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Theorem xppreima 29776
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 6131 . . . . 5 (Fun 𝐹𝐹 Fn dom 𝐹)
2 fncnvima2 6561 . . . . 5 (𝐹 Fn dom 𝐹 → (𝐹 “ (𝑌 × 𝑍)) = {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) ∈ (𝑌 × 𝑍)})
31, 2sylbi 208 . . . 4 (Fun 𝐹 → (𝐹 “ (𝑌 × 𝑍)) = {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) ∈ (𝑌 × 𝑍)})
43adantr 468 . . 3 ((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) → (𝐹 “ (𝑌 × 𝑍)) = {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) ∈ (𝑌 × 𝑍)})
5 fvco 6495 . . . . . . . . . 10 ((Fun 𝐹𝑥 ∈ dom 𝐹) → ((1st𝐹)‘𝑥) = (1st ‘(𝐹𝑥)))
6 fvco 6495 . . . . . . . . . 10 ((Fun 𝐹𝑥 ∈ dom 𝐹) → ((2nd𝐹)‘𝑥) = (2nd ‘(𝐹𝑥)))
75, 6opeq12d 4603 . . . . . . . . 9 ((Fun 𝐹𝑥 ∈ dom 𝐹) → ⟨((1st𝐹)‘𝑥), ((2nd𝐹)‘𝑥)⟩ = ⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩)
87eqeq2d 2816 . . . . . . . 8 ((Fun 𝐹𝑥 ∈ dom 𝐹) → ((𝐹𝑥) = ⟨((1st𝐹)‘𝑥), ((2nd𝐹)‘𝑥)⟩ ↔ (𝐹𝑥) = ⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩))
95eleq1d 2870 . . . . . . . . 9 ((Fun 𝐹𝑥 ∈ dom 𝐹) → (((1st𝐹)‘𝑥) ∈ 𝑌 ↔ (1st ‘(𝐹𝑥)) ∈ 𝑌))
106eleq1d 2870 . . . . . . . . 9 ((Fun 𝐹𝑥 ∈ dom 𝐹) → (((2nd𝐹)‘𝑥) ∈ 𝑍 ↔ (2nd ‘(𝐹𝑥)) ∈ 𝑍))
119, 10anbi12d 618 . . . . . . . 8 ((Fun 𝐹𝑥 ∈ dom 𝐹) → ((((1st𝐹)‘𝑥) ∈ 𝑌 ∧ ((2nd𝐹)‘𝑥) ∈ 𝑍) ↔ ((1st ‘(𝐹𝑥)) ∈ 𝑌 ∧ (2nd ‘(𝐹𝑥)) ∈ 𝑍)))
128, 11anbi12d 618 . . . . . . 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 7432 . . . . . . 7 ((𝐹𝑥) ∈ (𝑌 × 𝑍) ↔ ((𝐹𝑥) = ⟨(1st ‘(𝐹𝑥)), (2nd ‘(𝐹𝑥))⟩ ∧ ((1st ‘(𝐹𝑥)) ∈ 𝑌 ∧ (2nd ‘(𝐹𝑥)) ∈ 𝑍)))
1412, 13syl6rbbr 281 . . . . . 6 ((Fun 𝐹𝑥 ∈ dom 𝐹) → ((𝐹𝑥) ∈ (𝑌 × 𝑍) ↔ ((𝐹𝑥) = ⟨((1st𝐹)‘𝑥), ((2nd𝐹)‘𝑥)⟩ ∧ (((1st𝐹)‘𝑥) ∈ 𝑌 ∧ ((2nd𝐹)‘𝑥) ∈ 𝑍))))
1514adantlr 697 . . . . 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 29775 . . . . . 6 (((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) ∧ 𝑥 ∈ dom 𝐹) → (𝐹𝑥) = ⟨((1st𝐹)‘𝑥), ((2nd𝐹)‘𝑥)⟩)
1716biantrurd 524 . . . . 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 7418 . . . . . . . . . . 11 1st :V–onto→V
19 fofun 6332 . . . . . . . . . . 11 (1st :V–onto→V → Fun 1st )
2018, 19ax-mp 5 . . . . . . . . . 10 Fun 1st
21 funco 6141 . . . . . . . . . 10 ((Fun 1st ∧ Fun 𝐹) → Fun (1st𝐹))
2220, 21mpan 673 . . . . . . . . 9 (Fun 𝐹 → Fun (1st𝐹))
2322adantr 468 . . . . . . . 8 ((Fun 𝐹𝑥 ∈ dom 𝐹) → Fun (1st𝐹))
24 ssv 3822 . . . . . . . . . . . 12 (𝐹 “ dom 𝐹) ⊆ V
25 fof 6331 . . . . . . . . . . . . 13 (1st :V–onto→V → 1st :V⟶V)
26 fdm 6264 . . . . . . . . . . . . 13 (1st :V⟶V → dom 1st = V)
2718, 25, 26mp2b 10 . . . . . . . . . . . 12 dom 1st = V
2824, 27sseqtr4i 3835 . . . . . . . . . . 11 (𝐹 “ dom 𝐹) ⊆ dom 1st
29 ssid 3820 . . . . . . . . . . . 12 dom 𝐹 ⊆ dom 𝐹
30 funimass3 6555 . . . . . . . . . . . 12 ((Fun 𝐹 ∧ dom 𝐹 ⊆ dom 𝐹) → ((𝐹 “ dom 𝐹) ⊆ dom 1st ↔ dom 𝐹 ⊆ (𝐹 “ dom 1st )))
3129, 30mpan2 674 . . . . . . . . . . 11 (Fun 𝐹 → ((𝐹 “ dom 𝐹) ⊆ dom 1st ↔ dom 𝐹 ⊆ (𝐹 “ dom 1st )))
3228, 31mpbii 224 . . . . . . . . . 10 (Fun 𝐹 → dom 𝐹 ⊆ (𝐹 “ dom 1st ))
3332sselda 3798 . . . . . . . . 9 ((Fun 𝐹𝑥 ∈ dom 𝐹) → 𝑥 ∈ (𝐹 “ dom 1st ))
34 dmco 5857 . . . . . . . . 9 dom (1st𝐹) = (𝐹 “ dom 1st )
3533, 34syl6eleqr 2896 . . . . . . . 8 ((Fun 𝐹𝑥 ∈ dom 𝐹) → 𝑥 ∈ dom (1st𝐹))
36 fvimacnv 6554 . . . . . . . 8 ((Fun (1st𝐹) ∧ 𝑥 ∈ dom (1st𝐹)) → (((1st𝐹)‘𝑥) ∈ 𝑌𝑥 ∈ ((1st𝐹) “ 𝑌)))
3723, 35, 36syl2anc 575 . . . . . . 7 ((Fun 𝐹𝑥 ∈ dom 𝐹) → (((1st𝐹)‘𝑥) ∈ 𝑌𝑥 ∈ ((1st𝐹) “ 𝑌)))
38 fo2nd 7419 . . . . . . . . . . 11 2nd :V–onto→V
39 fofun 6332 . . . . . . . . . . 11 (2nd :V–onto→V → Fun 2nd )
4038, 39ax-mp 5 . . . . . . . . . 10 Fun 2nd
41 funco 6141 . . . . . . . . . 10 ((Fun 2nd ∧ Fun 𝐹) → Fun (2nd𝐹))
4240, 41mpan 673 . . . . . . . . 9 (Fun 𝐹 → Fun (2nd𝐹))
4342adantr 468 . . . . . . . 8 ((Fun 𝐹𝑥 ∈ dom 𝐹) → Fun (2nd𝐹))
44 fof 6331 . . . . . . . . . . . . 13 (2nd :V–onto→V → 2nd :V⟶V)
45 fdm 6264 . . . . . . . . . . . . 13 (2nd :V⟶V → dom 2nd = V)
4638, 44, 45mp2b 10 . . . . . . . . . . . 12 dom 2nd = V
4724, 46sseqtr4i 3835 . . . . . . . . . . 11 (𝐹 “ dom 𝐹) ⊆ dom 2nd
48 funimass3 6555 . . . . . . . . . . . 12 ((Fun 𝐹 ∧ dom 𝐹 ⊆ dom 𝐹) → ((𝐹 “ dom 𝐹) ⊆ dom 2nd ↔ dom 𝐹 ⊆ (𝐹 “ dom 2nd )))
4929, 48mpan2 674 . . . . . . . . . . 11 (Fun 𝐹 → ((𝐹 “ dom 𝐹) ⊆ dom 2nd ↔ dom 𝐹 ⊆ (𝐹 “ dom 2nd )))
5047, 49mpbii 224 . . . . . . . . . 10 (Fun 𝐹 → dom 𝐹 ⊆ (𝐹 “ dom 2nd ))
5150sselda 3798 . . . . . . . . 9 ((Fun 𝐹𝑥 ∈ dom 𝐹) → 𝑥 ∈ (𝐹 “ dom 2nd ))
52 dmco 5857 . . . . . . . . 9 dom (2nd𝐹) = (𝐹 “ dom 2nd )
5351, 52syl6eleqr 2896 . . . . . . . 8 ((Fun 𝐹𝑥 ∈ dom 𝐹) → 𝑥 ∈ dom (2nd𝐹))
54 fvimacnv 6554 . . . . . . . 8 ((Fun (2nd𝐹) ∧ 𝑥 ∈ dom (2nd𝐹)) → (((2nd𝐹)‘𝑥) ∈ 𝑍𝑥 ∈ ((2nd𝐹) “ 𝑍)))
5543, 53, 54syl2anc 575 . . . . . . 7 ((Fun 𝐹𝑥 ∈ dom 𝐹) → (((2nd𝐹)‘𝑥) ∈ 𝑍𝑥 ∈ ((2nd𝐹) “ 𝑍)))
5637, 55anbi12d 618 . . . . . 6 ((Fun 𝐹𝑥 ∈ dom 𝐹) → ((((1st𝐹)‘𝑥) ∈ 𝑌 ∧ ((2nd𝐹)‘𝑥) ∈ 𝑍) ↔ (𝑥 ∈ ((1st𝐹) “ 𝑌) ∧ 𝑥 ∈ ((2nd𝐹) “ 𝑍))))
5756adantlr 697 . . . . 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 298 . . . 4 (((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) ∧ 𝑥 ∈ dom 𝐹) → ((𝐹𝑥) ∈ (𝑌 × 𝑍) ↔ (𝑥 ∈ ((1st𝐹) “ 𝑌) ∧ 𝑥 ∈ ((2nd𝐹) “ 𝑍))))
5958rabbidva 3378 . . 3 ((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) → {𝑥 ∈ dom 𝐹 ∣ (𝐹𝑥) ∈ (𝑌 × 𝑍)} = {𝑥 ∈ dom 𝐹 ∣ (𝑥 ∈ ((1st𝐹) “ 𝑌) ∧ 𝑥 ∈ ((2nd𝐹) “ 𝑍))})
604, 59eqtrd 2840 . 2 ((Fun 𝐹 ∧ ran 𝐹 ⊆ (V × V)) → (𝐹 “ (𝑌 × 𝑍)) = {𝑥 ∈ dom 𝐹 ∣ (𝑥 ∈ ((1st𝐹) “ 𝑌) ∧ 𝑥 ∈ ((2nd𝐹) “ 𝑍))})
61 dfin5 3777 . . . 4 (dom 𝐹 ∩ ((1st𝐹) “ 𝑌)) = {𝑥 ∈ dom 𝐹𝑥 ∈ ((1st𝐹) “ 𝑌)}
62 dfin5 3777 . . . 4 (dom 𝐹 ∩ ((2nd𝐹) “ 𝑍)) = {𝑥 ∈ dom 𝐹𝑥 ∈ ((2nd𝐹) “ 𝑍)}
6361, 62ineq12i 4011 . . 3 ((dom 𝐹 ∩ ((1st𝐹) “ 𝑌)) ∩ (dom 𝐹 ∩ ((2nd𝐹) “ 𝑍))) = ({𝑥 ∈ dom 𝐹𝑥 ∈ ((1st𝐹) “ 𝑌)} ∩ {𝑥 ∈ dom 𝐹𝑥 ∈ ((2nd𝐹) “ 𝑍)})
64 cnvimass 5695 . . . . . 6 ((1st𝐹) “ 𝑌) ⊆ dom (1st𝐹)
65 dmcoss 5586 . . . . . 6 dom (1st𝐹) ⊆ dom 𝐹
6664, 65sstri 3807 . . . . 5 ((1st𝐹) “ 𝑌) ⊆ dom 𝐹
67 sseqin2 4016 . . . . 5 (((1st𝐹) “ 𝑌) ⊆ dom 𝐹 ↔ (dom 𝐹 ∩ ((1st𝐹) “ 𝑌)) = ((1st𝐹) “ 𝑌))
6866, 67mpbi 221 . . . 4 (dom 𝐹 ∩ ((1st𝐹) “ 𝑌)) = ((1st𝐹) “ 𝑌)
69 cnvimass 5695 . . . . . 6 ((2nd𝐹) “ 𝑍) ⊆ dom (2nd𝐹)
70 dmcoss 5586 . . . . . 6 dom (2nd𝐹) ⊆ dom 𝐹
7169, 70sstri 3807 . . . . 5 ((2nd𝐹) “ 𝑍) ⊆ dom 𝐹
72 sseqin2 4016 . . . . 5 (((2nd𝐹) “ 𝑍) ⊆ dom 𝐹 ↔ (dom 𝐹 ∩ ((2nd𝐹) “ 𝑍)) = ((2nd𝐹) “ 𝑍))
7371, 72mpbi 221 . . . 4 (dom 𝐹 ∩ ((2nd𝐹) “ 𝑍)) = ((2nd𝐹) “ 𝑍)
7468, 73ineq12i 4011 . . 3 ((dom 𝐹 ∩ ((1st𝐹) “ 𝑌)) ∩ (dom 𝐹 ∩ ((2nd𝐹) “ 𝑍))) = (((1st𝐹) “ 𝑌) ∩ ((2nd𝐹) “ 𝑍))
75 inrab 4100 . . 3 ({𝑥 ∈ dom 𝐹𝑥 ∈ ((1st𝐹) “ 𝑌)} ∩ {𝑥 ∈ dom 𝐹𝑥 ∈ ((2nd𝐹) “ 𝑍)}) = {𝑥 ∈ dom 𝐹 ∣ (𝑥 ∈ ((1st𝐹) “ 𝑌) ∧ 𝑥 ∈ ((2nd𝐹) “ 𝑍))}
7663, 74, 753eqtr3ri 2837 . 2 {𝑥 ∈ dom 𝐹 ∣ (𝑥 ∈ ((1st𝐹) “ 𝑌) ∧ 𝑥 ∈ ((2nd𝐹) “ 𝑍))} = (((1st𝐹) “ 𝑌) ∩ ((2nd𝐹) “ 𝑍))
7760, 76syl6eq 2856 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 197  wa 384   = wceq 1637  wcel 2156  {crab 3100  Vcvv 3391  cin 3768  wss 3769  cop 4376   × cxp 5309  ccnv 5310  dom cdm 5311  ran crn 5312  cima 5314  ccom 5315  Fun wfun 6095   Fn wfn 6096  wf 6097  ontowfo 6099  cfv 6101  1st c1st 7396  2nd c2nd 7397
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2068  ax-7 2104  ax-8 2158  ax-9 2165  ax-10 2185  ax-11 2201  ax-12 2214  ax-13 2420  ax-ext 2784  ax-sep 4975  ax-nul 4983  ax-pow 5035  ax-pr 5096  ax-un 7179
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2061  df-eu 2634  df-mo 2635  df-clab 2793  df-cleq 2799  df-clel 2802  df-nfc 2937  df-ne 2979  df-ral 3101  df-rex 3102  df-rab 3105  df-v 3393  df-sbc 3634  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4117  df-if 4280  df-sn 4371  df-pr 4373  df-op 4377  df-uni 4631  df-br 4845  df-opab 4907  df-mpt 4924  df-id 5219  df-xp 5317  df-rel 5318  df-cnv 5319  df-co 5320  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324  df-iota 6064  df-fun 6103  df-fn 6104  df-f 6105  df-fo 6107  df-fv 6109  df-1st 7398  df-2nd 7399
This theorem is referenced by:  xppreima2  29777
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