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

Theorem mptprop 30751
Description: Rewrite pairs of ordered pairs as mapping to functions. (Contributed by Thierry Arnoux, 24-Sep-2023.)
Hypotheses
Ref Expression
brprop.a (𝜑𝐴𝑉)
brprop.b (𝜑𝐵𝑊)
brprop.c (𝜑𝐶𝑉)
brprop.d (𝜑𝐷𝑊)
mptprop.1 (𝜑𝐴𝐶)
Assertion
Ref Expression
mptprop (𝜑 → {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = (𝑥 ∈ {𝐴, 𝐶} ↦ if(𝑥 = 𝐴, 𝐵, 𝐷)))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶   𝑥,𝐷   𝜑,𝑥
Allowed substitution hints:   𝑉(𝑥)   𝑊(𝑥)

Proof of Theorem mptprop
StepHypRef Expression
1 df-pr 4544 . 2 {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩})
2 brprop.a . . . . . . 7 (𝜑𝐴𝑉)
3 brprop.b . . . . . . 7 (𝜑𝐵𝑊)
4 fmptsn 6982 . . . . . . 7 ((𝐴𝑉𝐵𝑊) → {⟨𝐴, 𝐵⟩} = (𝑥 ∈ {𝐴} ↦ 𝐵))
52, 3, 4syl2anc 587 . . . . . 6 (𝜑 → {⟨𝐴, 𝐵⟩} = (𝑥 ∈ {𝐴} ↦ 𝐵))
6 incom 4115 . . . . . . . 8 ({𝐴} ∩ {𝐴, 𝐶}) = ({𝐴, 𝐶} ∩ {𝐴})
7 prid1g 4676 . . . . . . . . . 10 (𝐴𝑉𝐴 ∈ {𝐴, 𝐶})
8 snssi 4721 . . . . . . . . . 10 (𝐴 ∈ {𝐴, 𝐶} → {𝐴} ⊆ {𝐴, 𝐶})
92, 7, 83syl 18 . . . . . . . . 9 (𝜑 → {𝐴} ⊆ {𝐴, 𝐶})
10 df-ss 3883 . . . . . . . . 9 ({𝐴} ⊆ {𝐴, 𝐶} ↔ ({𝐴} ∩ {𝐴, 𝐶}) = {𝐴})
119, 10sylib 221 . . . . . . . 8 (𝜑 → ({𝐴} ∩ {𝐴, 𝐶}) = {𝐴})
126, 11eqtr3id 2792 . . . . . . 7 (𝜑 → ({𝐴, 𝐶} ∩ {𝐴}) = {𝐴})
1312mpteq1d 5144 . . . . . 6 (𝜑 → (𝑥 ∈ ({𝐴, 𝐶} ∩ {𝐴}) ↦ 𝐵) = (𝑥 ∈ {𝐴} ↦ 𝐵))
145, 13eqtr4d 2780 . . . . 5 (𝜑 → {⟨𝐴, 𝐵⟩} = (𝑥 ∈ ({𝐴, 𝐶} ∩ {𝐴}) ↦ 𝐵))
15 brprop.c . . . . . . 7 (𝜑𝐶𝑉)
16 brprop.d . . . . . . 7 (𝜑𝐷𝑊)
17 fmptsn 6982 . . . . . . 7 ((𝐶𝑉𝐷𝑊) → {⟨𝐶, 𝐷⟩} = (𝑥 ∈ {𝐶} ↦ 𝐷))
1815, 16, 17syl2anc 587 . . . . . 6 (𝜑 → {⟨𝐶, 𝐷⟩} = (𝑥 ∈ {𝐶} ↦ 𝐷))
19 mptprop.1 . . . . . . . 8 (𝜑𝐴𝐶)
20 difprsn1 4713 . . . . . . . 8 (𝐴𝐶 → ({𝐴, 𝐶} ∖ {𝐴}) = {𝐶})
2119, 20syl 17 . . . . . . 7 (𝜑 → ({𝐴, 𝐶} ∖ {𝐴}) = {𝐶})
2221mpteq1d 5144 . . . . . 6 (𝜑 → (𝑥 ∈ ({𝐴, 𝐶} ∖ {𝐴}) ↦ 𝐷) = (𝑥 ∈ {𝐶} ↦ 𝐷))
2318, 22eqtr4d 2780 . . . . 5 (𝜑 → {⟨𝐶, 𝐷⟩} = (𝑥 ∈ ({𝐴, 𝐶} ∖ {𝐴}) ↦ 𝐷))
2414, 23uneq12d 4078 . . . 4 (𝜑 → ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩}) = ((𝑥 ∈ ({𝐴, 𝐶} ∩ {𝐴}) ↦ 𝐵) ∪ (𝑥 ∈ ({𝐴, 𝐶} ∖ {𝐴}) ↦ 𝐷)))
25 partfun 6525 . . . 4 (𝑥 ∈ {𝐴, 𝐶} ↦ if(𝑥 ∈ {𝐴}, 𝐵, 𝐷)) = ((𝑥 ∈ ({𝐴, 𝐶} ∩ {𝐴}) ↦ 𝐵) ∪ (𝑥 ∈ ({𝐴, 𝐶} ∖ {𝐴}) ↦ 𝐷))
2624, 25eqtr4di 2796 . . 3 (𝜑 → ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩}) = (𝑥 ∈ {𝐴, 𝐶} ↦ if(𝑥 ∈ {𝐴}, 𝐵, 𝐷)))
27 elsn2g 4579 . . . . . 6 (𝐴𝑉 → (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴))
282, 27syl 17 . . . . 5 (𝜑 → (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴))
2928ifbid 4462 . . . 4 (𝜑 → if(𝑥 ∈ {𝐴}, 𝐵, 𝐷) = if(𝑥 = 𝐴, 𝐵, 𝐷))
3029mpteq2dv 5151 . . 3 (𝜑 → (𝑥 ∈ {𝐴, 𝐶} ↦ if(𝑥 ∈ {𝐴}, 𝐵, 𝐷)) = (𝑥 ∈ {𝐴, 𝐶} ↦ if(𝑥 = 𝐴, 𝐵, 𝐷)))
3126, 30eqtrd 2777 . 2 (𝜑 → ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩}) = (𝑥 ∈ {𝐴, 𝐶} ↦ if(𝑥 = 𝐴, 𝐵, 𝐷)))
321, 31syl5eq 2790 1 (𝜑 → {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = (𝑥 ∈ {𝐴, 𝐶} ↦ if(𝑥 = 𝐴, 𝐵, 𝐷)))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209   = wceq 1543  wcel 2110  wne 2940  cdif 3863  cun 3864  cin 3865  wss 3866  ifcif 4439  {csn 4541  {cpr 4543  cop 4547  cmpt 5135
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2708  ax-sep 5192  ax-nul 5199  ax-pr 5322
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2071  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2886  df-ne 2941  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3410  df-dif 3869  df-un 3871  df-in 3873  df-ss 3883  df-nul 4238  df-if 4440  df-sn 4542  df-pr 4544  df-op 4548  df-br 5054  df-opab 5116  df-mpt 5136  df-id 5455  df-xp 5557  df-rel 5558  df-cnv 5559  df-co 5560  df-dm 5561  df-rn 5562  df-fun 6382  df-fn 6383  df-f 6384  df-f1 6385  df-fo 6386  df-f1o 6387
This theorem is referenced by:  cycpm2tr  31105
  Copyright terms: Public domain W3C validator