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Theorem mptprop 32713
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 4634 . 2 {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩})
2 brprop.a . . . . . . 7 (𝜑𝐴𝑉)
3 brprop.b . . . . . . 7 (𝜑𝐵𝑊)
4 fmptsn 7187 . . . . . . 7 ((𝐴𝑉𝐵𝑊) → {⟨𝐴, 𝐵⟩} = (𝑥 ∈ {𝐴} ↦ 𝐵))
52, 3, 4syl2anc 584 . . . . . 6 (𝜑 → {⟨𝐴, 𝐵⟩} = (𝑥 ∈ {𝐴} ↦ 𝐵))
6 incom 4217 . . . . . . . 8 ({𝐴} ∩ {𝐴, 𝐶}) = ({𝐴, 𝐶} ∩ {𝐴})
7 prid1g 4765 . . . . . . . . . 10 (𝐴𝑉𝐴 ∈ {𝐴, 𝐶})
8 snssi 4813 . . . . . . . . . 10 (𝐴 ∈ {𝐴, 𝐶} → {𝐴} ⊆ {𝐴, 𝐶})
92, 7, 83syl 18 . . . . . . . . 9 (𝜑 → {𝐴} ⊆ {𝐴, 𝐶})
10 dfss2 3981 . . . . . . . . 9 ({𝐴} ⊆ {𝐴, 𝐶} ↔ ({𝐴} ∩ {𝐴, 𝐶}) = {𝐴})
119, 10sylib 218 . . . . . . . 8 (𝜑 → ({𝐴} ∩ {𝐴, 𝐶}) = {𝐴})
126, 11eqtr3id 2789 . . . . . . 7 (𝜑 → ({𝐴, 𝐶} ∩ {𝐴}) = {𝐴})
1312mpteq1d 5243 . . . . . 6 (𝜑 → (𝑥 ∈ ({𝐴, 𝐶} ∩ {𝐴}) ↦ 𝐵) = (𝑥 ∈ {𝐴} ↦ 𝐵))
145, 13eqtr4d 2778 . . . . 5 (𝜑 → {⟨𝐴, 𝐵⟩} = (𝑥 ∈ ({𝐴, 𝐶} ∩ {𝐴}) ↦ 𝐵))
15 brprop.c . . . . . . 7 (𝜑𝐶𝑉)
16 brprop.d . . . . . . 7 (𝜑𝐷𝑊)
17 fmptsn 7187 . . . . . . 7 ((𝐶𝑉𝐷𝑊) → {⟨𝐶, 𝐷⟩} = (𝑥 ∈ {𝐶} ↦ 𝐷))
1815, 16, 17syl2anc 584 . . . . . 6 (𝜑 → {⟨𝐶, 𝐷⟩} = (𝑥 ∈ {𝐶} ↦ 𝐷))
19 mptprop.1 . . . . . . . 8 (𝜑𝐴𝐶)
20 difprsn1 4805 . . . . . . . 8 (𝐴𝐶 → ({𝐴, 𝐶} ∖ {𝐴}) = {𝐶})
2119, 20syl 17 . . . . . . 7 (𝜑 → ({𝐴, 𝐶} ∖ {𝐴}) = {𝐶})
2221mpteq1d 5243 . . . . . 6 (𝜑 → (𝑥 ∈ ({𝐴, 𝐶} ∖ {𝐴}) ↦ 𝐷) = (𝑥 ∈ {𝐶} ↦ 𝐷))
2318, 22eqtr4d 2778 . . . . 5 (𝜑 → {⟨𝐶, 𝐷⟩} = (𝑥 ∈ ({𝐴, 𝐶} ∖ {𝐴}) ↦ 𝐷))
2414, 23uneq12d 4179 . . . 4 (𝜑 → ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩}) = ((𝑥 ∈ ({𝐴, 𝐶} ∩ {𝐴}) ↦ 𝐵) ∪ (𝑥 ∈ ({𝐴, 𝐶} ∖ {𝐴}) ↦ 𝐷)))
25 partfun 6716 . . . 4 (𝑥 ∈ {𝐴, 𝐶} ↦ if(𝑥 ∈ {𝐴}, 𝐵, 𝐷)) = ((𝑥 ∈ ({𝐴, 𝐶} ∩ {𝐴}) ↦ 𝐵) ∪ (𝑥 ∈ ({𝐴, 𝐶} ∖ {𝐴}) ↦ 𝐷))
2624, 25eqtr4di 2793 . . 3 (𝜑 → ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩}) = (𝑥 ∈ {𝐴, 𝐶} ↦ if(𝑥 ∈ {𝐴}, 𝐵, 𝐷)))
27 elsn2g 4669 . . . . . 6 (𝐴𝑉 → (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴))
282, 27syl 17 . . . . 5 (𝜑 → (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴))
2928ifbid 4554 . . . 4 (𝜑 → if(𝑥 ∈ {𝐴}, 𝐵, 𝐷) = if(𝑥 = 𝐴, 𝐵, 𝐷))
3029mpteq2dv 5250 . . 3 (𝜑 → (𝑥 ∈ {𝐴, 𝐶} ↦ if(𝑥 ∈ {𝐴}, 𝐵, 𝐷)) = (𝑥 ∈ {𝐴, 𝐶} ↦ if(𝑥 = 𝐴, 𝐵, 𝐷)))
3126, 30eqtrd 2775 . 2 (𝜑 → ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩}) = (𝑥 ∈ {𝐴, 𝐶} ↦ if(𝑥 = 𝐴, 𝐵, 𝐷)))
321, 31eqtrid 2787 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 206   = wceq 1537  wcel 2106  wne 2938  cdif 3960  cun 3961  cin 3962  wss 3963  ifcif 4531  {csn 4631  {cpr 4633  cop 4637  cmpt 5231
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-sep 5302  ax-nul 5312  ax-pr 5438
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-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-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-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570
This theorem is referenced by:  cycpm2tr  33122
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