Theorem mptprop 32955

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 4588	. 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 7155	. . . . . . 7 ⊢ ((𝐴 ∈ 𝑉 ∧ 𝐵 ∈ 𝑊) → {⟨𝐴, 𝐵⟩} = (𝑥 ∈ {𝐴} ↦ 𝐵))
5	2, 3, 4	syl2anc 595	. . . . . 6 ⊢ (𝜑 → {⟨𝐴, 𝐵⟩} = (𝑥 ∈ {𝐴} ↦ 𝐵))
6		incom 4164	. . . . . . . 8 ⊢ ({𝐴} ∩ {𝐴, 𝐶}) = ({𝐴, 𝐶} ∩ {𝐴})
7		prid1g 4722	. . . . . . . . . 10 ⊢ (𝐴 ∈ 𝑉 → 𝐴 ∈ {𝐴, 𝐶})
8		snssi 4747	. . . . . . . . . 10 ⊢ (𝐴 ∈ {𝐴, 𝐶} → {𝐴} ⊆ {𝐴, 𝐶})
9	2, 7, 8	3syl 19	. . . . . . . . 9 ⊢ (𝜑 → {𝐴} ⊆ {𝐴, 𝐶})
10		dfss2 3925	. . . . . . . . 9 ⊢ ({𝐴} ⊆ {𝐴, 𝐶} ↔ ({𝐴} ∩ {𝐴, 𝐶}) = {𝐴})
11	9, 10	sylib 221	. . . . . . . 8 ⊢ (𝜑 → ({𝐴} ∩ {𝐴, 𝐶}) = {𝐴})
12	6, 11	eqtr3id 2814	. . . . . . 7 ⊢ (𝜑 → ({𝐴, 𝐶} ∩ {𝐴}) = {𝐴})
13	12	mpteq1d 5195	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ({𝐴, 𝐶} ∩ {𝐴}) ↦ 𝐵) = (𝑥 ∈ {𝐴} ↦ 𝐵))
14	5, 13	eqtr4d 2803	. . . . 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 7155	. . . . . . 7 ⊢ ((𝐶 ∈ 𝑉 ∧ 𝐷 ∈ 𝑊) → {⟨𝐶, 𝐷⟩} = (𝑥 ∈ {𝐶} ↦ 𝐷))
18	15, 16, 17	syl2anc 595	. . . . . 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 5195	. . . . . 6 ⊢ (𝜑 → (𝑥 ∈ ({𝐴, 𝐶} ∖ {𝐴}) ↦ 𝐷) = (𝑥 ∈ {𝐶} ↦ 𝐷))
23	18, 22	eqtr4d 2803	. . . . 5 ⊢ (𝜑 → {⟨𝐶, 𝐷⟩} = (𝑥 ∈ ({𝐴, 𝐶} ∖ {𝐴}) ↦ 𝐷))
24	14, 23	uneq12d 4125	. . . 4 ⊢ (𝜑 → ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩}) = ((𝑥 ∈ ({𝐴, 𝐶} ∩ {𝐴}) ↦ 𝐵) ∪ (𝑥 ∈ ({𝐴, 𝐶} ∖ {𝐴}) ↦ 𝐷)))
25		partfun 6672	. . . 4 ⊢ (𝑥 ∈ {𝐴, 𝐶} ↦ if(𝑥 ∈ {𝐴}, 𝐵, 𝐷)) = ((𝑥 ∈ ({𝐴, 𝐶} ∩ {𝐴}) ↦ 𝐵) ∪ (𝑥 ∈ ({𝐴, 𝐶} ∖ {𝐴}) ↦ 𝐷))
26	24, 25	eqtr4di 2818	. . 3 ⊢ (𝜑 → ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩}) = (𝑥 ∈ {𝐴, 𝐶} ↦ if(𝑥 ∈ {𝐴}, 𝐵, 𝐷)))
27		elsn2g 4626	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴))
28	2, 27	syl 18	. . . . 5 ⊢ (𝜑 → (𝑥 ∈ {𝐴} ↔ 𝑥 = 𝐴))
29	28	ifbid 4507	. . . 4 ⊢ (𝜑 → if(𝑥 ∈ {𝐴}, 𝐵, 𝐷) = if(𝑥 = 𝐴, 𝐵, 𝐷))
30	29	mpteq2dv 5199	. . 3 ⊢ (𝜑 → (𝑥 ∈ {𝐴, 𝐶} ↦ if(𝑥 ∈ {𝐴}, 𝐵, 𝐷)) = (𝑥 ∈ {𝐴, 𝐶} ↦ if(𝑥 = 𝐴, 𝐵, 𝐷)))
31	26, 30	eqtrd 2800	. 2 ⊢ (𝜑 → ({⟨𝐴, 𝐵⟩} ∪ {⟨𝐶, 𝐷⟩}) = (𝑥 ∈ {𝐴, 𝐶} ↦ if(𝑥 = 𝐴, 𝐵, 𝐷)))
32	1, 31	eqtrid 2812	1 ⊢ (𝜑 → {⟨𝐴, 𝐵⟩, ⟨𝐶, 𝐷⟩} = (𝑥 ∈ {𝐴, 𝐶} ↦ if(𝑥 = 𝐴, 𝐵, 𝐷)))

Syntax hints:   → wi 4   ↔ wb 209   = wceq 1563   ∈ wcel 2145   ≠ wne 2960   ∖ cdif 3904   ∪ cun 3905   ∩ cin 3906   ⊆ wss 3907  ifcif 4483  {csn 4585  {cpr 4587  ⟨cop 4591   ↦ cmpt 5186

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2737  ax-sep 5251  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-nf 1807  df-sb 2094  df-mo 2569  df-eu 2599  df-clab 2744  df-cleq 2757  df-clel 2840  df-nfc 2914  df-ne 2961  df-ral 3080  df-rex 3090  df-reu 3371  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-br 5106  df-opab 5168  df-mpt 5187  df-id 5547  df-xp 5658  df-rel 5659  df-cnv 5660  df-co 5661  df-dm 5662  df-rn 5663  df-fun 6527  df-fn 6528  df-f 6529  df-f1 6530  df-fo 6531  df-f1o 6532

This theorem is referenced by:  cycpm2tr  33352