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Theorem mptprop 32710
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 4651 . 2 {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩})
2 brprop.a . . . . . . 7 (𝜑𝐴𝑉)
3 brprop.b . . . . . . 7 (𝜑𝐵𝑊)
4 fmptsn 7201 . . . . . . 7 ((𝐴𝑉𝐵𝑊) → {⟨𝐴, 𝐵⟩} = (𝑥 ∈ {𝐴} ↦ 𝐵))
52, 3, 4syl2anc 583 . . . . . 6 (𝜑 → {⟨𝐴, 𝐵⟩} = (𝑥 ∈ {𝐴} ↦ 𝐵))
6 incom 4230 . . . . . . . 8 ({𝐴} ∩ {𝐴, 𝐶}) = ({𝐴, 𝐶} ∩ {𝐴})
7 prid1g 4785 . . . . . . . . . 10 (𝐴𝑉𝐴 ∈ {𝐴, 𝐶})
8 snssi 4833 . . . . . . . . . 10 (𝐴 ∈ {𝐴, 𝐶} → {𝐴} ⊆ {𝐴, 𝐶})
92, 7, 83syl 18 . . . . . . . . 9 (𝜑 → {𝐴} ⊆ {𝐴, 𝐶})
10 dfss2 3994 . . . . . . . . 9 ({𝐴} ⊆ {𝐴, 𝐶} ↔ ({𝐴} ∩ {𝐴, 𝐶}) = {𝐴})
119, 10sylib 218 . . . . . . . 8 (𝜑 → ({𝐴} ∩ {𝐴, 𝐶}) = {𝐴})
126, 11eqtr3id 2794 . . . . . . 7 (𝜑 → ({𝐴, 𝐶} ∩ {𝐴}) = {𝐴})
1312mpteq1d 5261 . . . . . 6 (𝜑 → (𝑥 ∈ ({𝐴, 𝐶} ∩ {𝐴}) ↦ 𝐵) = (𝑥 ∈ {𝐴} ↦ 𝐵))
145, 13eqtr4d 2783 . . . . 5 (𝜑 → {⟨𝐴, 𝐵⟩} = (𝑥 ∈ ({𝐴, 𝐶} ∩ {𝐴}) ↦ 𝐵))
15 brprop.c . . . . . . 7 (𝜑𝐶𝑉)
16 brprop.d . . . . . . 7 (𝜑𝐷𝑊)
17 fmptsn 7201 . . . . . . 7 ((𝐶𝑉𝐷𝑊) → {⟨𝐶, 𝐷⟩} = (𝑥 ∈ {𝐶} ↦ 𝐷))
1815, 16, 17syl2anc 583 . . . . . 6 (𝜑 → {⟨𝐶, 𝐷⟩} = (𝑥 ∈ {𝐶} ↦ 𝐷))
19 mptprop.1 . . . . . . . 8 (𝜑𝐴𝐶)
20 difprsn1 4825 . . . . . . . 8 (𝐴𝐶 → ({𝐴, 𝐶} ∖ {𝐴}) = {𝐶})
2119, 20syl 17 . . . . . . 7 (𝜑 → ({𝐴, 𝐶} ∖ {𝐴}) = {𝐶})
2221mpteq1d 5261 . . . . . 6 (𝜑 → (𝑥 ∈ ({𝐴, 𝐶} ∖ {𝐴}) ↦ 𝐷) = (𝑥 ∈ {𝐶} ↦ 𝐷))
2318, 22eqtr4d 2783 . . . . 5 (𝜑 → {⟨𝐶, 𝐷⟩} = (𝑥 ∈ ({𝐴, 𝐶} ∖ {𝐴}) ↦ 𝐷))
2414, 23uneq12d 4192 . . . 4 (𝜑 → ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩}) = ((𝑥 ∈ ({𝐴, 𝐶} ∩ {𝐴}) ↦ 𝐵) ∪ (𝑥 ∈ ({𝐴, 𝐶} ∖ {𝐴}) ↦ 𝐷)))
25 partfun 6727 . . . 4 (𝑥 ∈ {𝐴, 𝐶} ↦ if(𝑥 ∈ {𝐴}, 𝐵, 𝐷)) = ((𝑥 ∈ ({𝐴, 𝐶} ∩ {𝐴}) ↦ 𝐵) ∪ (𝑥 ∈ ({𝐴, 𝐶} ∖ {𝐴}) ↦ 𝐷))
2624, 25eqtr4di 2798 . . 3 (𝜑 → ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩}) = (𝑥 ∈ {𝐴, 𝐶} ↦ if(𝑥 ∈ {𝐴}, 𝐵, 𝐷)))
27 elsn2g 4686 . . . . . 6 (𝐴𝑉 → (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴))
282, 27syl 17 . . . . 5 (𝜑 → (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴))
2928ifbid 4571 . . . 4 (𝜑 → if(𝑥 ∈ {𝐴}, 𝐵, 𝐷) = if(𝑥 = 𝐴, 𝐵, 𝐷))
3029mpteq2dv 5268 . . 3 (𝜑 → (𝑥 ∈ {𝐴, 𝐶} ↦ if(𝑥 ∈ {𝐴}, 𝐵, 𝐷)) = (𝑥 ∈ {𝐴, 𝐶} ↦ if(𝑥 = 𝐴, 𝐵, 𝐷)))
3126, 30eqtrd 2780 . 2 (𝜑 → ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩}) = (𝑥 ∈ {𝐴, 𝐶} ↦ if(𝑥 = 𝐴, 𝐵, 𝐷)))
321, 31eqtrid 2792 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 2108  wne 2946  cdif 3973  cun 3974  cin 3975  wss 3976  ifcif 4548  {csn 4648  {cpr 4650  cop 4654  cmpt 5249
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580
This theorem is referenced by:  cycpm2tr  33112
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