Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  ofpreima2 Structured version   Visualization version   GIF version

Theorem ofpreima2 30339
Description: Express the preimage of a function operation as a union of preimages. This version of ofpreima 30338 iterates the union over a smaller set. (Contributed by Thierry Arnoux, 8-Mar-2018.)
Hypotheses
Ref Expression
ofpreima.1 (𝜑𝐹:𝐴𝐵)
ofpreima.2 (𝜑𝐺:𝐴𝐶)
ofpreima.3 (𝜑𝐴𝑉)
ofpreima.4 (𝜑𝑅 Fn (𝐵 × 𝐶))
Assertion
Ref Expression
ofpreima2 (𝜑 → ((𝐹f 𝑅𝐺) “ 𝐷) = 𝑝 ∈ ((𝑅𝐷) ∩ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})))
Distinct variable groups:   𝐴,𝑝   𝐷,𝑝   𝐹,𝑝   𝐺,𝑝   𝑅,𝑝   𝜑,𝑝
Allowed substitution hints:   𝐵(𝑝)   𝐶(𝑝)   𝑉(𝑝)

Proof of Theorem ofpreima2
StepHypRef Expression
1 ofpreima.1 . . . 4 (𝜑𝐹:𝐴𝐵)
2 ofpreima.2 . . . 4 (𝜑𝐺:𝐴𝐶)
3 ofpreima.3 . . . 4 (𝜑𝐴𝑉)
4 ofpreima.4 . . . 4 (𝜑𝑅 Fn (𝐵 × 𝐶))
51, 2, 3, 4ofpreima 30338 . . 3 (𝜑 → ((𝐹f 𝑅𝐺) “ 𝐷) = 𝑝 ∈ (𝑅𝐷)((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})))
6 inundif 4423 . . . . 5 (((𝑅𝐷) ∩ (ran 𝐹 × ran 𝐺)) ∪ ((𝑅𝐷) ∖ (ran 𝐹 × ran 𝐺))) = (𝑅𝐷)
7 iuneq1 4926 . . . . 5 ((((𝑅𝐷) ∩ (ran 𝐹 × ran 𝐺)) ∪ ((𝑅𝐷) ∖ (ran 𝐹 × ran 𝐺))) = (𝑅𝐷) → 𝑝 ∈ (((𝑅𝐷) ∩ (ran 𝐹 × ran 𝐺)) ∪ ((𝑅𝐷) ∖ (ran 𝐹 × ran 𝐺)))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) = 𝑝 ∈ (𝑅𝐷)((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})))
86, 7ax-mp 5 . . . 4 𝑝 ∈ (((𝑅𝐷) ∩ (ran 𝐹 × ran 𝐺)) ∪ ((𝑅𝐷) ∖ (ran 𝐹 × ran 𝐺)))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) = 𝑝 ∈ (𝑅𝐷)((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)}))
9 iunxun 5007 . . . 4 𝑝 ∈ (((𝑅𝐷) ∩ (ran 𝐹 × ran 𝐺)) ∪ ((𝑅𝐷) ∖ (ran 𝐹 × ran 𝐺)))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) = ( 𝑝 ∈ ((𝑅𝐷) ∩ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) ∪ 𝑝 ∈ ((𝑅𝐷) ∖ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})))
108, 9eqtr3i 2843 . . 3 𝑝 ∈ (𝑅𝐷)((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) = ( 𝑝 ∈ ((𝑅𝐷) ∩ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) ∪ 𝑝 ∈ ((𝑅𝐷) ∖ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})))
115, 10syl6eq 2869 . 2 (𝜑 → ((𝐹f 𝑅𝐺) “ 𝐷) = ( 𝑝 ∈ ((𝑅𝐷) ∩ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) ∪ 𝑝 ∈ ((𝑅𝐷) ∖ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)}))))
12 simpr 485 . . . . . . . . . . 11 ((𝜑𝑝 ∈ ((𝑅𝐷) ∖ (ran 𝐹 × ran 𝐺))) → 𝑝 ∈ ((𝑅𝐷) ∖ (ran 𝐹 × ran 𝐺)))
1312eldifbd 3946 . . . . . . . . . 10 ((𝜑𝑝 ∈ ((𝑅𝐷) ∖ (ran 𝐹 × ran 𝐺))) → ¬ 𝑝 ∈ (ran 𝐹 × ran 𝐺))
14 cnvimass 5942 . . . . . . . . . . . . . 14 (𝑅𝐷) ⊆ dom 𝑅
15 fndm 6448 . . . . . . . . . . . . . . 15 (𝑅 Fn (𝐵 × 𝐶) → dom 𝑅 = (𝐵 × 𝐶))
164, 15syl 17 . . . . . . . . . . . . . 14 (𝜑 → dom 𝑅 = (𝐵 × 𝐶))
1714, 16sseqtrid 4016 . . . . . . . . . . . . 13 (𝜑 → (𝑅𝐷) ⊆ (𝐵 × 𝐶))
1817ssdifssd 4116 . . . . . . . . . . . 12 (𝜑 → ((𝑅𝐷) ∖ (ran 𝐹 × ran 𝐺)) ⊆ (𝐵 × 𝐶))
1918sselda 3964 . . . . . . . . . . 11 ((𝜑𝑝 ∈ ((𝑅𝐷) ∖ (ran 𝐹 × ran 𝐺))) → 𝑝 ∈ (𝐵 × 𝐶))
20 1st2nd2 7717 . . . . . . . . . . 11 (𝑝 ∈ (𝐵 × 𝐶) → 𝑝 = ⟨(1st𝑝), (2nd𝑝)⟩)
21 elxp6 7712 . . . . . . . . . . . 12 (𝑝 ∈ (ran 𝐹 × ran 𝐺) ↔ (𝑝 = ⟨(1st𝑝), (2nd𝑝)⟩ ∧ ((1st𝑝) ∈ ran 𝐹 ∧ (2nd𝑝) ∈ ran 𝐺)))
2221simplbi2 501 . . . . . . . . . . 11 (𝑝 = ⟨(1st𝑝), (2nd𝑝)⟩ → (((1st𝑝) ∈ ran 𝐹 ∧ (2nd𝑝) ∈ ran 𝐺) → 𝑝 ∈ (ran 𝐹 × ran 𝐺)))
2319, 20, 223syl 18 . . . . . . . . . 10 ((𝜑𝑝 ∈ ((𝑅𝐷) ∖ (ran 𝐹 × ran 𝐺))) → (((1st𝑝) ∈ ran 𝐹 ∧ (2nd𝑝) ∈ ran 𝐺) → 𝑝 ∈ (ran 𝐹 × ran 𝐺)))
2413, 23mtod 199 . . . . . . . . 9 ((𝜑𝑝 ∈ ((𝑅𝐷) ∖ (ran 𝐹 × ran 𝐺))) → ¬ ((1st𝑝) ∈ ran 𝐹 ∧ (2nd𝑝) ∈ ran 𝐺))
25 ianor 975 . . . . . . . . 9 (¬ ((1st𝑝) ∈ ran 𝐹 ∧ (2nd𝑝) ∈ ran 𝐺) ↔ (¬ (1st𝑝) ∈ ran 𝐹 ∨ ¬ (2nd𝑝) ∈ ran 𝐺))
2624, 25sylib 219 . . . . . . . 8 ((𝜑𝑝 ∈ ((𝑅𝐷) ∖ (ran 𝐹 × ran 𝐺))) → (¬ (1st𝑝) ∈ ran 𝐹 ∨ ¬ (2nd𝑝) ∈ ran 𝐺))
27 disjsn 4639 . . . . . . . . 9 ((ran 𝐹 ∩ {(1st𝑝)}) = ∅ ↔ ¬ (1st𝑝) ∈ ran 𝐹)
28 disjsn 4639 . . . . . . . . 9 ((ran 𝐺 ∩ {(2nd𝑝)}) = ∅ ↔ ¬ (2nd𝑝) ∈ ran 𝐺)
2927, 28orbi12i 908 . . . . . . . 8 (((ran 𝐹 ∩ {(1st𝑝)}) = ∅ ∨ (ran 𝐺 ∩ {(2nd𝑝)}) = ∅) ↔ (¬ (1st𝑝) ∈ ran 𝐹 ∨ ¬ (2nd𝑝) ∈ ran 𝐺))
3026, 29sylibr 235 . . . . . . 7 ((𝜑𝑝 ∈ ((𝑅𝐷) ∖ (ran 𝐹 × ran 𝐺))) → ((ran 𝐹 ∩ {(1st𝑝)}) = ∅ ∨ (ran 𝐺 ∩ {(2nd𝑝)}) = ∅))
311ffnd 6508 . . . . . . . . 9 (𝜑𝐹 Fn 𝐴)
32 dffn3 6518 . . . . . . . . 9 (𝐹 Fn 𝐴𝐹:𝐴⟶ran 𝐹)
3331, 32sylib 219 . . . . . . . 8 (𝜑𝐹:𝐴⟶ran 𝐹)
342ffnd 6508 . . . . . . . . . 10 (𝜑𝐺 Fn 𝐴)
35 dffn3 6518 . . . . . . . . . 10 (𝐺 Fn 𝐴𝐺:𝐴⟶ran 𝐺)
3634, 35sylib 219 . . . . . . . . 9 (𝜑𝐺:𝐴⟶ran 𝐺)
3736adantr 481 . . . . . . . 8 ((𝜑𝑝 ∈ ((𝑅𝐷) ∖ (ran 𝐹 × ran 𝐺))) → 𝐺:𝐴⟶ran 𝐺)
38 fimacnvdisj 6550 . . . . . . . . . . 11 ((𝐹:𝐴⟶ran 𝐹 ∧ (ran 𝐹 ∩ {(1st𝑝)}) = ∅) → (𝐹 “ {(1st𝑝)}) = ∅)
39 ineq1 4178 . . . . . . . . . . . 12 ((𝐹 “ {(1st𝑝)}) = ∅ → ((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) = (∅ ∩ (𝐺 “ {(2nd𝑝)})))
40 0in 4344 . . . . . . . . . . . 12 (∅ ∩ (𝐺 “ {(2nd𝑝)})) = ∅
4139, 40syl6eq 2869 . . . . . . . . . . 11 ((𝐹 “ {(1st𝑝)}) = ∅ → ((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) = ∅)
4238, 41syl 17 . . . . . . . . . 10 ((𝐹:𝐴⟶ran 𝐹 ∧ (ran 𝐹 ∩ {(1st𝑝)}) = ∅) → ((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) = ∅)
4342ex 413 . . . . . . . . 9 (𝐹:𝐴⟶ran 𝐹 → ((ran 𝐹 ∩ {(1st𝑝)}) = ∅ → ((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) = ∅))
44 fimacnvdisj 6550 . . . . . . . . . . 11 ((𝐺:𝐴⟶ran 𝐺 ∧ (ran 𝐺 ∩ {(2nd𝑝)}) = ∅) → (𝐺 “ {(2nd𝑝)}) = ∅)
45 ineq2 4180 . . . . . . . . . . . 12 ((𝐺 “ {(2nd𝑝)}) = ∅ → ((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) = ((𝐹 “ {(1st𝑝)}) ∩ ∅))
46 in0 4342 . . . . . . . . . . . 12 ((𝐹 “ {(1st𝑝)}) ∩ ∅) = ∅
4745, 46syl6eq 2869 . . . . . . . . . . 11 ((𝐺 “ {(2nd𝑝)}) = ∅ → ((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) = ∅)
4844, 47syl 17 . . . . . . . . . 10 ((𝐺:𝐴⟶ran 𝐺 ∧ (ran 𝐺 ∩ {(2nd𝑝)}) = ∅) → ((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) = ∅)
4948ex 413 . . . . . . . . 9 (𝐺:𝐴⟶ran 𝐺 → ((ran 𝐺 ∩ {(2nd𝑝)}) = ∅ → ((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) = ∅))
5043, 49jaao 948 . . . . . . . 8 ((𝐹:𝐴⟶ran 𝐹𝐺:𝐴⟶ran 𝐺) → (((ran 𝐹 ∩ {(1st𝑝)}) = ∅ ∨ (ran 𝐺 ∩ {(2nd𝑝)}) = ∅) → ((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) = ∅))
5133, 37, 50syl2an2r 681 . . . . . . 7 ((𝜑𝑝 ∈ ((𝑅𝐷) ∖ (ran 𝐹 × ran 𝐺))) → (((ran 𝐹 ∩ {(1st𝑝)}) = ∅ ∨ (ran 𝐺 ∩ {(2nd𝑝)}) = ∅) → ((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) = ∅))
5230, 51mpd 15 . . . . . 6 ((𝜑𝑝 ∈ ((𝑅𝐷) ∖ (ran 𝐹 × ran 𝐺))) → ((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) = ∅)
5352iuneq2dv 4934 . . . . 5 (𝜑 𝑝 ∈ ((𝑅𝐷) ∖ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) = 𝑝 ∈ ((𝑅𝐷) ∖ (ran 𝐹 × ran 𝐺))∅)
54 iun0 4976 . . . . 5 𝑝 ∈ ((𝑅𝐷) ∖ (ran 𝐹 × ran 𝐺))∅ = ∅
5553, 54syl6eq 2869 . . . 4 (𝜑 𝑝 ∈ ((𝑅𝐷) ∖ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) = ∅)
5655uneq2d 4136 . . 3 (𝜑 → ( 𝑝 ∈ ((𝑅𝐷) ∩ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) ∪ 𝑝 ∈ ((𝑅𝐷) ∖ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)}))) = ( 𝑝 ∈ ((𝑅𝐷) ∩ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) ∪ ∅))
57 un0 4341 . . 3 ( 𝑝 ∈ ((𝑅𝐷) ∩ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) ∪ ∅) = 𝑝 ∈ ((𝑅𝐷) ∩ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)}))
5856, 57syl6eq 2869 . 2 (𝜑 → ( 𝑝 ∈ ((𝑅𝐷) ∩ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) ∪ 𝑝 ∈ ((𝑅𝐷) ∖ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)}))) = 𝑝 ∈ ((𝑅𝐷) ∩ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})))
5911, 58eqtrd 2853 1 (𝜑 → ((𝐹f 𝑅𝐺) “ 𝐷) = 𝑝 ∈ ((𝑅𝐷) ∩ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  wo 841   = wceq 1528  wcel 2105  cdif 3930  cun 3931  cin 3932  c0 4288  {csn 4557  cop 4563   ciun 4910   × cxp 5546  ccnv 5547  dom cdm 5548  ran crn 5549  cima 5551   Fn wfn 6343  wf 6344  cfv 6348  (class class class)co 7145  f cof 7396  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-rep 5181  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-reu 3142  df-rab 3144  df-v 3494  df-sbc 3770  df-csb 3881  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-iun 4912  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-f1 6353  df-fo 6354  df-f1o 6355  df-fv 6356  df-ov 7148  df-oprab 7149  df-mpo 7150  df-of 7398  df-1st 7678  df-2nd 7679
This theorem is referenced by:  sibfof  31497
  Copyright terms: Public domain W3C validator