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Theorem mptprop 30348
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 4566 . 2 {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩})
2 brprop.a . . . . . . 7 (𝜑𝐴𝑉)
3 brprop.b . . . . . . 7 (𝜑𝐵𝑊)
4 fmptsn 6924 . . . . . . 7 ((𝐴𝑉𝐵𝑊) → {⟨𝐴, 𝐵⟩} = (𝑥 ∈ {𝐴} ↦ 𝐵))
52, 3, 4syl2anc 584 . . . . . 6 (𝜑 → {⟨𝐴, 𝐵⟩} = (𝑥 ∈ {𝐴} ↦ 𝐵))
6 incom 4181 . . . . . . . 8 ({𝐴} ∩ {𝐴, 𝐶}) = ({𝐴, 𝐶} ∩ {𝐴})
7 prid1g 4694 . . . . . . . . . 10 (𝐴𝑉𝐴 ∈ {𝐴, 𝐶})
8 snssi 4739 . . . . . . . . . 10 (𝐴 ∈ {𝐴, 𝐶} → {𝐴} ⊆ {𝐴, 𝐶})
92, 7, 83syl 18 . . . . . . . . 9 (𝜑 → {𝐴} ⊆ {𝐴, 𝐶})
10 df-ss 3955 . . . . . . . . 9 ({𝐴} ⊆ {𝐴, 𝐶} ↔ ({𝐴} ∩ {𝐴, 𝐶}) = {𝐴})
119, 10sylib 219 . . . . . . . 8 (𝜑 → ({𝐴} ∩ {𝐴, 𝐶}) = {𝐴})
126, 11syl5eqr 2874 . . . . . . 7 (𝜑 → ({𝐴, 𝐶} ∩ {𝐴}) = {𝐴})
1312mpteq1d 5151 . . . . . 6 (𝜑 → (𝑥 ∈ ({𝐴, 𝐶} ∩ {𝐴}) ↦ 𝐵) = (𝑥 ∈ {𝐴} ↦ 𝐵))
145, 13eqtr4d 2863 . . . . 5 (𝜑 → {⟨𝐴, 𝐵⟩} = (𝑥 ∈ ({𝐴, 𝐶} ∩ {𝐴}) ↦ 𝐵))
15 brprop.c . . . . . . 7 (𝜑𝐶𝑉)
16 brprop.d . . . . . . 7 (𝜑𝐷𝑊)
17 fmptsn 6924 . . . . . . 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 4731 . . . . . . . 8 (𝐴𝐶 → ({𝐴, 𝐶} ∖ {𝐴}) = {𝐶})
2119, 20syl 17 . . . . . . 7 (𝜑 → ({𝐴, 𝐶} ∖ {𝐴}) = {𝐶})
2221mpteq1d 5151 . . . . . 6 (𝜑 → (𝑥 ∈ ({𝐴, 𝐶} ∖ {𝐴}) ↦ 𝐷) = (𝑥 ∈ {𝐶} ↦ 𝐷))
2318, 22eqtr4d 2863 . . . . 5 (𝜑 → {⟨𝐶, 𝐷⟩} = (𝑥 ∈ ({𝐴, 𝐶} ∖ {𝐴}) ↦ 𝐷))
2414, 23uneq12d 4143 . . . 4 (𝜑 → ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩}) = ((𝑥 ∈ ({𝐴, 𝐶} ∩ {𝐴}) ↦ 𝐵) ∪ (𝑥 ∈ ({𝐴, 𝐶} ∖ {𝐴}) ↦ 𝐷)))
25 partfun 30337 . . . 4 (𝑥 ∈ {𝐴, 𝐶} ↦ if(𝑥 ∈ {𝐴}, 𝐵, 𝐷)) = ((𝑥 ∈ ({𝐴, 𝐶} ∩ {𝐴}) ↦ 𝐵) ∪ (𝑥 ∈ ({𝐴, 𝐶} ∖ {𝐴}) ↦ 𝐷))
2624, 25syl6eqr 2878 . . 3 (𝜑 → ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩}) = (𝑥 ∈ {𝐴, 𝐶} ↦ if(𝑥 ∈ {𝐴}, 𝐵, 𝐷)))
27 elsn2g 4599 . . . . . 6 (𝐴𝑉 → (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴))
282, 27syl 17 . . . . 5 (𝜑 → (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴))
2928ifbid 4491 . . . 4 (𝜑 → if(𝑥 ∈ {𝐴}, 𝐵, 𝐷) = if(𝑥 = 𝐴, 𝐵, 𝐷))
3029mpteq2dv 5158 . . 3 (𝜑 → (𝑥 ∈ {𝐴, 𝐶} ↦ if(𝑥 ∈ {𝐴}, 𝐵, 𝐷)) = (𝑥 ∈ {𝐴, 𝐶} ↦ if(𝑥 = 𝐴, 𝐵, 𝐷)))
3126, 30eqtrd 2860 . 2 (𝜑 → ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩}) = (𝑥 ∈ {𝐴, 𝐶} ↦ if(𝑥 = 𝐴, 𝐵, 𝐷)))
321, 31syl5eq 2872 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 207   = wceq 1530  wcel 2106  wne 3020  cdif 3936  cun 3937  cin 3938  wss 3939  ifcif 4469  {csn 4563  {cpr 4565  cop 4569  cmpt 5142
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2152  ax-12 2167  ax-ext 2796  ax-sep 5199  ax-nul 5206  ax-pr 5325
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2615  df-eu 2649  df-clab 2803  df-cleq 2817  df-clel 2897  df-nfc 2967  df-ne 3021  df-ral 3147  df-rex 3148  df-reu 3149  df-rab 3151  df-v 3501  df-dif 3942  df-un 3944  df-in 3946  df-ss 3955  df-nul 4295  df-if 4470  df-sn 4564  df-pr 4566  df-op 4570  df-br 5063  df-opab 5125  df-mpt 5143  df-id 5458  df-xp 5559  df-rel 5560  df-cnv 5561  df-co 5562  df-dm 5563  df-rn 5564  df-fun 6353  df-fn 6354  df-f 6355  df-f1 6356  df-fo 6357  df-f1o 6358
This theorem is referenced by:  cycpm2tr  30676
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