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Theorem mptprop 30465
 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 4528 . 2 {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩})
2 brprop.a . . . . . . 7 (𝜑𝐴𝑉)
3 brprop.b . . . . . . 7 (𝜑𝐵𝑊)
4 fmptsn 6906 . . . . . . 7 ((𝐴𝑉𝐵𝑊) → {⟨𝐴, 𝐵⟩} = (𝑥 ∈ {𝐴} ↦ 𝐵))
52, 3, 4syl2anc 587 . . . . . 6 (𝜑 → {⟨𝐴, 𝐵⟩} = (𝑥 ∈ {𝐴} ↦ 𝐵))
6 incom 4128 . . . . . . . 8 ({𝐴} ∩ {𝐴, 𝐶}) = ({𝐴, 𝐶} ∩ {𝐴})
7 prid1g 4656 . . . . . . . . . 10 (𝐴𝑉𝐴 ∈ {𝐴, 𝐶})
8 snssi 4701 . . . . . . . . . 10 (𝐴 ∈ {𝐴, 𝐶} → {𝐴} ⊆ {𝐴, 𝐶})
92, 7, 83syl 18 . . . . . . . . 9 (𝜑 → {𝐴} ⊆ {𝐴, 𝐶})
10 df-ss 3898 . . . . . . . . 9 ({𝐴} ⊆ {𝐴, 𝐶} ↔ ({𝐴} ∩ {𝐴, 𝐶}) = {𝐴})
119, 10sylib 221 . . . . . . . 8 (𝜑 → ({𝐴} ∩ {𝐴, 𝐶}) = {𝐴})
126, 11syl5eqr 2847 . . . . . . 7 (𝜑 → ({𝐴, 𝐶} ∩ {𝐴}) = {𝐴})
1312mpteq1d 5119 . . . . . 6 (𝜑 → (𝑥 ∈ ({𝐴, 𝐶} ∩ {𝐴}) ↦ 𝐵) = (𝑥 ∈ {𝐴} ↦ 𝐵))
145, 13eqtr4d 2836 . . . . 5 (𝜑 → {⟨𝐴, 𝐵⟩} = (𝑥 ∈ ({𝐴, 𝐶} ∩ {𝐴}) ↦ 𝐵))
15 brprop.c . . . . . . 7 (𝜑𝐶𝑉)
16 brprop.d . . . . . . 7 (𝜑𝐷𝑊)
17 fmptsn 6906 . . . . . . 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 4693 . . . . . . . 8 (𝐴𝐶 → ({𝐴, 𝐶} ∖ {𝐴}) = {𝐶})
2119, 20syl 17 . . . . . . 7 (𝜑 → ({𝐴, 𝐶} ∖ {𝐴}) = {𝐶})
2221mpteq1d 5119 . . . . . 6 (𝜑 → (𝑥 ∈ ({𝐴, 𝐶} ∖ {𝐴}) ↦ 𝐷) = (𝑥 ∈ {𝐶} ↦ 𝐷))
2318, 22eqtr4d 2836 . . . . 5 (𝜑 → {⟨𝐶, 𝐷⟩} = (𝑥 ∈ ({𝐴, 𝐶} ∖ {𝐴}) ↦ 𝐷))
2414, 23uneq12d 4091 . . . 4 (𝜑 → ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩}) = ((𝑥 ∈ ({𝐴, 𝐶} ∩ {𝐴}) ↦ 𝐵) ∪ (𝑥 ∈ ({𝐴, 𝐶} ∖ {𝐴}) ↦ 𝐷)))
25 partfun 6467 . . . 4 (𝑥 ∈ {𝐴, 𝐶} ↦ if(𝑥 ∈ {𝐴}, 𝐵, 𝐷)) = ((𝑥 ∈ ({𝐴, 𝐶} ∩ {𝐴}) ↦ 𝐵) ∪ (𝑥 ∈ ({𝐴, 𝐶} ∖ {𝐴}) ↦ 𝐷))
2624, 25eqtr4di 2851 . . 3 (𝜑 → ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩}) = (𝑥 ∈ {𝐴, 𝐶} ↦ if(𝑥 ∈ {𝐴}, 𝐵, 𝐷)))
27 elsn2g 4563 . . . . . 6 (𝐴𝑉 → (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴))
282, 27syl 17 . . . . 5 (𝜑 → (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴))
2928ifbid 4447 . . . 4 (𝜑 → if(𝑥 ∈ {𝐴}, 𝐵, 𝐷) = if(𝑥 = 𝐴, 𝐵, 𝐷))
3029mpteq2dv 5126 . . 3 (𝜑 → (𝑥 ∈ {𝐴, 𝐶} ↦ if(𝑥 ∈ {𝐴}, 𝐵, 𝐷)) = (𝑥 ∈ {𝐴, 𝐶} ↦ if(𝑥 = 𝐴, 𝐵, 𝐷)))
3126, 30eqtrd 2833 . 2 (𝜑 → ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩}) = (𝑥 ∈ {𝐴, 𝐶} ↦ if(𝑥 = 𝐴, 𝐵, 𝐷)))
321, 31syl5eq 2845 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 1538   ∈ wcel 2111   ≠ wne 2987   ∖ cdif 3878   ∪ cun 3879   ∩ cin 3880   ⊆ wss 3881  ifcif 4425  {csn 4525  {cpr 4527  ⟨cop 4531   ↦ cmpt 5110 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-reu 3113  df-rab 3115  df-v 3443  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-nul 4244  df-if 4426  df-sn 4526  df-pr 4528  df-op 4532  df-br 5031  df-opab 5093  df-mpt 5111  df-id 5425  df-xp 5525  df-rel 5526  df-cnv 5527  df-co 5528  df-dm 5529  df-rn 5530  df-fun 6326  df-fn 6327  df-f 6328  df-f1 6329  df-fo 6330  df-f1o 6331 This theorem is referenced by:  cycpm2tr  30818
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