Theorem mptprop 33365

Description: Rewrite pairs of ordered pairs as mapping to functions. (Contributed by Thierry Arnoux, 24-Sep-2023.)

Hypotheses

Ref	Expression
brprop.a	⊢ (𝜑 → 𝐴 ∈ 𝑉)
brprop.b	⊢ (𝜑 → 𝐵 ∈ 𝑊)
brprop.c	⊢ (𝜑 → 𝐶 ∈ 𝑉)
brprop.d	⊢ (𝜑 → 𝐷 ∈ 𝑊)
mptprop.1	⊢ (𝜑 → 𝐴 ≠ 𝐶)

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

Step	Hyp	Ref	Expression
1		df-pr 4591	. 2 ⊢ {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩})
2		brprop.a	. . . . . . 7 ⊢ (𝜑 → 𝐴 ∈ 𝑉)
3		brprop.b	. . . . . . 7 ⊢ (𝜑 → 𝐵 ∈ 𝑊)
4		fmptsn 7198	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {⟨𝐴, 𝐵⟩} = (𝑥 ∈ {𝐴} ↦ 𝐵))
5	2, 3, 4	syl2anc 596	. . . . . 6 ⊢ (𝜑 → {⟨𝐴, 𝐵⟩} = (𝑥 ∈ {𝐴} ↦ 𝐵))
6		incom 4109	. . . . . . . 8 ⊢ ({𝐴} ∩ {𝐴, 𝐶}) = ({𝐴, 𝐶} ∩ {𝐴})
7		prid1g 4721	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴, 𝐶})
8		snssi 4746	. . . . . . . . . 10 ⊢ (𝐴 ∈ {𝐴, 𝐶} → {𝐴} ⊆ {𝐴, 𝐶})
9	2, 7, 8	3syl 19	. . . . . . . . 9 ⊢ (𝜑 → {𝐴} ⊆ {𝐴, 𝐶})
10		dfss2 4097	. . . . . . . . 9 ⊢ ({𝐴} ⊆ {𝐴, 𝐶} ↔ ({𝐴} ∩ {𝐴, 𝐶}) = {𝐴})
11	9, 10	sylib 221	. . . . . . . 8 ⊢ (𝜑 → ({𝐴} ∩ {𝐴, 𝐶}) = {𝐴})
12	6, 11	eqtr3id 2812	. . . . . . 7 ⊢ (𝜑 → ({𝐴, 𝐶} ∩ {𝐴}) = {𝐴})
13	12	mpteq1d 5199	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ({𝐴, 𝐶} ∩ {𝐴}) ↦ 𝐵) = (𝑥 ∈ {𝐴} ↦ 𝐵))
14	5, 13	eqtr4d 2801	. . . . 5 ⊢ (𝜑 → {⟨𝐴, 𝐵⟩} = (𝑥 ∈ ({𝐴, 𝐶} ∩ {𝐴}) ↦ 𝐵))
15		brprop.c	. . . . . . 7 ⊢ (𝜑 → 𝐶 ∈ 𝑉)
16		brprop.d	. . . . . . 7 ⊢ (𝜑 → 𝐷 ∈ 𝑊)
17		fmptsn 7198	. . . . . . 7 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) → {⟨𝐶, 𝐷⟩} = (𝑥 ∈ {𝐶} ↦ 𝐷))
18	15, 16, 17	syl2anc 596	. . . . . 6 ⊢ (𝜑 → {⟨𝐶, 𝐷⟩} = (𝑥 ∈ {𝐶} ↦ 𝐷))
19		mptprop.1	. . . . . . . 8 ⊢ (𝜑 → 𝐴 ≠ 𝐶)
20		difprsn1 4763	. . . . . . . 8 ⊢ (𝐴 ≠ 𝐶 → ({𝐴, 𝐶} ∖ {𝐴}) = {𝐶})
21	19, 20	syl 18	. . . . . . 7 ⊢ (𝜑 → ({𝐴, 𝐶} ∖ {𝐴}) = {𝐶})
22	21	mpteq1d 5199	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ({𝐴, 𝐶} ∖ {𝐴}) ↦ 𝐷) = (𝑥 ∈ {𝐶} ↦ 𝐷))
23	18, 22	eqtr4d 2801	. . . . 5 ⊢ (𝜑 → {⟨𝐶, 𝐷⟩} = (𝑥 ∈ ({𝐴, 𝐶} ∖ {𝐴}) ↦ 𝐷))
24	14, 23	uneq12d 4171	. . . 4 ⊢ (𝜑 → ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩}) = ((𝑥 ∈ ({𝐴, 𝐶} ∩ {𝐴}) ↦ 𝐵) ∪ (𝑥 ∈ ({𝐴, 𝐶} ∖ {𝐴}) ↦ 𝐷)))
25		partfun 6711	. . . 4 ⊢ (𝑥 ∈ {𝐴, 𝐶} ↦ if(𝑥 ∈ {𝐴}, 𝐵, 𝐷)) = ((𝑥 ∈ ({𝐴, 𝐶} ∩ {𝐴}) ↦ 𝐵) ∪ (𝑥 ∈ ({𝐴, 𝐶} ∖ {𝐴}) ↦ 𝐷))
26	24, 25	eqtr4di 2816	. . 3 ⊢ (𝜑 → ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩}) = (𝑥 ∈ {𝐴, 𝐶} ↦ if(𝑥 ∈ {𝐴}, 𝐵, 𝐷)))
27		elsn2g 4625	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴))
28	2, 27	syl 18	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴))
29	28	ifbid 4509	. . . 4 ⊢ (𝜑 → if(𝑥 ∈ {𝐴}, 𝐵, 𝐷) = if(𝑥 = 𝐴, 𝐵, 𝐷))
30	29	mpteq2dv 5203	. . 3 ⊢ (𝜑 → (𝑥 ∈ {𝐴, 𝐶} ↦ if(𝑥 ∈ {𝐴}, 𝐵, 𝐷)) = (𝑥 ∈ {𝐴, 𝐶} ↦ if(𝑥 = 𝐴, 𝐵, 𝐷)))
31	26, 30	eqtrd 2798	. 2 ⊢ (𝜑 → ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩}) = (𝑥 ∈ {𝐴, 𝐶} ↦ if(𝑥 = 𝐴, 𝐵, 𝐷)))
32	1, 31	eqtrid 2810	1 ⊢ (𝜑 → {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = (𝑥 ∈ {𝐴, 𝐶} ↦ if(𝑥 = 𝐴, 𝐵, 𝐷)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2147   ≠ wne 2959   ⊆ wss 3900   ∖ cdif 4048   ∩ cin 4092   ∪ cun 4152  ifcif 4485  {csn 4588  {cpr 4590  ⟨cop 4838   ↦ cmpt 5190

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735  ax-sep 5253  ax-pr 5396

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3081  df-rex 3091  df-reu 3370  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-br 5107  df-opab 5172  df-mpt 5191  df-id 5551  df-xp 5662  df-rel 5663  df-cnv 5664  df-co 5665  df-dm 5666  df-rn 5667  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568

This theorem is used by:  cycpm2tr  33756