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 32683
Description: Express the preimage of a function operation as a union of preimages. This version of ofpreima 32682 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 32682 . . 3 (𝜑 → ((𝐹f 𝑅𝐺) “ 𝐷) = 𝑝 ∈ (𝑅𝐷)((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})))
6 inundif 4485 . . . . 5 (((𝑅𝐷) ∩ (ran 𝐹 × ran 𝐺)) ∪ ((𝑅𝐷) ∖ (ran 𝐹 × ran 𝐺))) = (𝑅𝐷)
7 iuneq1 5013 . . . . 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 5099 . . . 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 2765 . . 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, 10eqtrdi 2791 . 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 484 . . . . . . . . . . 11 ((𝜑𝑝 ∈ ((𝑅𝐷) ∖ (ran 𝐹 × ran 𝐺))) → 𝑝 ∈ ((𝑅𝐷) ∖ (ran 𝐹 × ran 𝐺)))
1312eldifbd 3976 . . . . . . . . . 10 ((𝜑𝑝 ∈ ((𝑅𝐷) ∖ (ran 𝐹 × ran 𝐺))) → ¬ 𝑝 ∈ (ran 𝐹 × ran 𝐺))
14 cnvimass 6102 . . . . . . . . . . . . . 14 (𝑅𝐷) ⊆ dom 𝑅
154fndmd 6674 . . . . . . . . . . . . . 14 (𝜑 → dom 𝑅 = (𝐵 × 𝐶))
1614, 15sseqtrid 4048 . . . . . . . . . . . . 13 (𝜑 → (𝑅𝐷) ⊆ (𝐵 × 𝐶))
1716ssdifssd 4157 . . . . . . . . . . . 12 (𝜑 → ((𝑅𝐷) ∖ (ran 𝐹 × ran 𝐺)) ⊆ (𝐵 × 𝐶))
1817sselda 3995 . . . . . . . . . . 11 ((𝜑𝑝 ∈ ((𝑅𝐷) ∖ (ran 𝐹 × ran 𝐺))) → 𝑝 ∈ (𝐵 × 𝐶))
19 1st2nd2 8052 . . . . . . . . . . 11 (𝑝 ∈ (𝐵 × 𝐶) → 𝑝 = ⟨(1st𝑝), (2nd𝑝)⟩)
20 elxp6 8047 . . . . . . . . . . . 12 (𝑝 ∈ (ran 𝐹 × ran 𝐺) ↔ (𝑝 = ⟨(1st𝑝), (2nd𝑝)⟩ ∧ ((1st𝑝) ∈ ran 𝐹 ∧ (2nd𝑝) ∈ ran 𝐺)))
2120simplbi2 500 . . . . . . . . . . 11 (𝑝 = ⟨(1st𝑝), (2nd𝑝)⟩ → (((1st𝑝) ∈ ran 𝐹 ∧ (2nd𝑝) ∈ ran 𝐺) → 𝑝 ∈ (ran 𝐹 × ran 𝐺)))
2218, 19, 213syl 18 . . . . . . . . . 10 ((𝜑𝑝 ∈ ((𝑅𝐷) ∖ (ran 𝐹 × ran 𝐺))) → (((1st𝑝) ∈ ran 𝐹 ∧ (2nd𝑝) ∈ ran 𝐺) → 𝑝 ∈ (ran 𝐹 × ran 𝐺)))
2313, 22mtod 198 . . . . . . . . 9 ((𝜑𝑝 ∈ ((𝑅𝐷) ∖ (ran 𝐹 × ran 𝐺))) → ¬ ((1st𝑝) ∈ ran 𝐹 ∧ (2nd𝑝) ∈ ran 𝐺))
24 ianor 983 . . . . . . . . 9 (¬ ((1st𝑝) ∈ ran 𝐹 ∧ (2nd𝑝) ∈ ran 𝐺) ↔ (¬ (1st𝑝) ∈ ran 𝐹 ∨ ¬ (2nd𝑝) ∈ ran 𝐺))
2523, 24sylib 218 . . . . . . . 8 ((𝜑𝑝 ∈ ((𝑅𝐷) ∖ (ran 𝐹 × ran 𝐺))) → (¬ (1st𝑝) ∈ ran 𝐹 ∨ ¬ (2nd𝑝) ∈ ran 𝐺))
26 disjsn 4716 . . . . . . . . 9 ((ran 𝐹 ∩ {(1st𝑝)}) = ∅ ↔ ¬ (1st𝑝) ∈ ran 𝐹)
27 disjsn 4716 . . . . . . . . 9 ((ran 𝐺 ∩ {(2nd𝑝)}) = ∅ ↔ ¬ (2nd𝑝) ∈ ran 𝐺)
2826, 27orbi12i 914 . . . . . . . 8 (((ran 𝐹 ∩ {(1st𝑝)}) = ∅ ∨ (ran 𝐺 ∩ {(2nd𝑝)}) = ∅) ↔ (¬ (1st𝑝) ∈ ran 𝐹 ∨ ¬ (2nd𝑝) ∈ ran 𝐺))
2925, 28sylibr 234 . . . . . . 7 ((𝜑𝑝 ∈ ((𝑅𝐷) ∖ (ran 𝐹 × ran 𝐺))) → ((ran 𝐹 ∩ {(1st𝑝)}) = ∅ ∨ (ran 𝐺 ∩ {(2nd𝑝)}) = ∅))
301ffnd 6738 . . . . . . . . 9 (𝜑𝐹 Fn 𝐴)
31 dffn3 6749 . . . . . . . . 9 (𝐹 Fn 𝐴𝐹:𝐴⟶ran 𝐹)
3230, 31sylib 218 . . . . . . . 8 (𝜑𝐹:𝐴⟶ran 𝐹)
332ffnd 6738 . . . . . . . . . 10 (𝜑𝐺 Fn 𝐴)
34 dffn3 6749 . . . . . . . . . 10 (𝐺 Fn 𝐴𝐺:𝐴⟶ran 𝐺)
3533, 34sylib 218 . . . . . . . . 9 (𝜑𝐺:𝐴⟶ran 𝐺)
3635adantr 480 . . . . . . . 8 ((𝜑𝑝 ∈ ((𝑅𝐷) ∖ (ran 𝐹 × ran 𝐺))) → 𝐺:𝐴⟶ran 𝐺)
37 fimacnvdisj 6787 . . . . . . . . . . 11 ((𝐹:𝐴⟶ran 𝐹 ∧ (ran 𝐹 ∩ {(1st𝑝)}) = ∅) → (𝐹 “ {(1st𝑝)}) = ∅)
38 ineq1 4221 . . . . . . . . . . . 12 ((𝐹 “ {(1st𝑝)}) = ∅ → ((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) = (∅ ∩ (𝐺 “ {(2nd𝑝)})))
39 0in 4403 . . . . . . . . . . . 12 (∅ ∩ (𝐺 “ {(2nd𝑝)})) = ∅
4038, 39eqtrdi 2791 . . . . . . . . . . 11 ((𝐹 “ {(1st𝑝)}) = ∅ → ((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) = ∅)
4137, 40syl 17 . . . . . . . . . 10 ((𝐹:𝐴⟶ran 𝐹 ∧ (ran 𝐹 ∩ {(1st𝑝)}) = ∅) → ((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) = ∅)
4241ex 412 . . . . . . . . 9 (𝐹:𝐴⟶ran 𝐹 → ((ran 𝐹 ∩ {(1st𝑝)}) = ∅ → ((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) = ∅))
43 fimacnvdisj 6787 . . . . . . . . . . 11 ((𝐺:𝐴⟶ran 𝐺 ∧ (ran 𝐺 ∩ {(2nd𝑝)}) = ∅) → (𝐺 “ {(2nd𝑝)}) = ∅)
44 ineq2 4222 . . . . . . . . . . . 12 ((𝐺 “ {(2nd𝑝)}) = ∅ → ((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) = ((𝐹 “ {(1st𝑝)}) ∩ ∅))
45 in0 4401 . . . . . . . . . . . 12 ((𝐹 “ {(1st𝑝)}) ∩ ∅) = ∅
4644, 45eqtrdi 2791 . . . . . . . . . . 11 ((𝐺 “ {(2nd𝑝)}) = ∅ → ((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) = ∅)
4743, 46syl 17 . . . . . . . . . 10 ((𝐺:𝐴⟶ran 𝐺 ∧ (ran 𝐺 ∩ {(2nd𝑝)}) = ∅) → ((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) = ∅)
4847ex 412 . . . . . . . . 9 (𝐺:𝐴⟶ran 𝐺 → ((ran 𝐺 ∩ {(2nd𝑝)}) = ∅ → ((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) = ∅))
4942, 48jaao 956 . . . . . . . 8 ((𝐹:𝐴⟶ran 𝐹𝐺:𝐴⟶ran 𝐺) → (((ran 𝐹 ∩ {(1st𝑝)}) = ∅ ∨ (ran 𝐺 ∩ {(2nd𝑝)}) = ∅) → ((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) = ∅))
5032, 36, 49syl2an2r 685 . . . . . . 7 ((𝜑𝑝 ∈ ((𝑅𝐷) ∖ (ran 𝐹 × ran 𝐺))) → (((ran 𝐹 ∩ {(1st𝑝)}) = ∅ ∨ (ran 𝐺 ∩ {(2nd𝑝)}) = ∅) → ((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) = ∅))
5129, 50mpd 15 . . . . . 6 ((𝜑𝑝 ∈ ((𝑅𝐷) ∖ (ran 𝐹 × ran 𝐺))) → ((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) = ∅)
5251iuneq2dv 5021 . . . . 5 (𝜑 𝑝 ∈ ((𝑅𝐷) ∖ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) = 𝑝 ∈ ((𝑅𝐷) ∖ (ran 𝐹 × ran 𝐺))∅)
53 iun0 5067 . . . . 5 𝑝 ∈ ((𝑅𝐷) ∖ (ran 𝐹 × ran 𝐺))∅ = ∅
5452, 53eqtrdi 2791 . . . 4 (𝜑 𝑝 ∈ ((𝑅𝐷) ∖ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) = ∅)
5554uneq2d 4178 . . 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)})) ∪ ∅))
56 un0 4400 . . 3 ( 𝑝 ∈ ((𝑅𝐷) ∩ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)})) ∪ ∅) = 𝑝 ∈ ((𝑅𝐷) ∩ (ran 𝐹 × ran 𝐺))((𝐹 “ {(1st𝑝)}) ∩ (𝐺 “ {(2nd𝑝)}))
5755, 56eqtrdi 2791 . 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)})))
5811, 57eqtrd 2775 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 395  wo 847   = wceq 1537  wcel 2106  cdif 3960  cun 3961  cin 3962  c0 4339  {csn 4631  cop 4637   ciun 4996   × cxp 5687  ccnv 5688  dom cdm 5689  ran crn 5690  cima 5692   Fn wfn 6558  wf 6559  cfv 6563  (class class class)co 7431  f cof 7695  1st c1st 8011  2nd c2nd 8012
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-ov 7434  df-oprab 7435  df-mpo 7436  df-of 7697  df-1st 8013  df-2nd 8014
This theorem is referenced by:  sibfof  34322
  Copyright terms: Public domain W3C validator