((𝐹𝑥) ∈ (𝑌 × 𝑍) ↔ ((𝐹𝑥) = ⟨((1st𝐹)‘𝑥), ((2nd𝐹)‘𝑥)⟩ ∧ (((1st𝐹)‘𝑥) ∈ 𝑌 ∧ ((2nd𝐹)‘𝑥) ∈ 𝑍))))
16 opfv 30321 . . . . . 6 (((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) ∧ 𝑥 ∈ dom 𝐹) → (𝐹𝑥) = ⟨((1st𝐹)‘𝑥), ((2nd𝐹)‘𝑥)⟩)
1716biantrurd 533 . . . . 5 (((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) ∧ 𝑥 ∈ dom 𝐹) → ((((1st𝐹)‘𝑥) ∈ 𝑌 ∧ ((2nd𝐹)‘𝑥) ∈ 𝑍) ↔ ((𝐹𝑥) = ⟨((1st𝐹)‘𝑥), ((2nd𝐹)‘𝑥)⟩ ∧ (((1st𝐹)‘𝑥) ∈ 𝑌 ∧ ((2nd𝐹)‘𝑥) ∈ 𝑍))))
18 fo1st 7698 . . . . . . . . . . 11 1st :V–onto→V
19 fofun 6584 . . . . . . . . . . 11 (1st :V–onto→V → Fun 1st )
2018, 19ax-mp 5 . . . . . . . . . 10 Fun 1st
21 funco 6388 . . . . . . . . . 10 ((Fun 1st ∧ Fun 𝐹) → Fun (1st𝐹))
2220, 21mpan 686 . . . . . . . . 9 (Fun 𝐹 → Fun (1st𝐹))
2322adantr 481 . . . . . . . 8 ((Fun 𝐹𝑥 ∈ dom 𝐹) → Fun (1st𝐹))
24 ssv 3988 . . . . . . . . . . . 12 (𝐹 “ dom 𝐹) ⊆ V
25 fof 6583 . . . . . . . . . . . . 13 (1st :V–onto→V → 1st :V⟶V)
26 fdm 6515 . . . . . . . . . . . . 13 (1st :V⟶V → dom 1st = V)
2718, 25, 26mp2b 10 . . . . . . . . . . . 12 dom 1st = V
2824, 27sseqtrri 4001 . . . . . . . . . . 11 (𝐹 “ dom 𝐹) ⊆ dom 1st
29 ssid 3986 . . . . . . . . . . . 12 dom 𝐹 ⊆ dom 𝐹
30 funimass3 6816 . . . . . . . . . . . 12 ((Fun 𝐹 ∧ dom 𝐹 ⊆ dom 𝐹) → ((𝐹 “ dom 𝐹) ⊆ dom 1st ↔ dom 𝐹 ⊆ (𝐹 “ dom 1st )))
3129, 30mpan2 687 . . . . . . . . . . 11 (Fun 𝐹 → ((𝐹 “ dom 𝐹) ⊆ dom 1st ↔ dom 𝐹 ⊆ (𝐹 “ dom 1st )))
3228, 31mpbii 234 . . . . . . . . . 10 (Fun 𝐹 → dom 𝐹 ⊆ (𝐹 “ dom 1st ))
3332sselda 3964 . . . . . . . . 9 ((Fun 𝐹𝑥 ∈ dom 𝐹) → 𝑥 ∈ (𝐹 “ dom 1st ))
34 dmco 6100 . . . . . . . . 9 dom (1st𝐹) = (𝐹 “ dom 1st )
3533, 34eleqtrrdi 2921 . . . . . . . 8 ((Fun 𝐹𝑥 ∈ dom 𝐹) → 𝑥 ∈ dom (1st𝐹))
36 fvimacnv 6815 . . . . . . . 8 ((Fun (1st𝐹) ∧ 𝑥 ∈ dom (1st𝐹)) → (((1st𝐹)‘𝑥) ∈ 𝑌𝑥 ∈ ((1st𝐹) “ 𝑌)))
3723, 35, 36syl2anc 584 . . . . . . 7 ((Fun 𝐹𝑥 ∈ dom 𝐹) → (((1st𝐹)‘𝑥) ∈ 𝑌𝑥 ∈ ((1st𝐹) “ 𝑌)))
38 fo2nd 7699 . . . . . . . . . . 11 2nd :V–onto→V
39 fofun 6584 . . . . . . . . . . 11 (2nd :V–onto→V → Fun 2nd )
4038, 39ax-mp 5 . . . . . . . . . 10 Fun 2nd
41 funco 6388 . . . . . . . . . 10 ((Fun 2nd ∧ Fun 𝐹) → Fun (2nd𝐹))
4240, 41mpan 686 . . . . . . . . 9 (Fun 𝐹 → Fun (2nd𝐹))
4342adantr 481 . . . . . . . 8 ((Fun 𝐹𝑥 ∈ dom 𝐹) → Fun (2nd𝐹))
44 fof 6583 . . . . . . . . . . . . 13 (2nd :V–onto→V → 2nd :V⟶V)
45 fdm 6515 . . . . . . . . . . . . 13 (2nd :V⟶V → dom 2nd = V)
4638, 44, 45mp2b 10 . . . . . . . . . . . 12 dom 2nd = V
4724, 46sseqtrri 4001 . . . . . . . . . . 11 (𝐹 “ dom 𝐹) ⊆ dom 2nd
48 funimass3 6816 . . . . . . . . . . . 12 ((Fun 𝐹 ∧ dom 𝐹 ⊆ dom 𝐹) → ((𝐹 “ dom 𝐹) ⊆ dom 2nd ↔ dom 𝐹 ⊆ (𝐹 “ dom 2nd )))
4929, 48mpan2 687 . . . . . . . . . . 11 (Fun 𝐹 → ((𝐹 “ dom 𝐹) ⊆ dom 2nd ↔ dom 𝐹 ⊆ (𝐹 “ dom 2nd )))
5047, 49mpbii 234 . . . . . . . . . 10 (Fun 𝐹 → dom 𝐹 ⊆ (𝐹 “ dom 2nd ))
5150sselda 3964 . . . . . . . . 9 ((Fun 𝐹𝑥 ∈ dom 𝐹) → 𝑥 ∈ (𝐹 “ dom 2nd ))
52 dmco 6100 . . . . . . . . 9 dom (2nd𝐹) = (𝐹 “ dom 2nd )
5351, 52eleqtrrdi 2921 . . . . . . . 8 ((Fun 𝐹𝑥 ∈ dom 𝐹) → 𝑥 ∈ dom (2nd𝐹))
54 fvimacnv 6815 . . . . . . . 8 ((Fun (2nd𝐹) ∧ 𝑥 ∈ dom (2nd𝐹)) → (((2nd𝐹)‘𝑥) ∈ 𝑍𝑥 ∈ ((2nd𝐹) “ 𝑍)))
5543, 53, 54syl2anc 584 . . . . . . 7 ((Fun 𝐹𝑥 ∈ dom 𝐹) → (((2nd𝐹)‘𝑥) ∈ 𝑍𝑥 ∈ ((2nd𝐹) “ 𝑍)))
5637, 55anbi12d 630 . . . . . 6 ((Fun 𝐹𝑥 ∈ dom 𝐹) → ((((1st𝐹)‘𝑥) ∈ 𝑌 ∧ ((2nd𝐹)‘𝑥) ∈ 𝑍) ↔ (𝑥 ∈ ((1st𝐹) “ 𝑌) ∧ 𝑥 ∈ ((2nd𝐹) “ 𝑍))))
5756adantlr 711 . . . . 5 (((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) ∧ 𝑥 ∈ dom 𝐹) → ((((1st𝐹)‘𝑥) ∈ 𝑌 ∧ ((2nd𝐹)‘𝑥) ∈ 𝑍) ↔ (𝑥 ∈ ((1st𝐹) “ 𝑌) ∧ 𝑥 ∈ ((2nd𝐹) “ 𝑍))))
5815, 17, 573bitr2d 308 . . . 4 (((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) ∧ 𝑥 ∈ dom 𝐹) → ((𝐹𝑥) ∈ (𝑌 × 𝑍) ↔ (𝑥 ∈ ((1st𝐹) “ 𝑌) ∧ 𝑥 ∈ ((2nd𝐹) “ 𝑍))))
5958rabbidva 3476 . . 3 ((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) → {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) ∈ (𝑌 × 𝑍)} = {𝑥 ∈ dom 𝐹 ∣ (𝑥 ∈ ((1st𝐹) “ 𝑌) ∧ 𝑥 ∈ ((2nd𝐹) “ 𝑍))})
604, 59eqtrd 2853 . 2 ((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) → (𝐹 “ (𝑌 × 𝑍)) = {𝑥 ∈ dom 𝐹 ∣ (𝑥 ∈ ((1st𝐹) “ 𝑌) ∧ 𝑥 ∈ ((2nd𝐹) “ 𝑍))})
61 dfin5 3941 . . . 4 (dom 𝐹 ∩ ((1st𝐹) “ 𝑌)) = {𝑥 ∈ dom 𝐹𝑥 ∈ ((1st𝐹) “ 𝑌)}
62 dfin5 3941 . . . 4 (dom 𝐹 ∩ ((2nd𝐹) “ 𝑍)) = {𝑥 ∈ dom 𝐹𝑥 ∈ ((2nd𝐹) “ 𝑍)}
6361, 62ineq12i 4184 . . 3 ((dom 𝐹 ∩ ((1st𝐹) “ 𝑌)) ∩ (dom 𝐹 ∩ ((2nd𝐹) “ 𝑍))) = ({𝑥 ∈ dom 𝐹𝑥 ∈ ((1st𝐹) “ 𝑌)} ∩ {𝑥 ∈ dom 𝐹𝑥 ∈ ((2nd𝐹) “ 𝑍)})
64 cnvimass 5942 . . . . . 6 ((1st𝐹) “ 𝑌) ⊆ dom (1st𝐹)
65 dmcoss 5835 . . . . . 6 dom (1st𝐹) ⊆ dom 𝐹
6664, 65sstri 3973 . . . . 5 ((1st𝐹) “ 𝑌) ⊆ dom 𝐹
67 sseqin2 4189 . . . . 5 (((1st𝐹) “ 𝑌) ⊆ dom 𝐹 ↔ (dom 𝐹 ∩ ((1st𝐹) “ 𝑌)) = ((1st𝐹) “ 𝑌))
6866, 67mpbi 231 . . . 4 (dom 𝐹 ∩ ((1st𝐹) “ 𝑌)) = ((1st𝐹) “ 𝑌)
69 cnvimass 5942 . . . . . 6 ((2nd𝐹) “ 𝑍) ⊆ dom (2nd𝐹)
70 dmcoss 5835 . . . . . 6 dom (2nd𝐹) ⊆ dom 𝐹
7169, 70sstri 3973 . . . . 5 ((2nd𝐹) “ 𝑍) ⊆ dom 𝐹
72 sseqin2 4189 . . . . 5 (((2nd𝐹) “ 𝑍) ⊆ dom 𝐹 ↔ (dom 𝐹 ∩ ((2nd𝐹) “ 𝑍)) = ((2nd𝐹) “ 𝑍))
7371, 72mpbi 231 . . . 4 (dom 𝐹 ∩ ((2nd𝐹) “ 𝑍)) = ((2nd𝐹) “ 𝑍)
7468, 73ineq12i 4184 . . 3 ((dom 𝐹 ∩ ((1st𝐹) “ 𝑌)) ∩ (dom 𝐹 ∩ ((2nd𝐹) “ 𝑍))) = (((1st𝐹) “ 𝑌) ∩ ((2nd𝐹) “ 𝑍))
75 inrab 4272 . . 3 ({𝑥 ∈ dom 𝐹𝑥 ∈ ((1st𝐹) “ 𝑌)} ∩ {𝑥 ∈ dom 𝐹𝑥 ∈ ((2nd𝐹) “ 𝑍)}) = {𝑥 ∈ dom 𝐹 ∣ (𝑥 ∈ ((1st𝐹) “ 𝑌) ∧ 𝑥 ∈ ((2nd𝐹) “ 𝑍))}
7663, 74, 753eqtr3ri 2850 . 2 {𝑥 ∈ dom 𝐹 ∣ (𝑥 ∈ ((1st𝐹) “ 𝑌) ∧ 𝑥 ∈ ((2nd𝐹) “ 𝑍))} = (((1st𝐹) “ 𝑌) ∩ ((2nd𝐹) “ 𝑍))
7760, 76syl6eq 2869 1 ((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) → (𝐹 “ (𝑌 × 𝑍)) = (((1st𝐹) “ 𝑌) ∩ ((2nd𝐹) “ 𝑍)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wa 396   = wceq 1528  wcel 2105  {crab 3139  Vcvv 3492  cin 3932  wss 3933  cop 4563   × cxp 5546  ccnv 5547  dom cdm 5548  ran crn 5549  cima 5551  ccom 5552  Fun wfun 6342   Fn wfn 6343  wf 6344  ontowfo 6346  cfv 6348  1st c1st 7676  2nd c2nd 7677
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pow 5257  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-op 4564  df-uni 4831  df-br 5058  df-opab 5120  df-mpt 5138  df-id 5453  df-xp 5554  df-rel 5555  df-cnv 5556  df-co 5557  df-dm 5558  df-rn 5559  df-res 5560  df-ima 5561  df-iota 6307  df-fun 6350  df-fn 6351  df-f 6352  df-fo 6354  df-fv 6356  df-1st 7678  df-2nd 7679
This theorem is referenced by:  xppreima2  30323
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