Theorem xpcomco 8991

Description: Composition with the bijection of xpcomf1o 8990 swaps the arguments to a mapping. (Contributed by Mario Carneiro, 30-May-2015.)

Hypotheses

Ref	Expression
xpcomf1o.1	⊢ 𝐹 = (𝑥 ∈ (𝐴 × 𝐵) ↦ ∪ ◡{𝑥})
xpcomco.1	⊢ 𝐺 = (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐴 ↦ 𝐶)

Assertion

Ref	Expression
xpcomco	⊢ (𝐺 ∘ 𝐹) = (𝑧 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)

Distinct variable groups:   𝑥,𝑦,𝑧,𝐴   𝑥,𝐵,𝑦,𝑧   𝑦,𝐹,𝑧

Allowed substitution hints:   𝐶(𝑥,𝑦,𝑧)   𝐹(𝑥)   𝐺(𝑥,𝑦,𝑧)

Proof of Theorem xpcomco

Dummy variables 𝑣 𝑢 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		xpcomf1o.1	. . . . . . . . . 10 ⊢ 𝐹 = (𝑥 ∈ (𝐴 × 𝐵) ↦ ∪ ◡{𝑥})
2	1	xpcomf1o 8990	. . . . . . . . 9 ⊢ 𝐹:(𝐴 × 𝐵)–1-1-onto→(𝐵 × 𝐴)
3		f1ofun 6770	. . . . . . . . 9 ⊢ (𝐹:(𝐴 × 𝐵)–1-1-onto→(𝐵 × 𝐴) → Fun 𝐹)
4		funbrfv2b 6884	. . . . . . . . 9 ⊢ (Fun 𝐹 → (𝑢𝐹𝑤 ↔ (𝑢 ∈ dom 𝐹 ∧ (𝐹‘𝑢) = 𝑤)))
5	2, 3, 4	mp2b 10	. . . . . . . 8 ⊢ (𝑢𝐹𝑤 ↔ (𝑢 ∈ dom 𝐹 ∧ (𝐹‘𝑢) = 𝑤))
6		ancom 460	. . . . . . . 8 ⊢ ((𝑢 ∈ dom 𝐹 ∧ (𝐹‘𝑢) = 𝑤) ↔ ((𝐹‘𝑢) = 𝑤 ∧ 𝑢 ∈ dom 𝐹))
7		eqcom 2736	. . . . . . . . 9 ⊢ ((𝐹‘𝑢) = 𝑤 ↔ 𝑤 = (𝐹‘𝑢))
8		f1odm 6772	. . . . . . . . . . 11 ⊢ (𝐹:(𝐴 × 𝐵)–1-1-onto→(𝐵 × 𝐴) → dom 𝐹 = (𝐴 × 𝐵))
9	2, 8	ax-mp 5	. . . . . . . . . 10 ⊢ dom 𝐹 = (𝐴 × 𝐵)
10	9	eleq2i 2820	. . . . . . . . 9 ⊢ (𝑢 ∈ dom 𝐹 ↔ 𝑢 ∈ (𝐴 × 𝐵))
11	7, 10	anbi12i 628	. . . . . . . 8 ⊢ (((𝐹‘𝑢) = 𝑤 ∧ 𝑢 ∈ dom 𝐹) ↔ (𝑤 = (𝐹‘𝑢) ∧ 𝑢 ∈ (𝐴 × 𝐵)))
12	5, 6, 11	3bitri 297	. . . . . . 7 ⊢ (𝑢𝐹𝑤 ↔ (𝑤 = (𝐹‘𝑢) ∧ 𝑢 ∈ (𝐴 × 𝐵)))
13	12	anbi1i 624	. . . . . 6 ⊢ ((𝑢𝐹𝑤 ∧ 𝑤𝐺𝑣) ↔ ((𝑤 = (𝐹‘𝑢) ∧ 𝑢 ∈ (𝐴 × 𝐵)) ∧ 𝑤𝐺𝑣))
14		anass 468	. . . . . 6 ⊢ (((𝑤 = (𝐹‘𝑢) ∧ 𝑢 ∈ (𝐴 × 𝐵)) ∧ 𝑤𝐺𝑣) ↔ (𝑤 = (𝐹‘𝑢) ∧ (𝑢 ∈ (𝐴 × 𝐵) ∧ 𝑤𝐺𝑣)))
15	13, 14	bitri 275	. . . . 5 ⊢ ((𝑢𝐹𝑤 ∧ 𝑤𝐺𝑣) ↔ (𝑤 = (𝐹‘𝑢) ∧ (𝑢 ∈ (𝐴 × 𝐵) ∧ 𝑤𝐺𝑣)))
16	15	exbii 1848	. . . 4 ⊢ (∃𝑤(𝑢𝐹𝑤 ∧ 𝑤𝐺𝑣) ↔ ∃𝑤(𝑤 = (𝐹‘𝑢) ∧ (𝑢 ∈ (𝐴 × 𝐵) ∧ 𝑤𝐺𝑣)))
17		fvex 6839	. . . . 5 ⊢ (𝐹‘𝑢) ∈ V
18		breq1 5098	. . . . . 6 ⊢ (𝑤 = (𝐹‘𝑢) → (𝑤𝐺𝑣 ↔ (𝐹‘𝑢)𝐺𝑣))
19	18	anbi2d 630	. . . . 5 ⊢ (𝑤 = (𝐹‘𝑢) → ((𝑢 ∈ (𝐴 × 𝐵) ∧ 𝑤𝐺𝑣) ↔ (𝑢 ∈ (𝐴 × 𝐵) ∧ (𝐹‘𝑢)𝐺𝑣)))
20	17, 19	ceqsexv 3489	. . . 4 ⊢ (∃𝑤(𝑤 = (𝐹‘𝑢) ∧ (𝑢 ∈ (𝐴 × 𝐵) ∧ 𝑤𝐺𝑣)) ↔ (𝑢 ∈ (𝐴 × 𝐵) ∧ (𝐹‘𝑢)𝐺𝑣))
21		elxp 5646	. . . . . 6 ⊢ (𝑢 ∈ (𝐴 × 𝐵) ↔ ∃𝑧∃𝑦(𝑢 = ⟨𝑧, 𝑦⟩ ∧ (𝑧 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)))
22	21	anbi1i 624	. . . . 5 ⊢ ((𝑢 ∈ (𝐴 × 𝐵) ∧ (𝐹‘𝑢)𝐺𝑣) ↔ (∃𝑧∃𝑦(𝑢 = ⟨𝑧, 𝑦⟩ ∧ (𝑧 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) ∧ (𝐹‘𝑢)𝐺𝑣))
23		nfcv 2891	. . . . . . 7 ⊢ Ⅎ𝑧(𝐹‘𝑢)
24		xpcomco.1	. . . . . . . 8 ⊢ 𝐺 = (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐴 ↦ 𝐶)
25		nfmpo2 7434	. . . . . . . 8 ⊢ Ⅎ𝑧(𝑦 ∈ 𝐵, 𝑧 ∈ 𝐴 ↦ 𝐶)
26	24, 25	nfcxfr 2889	. . . . . . 7 ⊢ Ⅎ𝑧𝐺
27		nfcv 2891	. . . . . . 7 ⊢ Ⅎ𝑧𝑣
28	23, 26, 27	nfbr 5142	. . . . . 6 ⊢ Ⅎ𝑧(𝐹‘𝑢)𝐺𝑣
29	28	19.41 2236	. . . . 5 ⊢ (∃𝑧(∃𝑦(𝑢 = ⟨𝑧, 𝑦⟩ ∧ (𝑧 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) ∧ (𝐹‘𝑢)𝐺𝑣) ↔ (∃𝑧∃𝑦(𝑢 = ⟨𝑧, 𝑦⟩ ∧ (𝑧 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) ∧ (𝐹‘𝑢)𝐺𝑣))
30		nfcv 2891	. . . . . . . . 9 ⊢ Ⅎ𝑦(𝐹‘𝑢)
31		nfmpo1 7433	. . . . . . . . . 10 ⊢ Ⅎ𝑦(𝑦 ∈ 𝐵, 𝑧 ∈ 𝐴 ↦ 𝐶)
32	24, 31	nfcxfr 2889	. . . . . . . . 9 ⊢ Ⅎ𝑦𝐺
33		nfcv 2891	. . . . . . . . 9 ⊢ Ⅎ𝑦𝑣
34	30, 32, 33	nfbr 5142	. . . . . . . 8 ⊢ Ⅎ𝑦(𝐹‘𝑢)𝐺𝑣
35	34	19.41 2236	. . . . . . 7 ⊢ (∃𝑦((𝑢 = ⟨𝑧, 𝑦⟩ ∧ (𝑧 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) ∧ (𝐹‘𝑢)𝐺𝑣) ↔ (∃𝑦(𝑢 = ⟨𝑧, 𝑦⟩ ∧ (𝑧 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) ∧ (𝐹‘𝑢)𝐺𝑣))
36		anass 468	. . . . . . . . 9 ⊢ (((𝑢 = ⟨𝑧, 𝑦⟩ ∧ (𝑧 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) ∧ (𝐹‘𝑢)𝐺𝑣) ↔ (𝑢 = ⟨𝑧, 𝑦⟩ ∧ ((𝑧 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝐹‘𝑢)𝐺𝑣)))
37		fveq2 6826	. . . . . . . . . . . . . 14 ⊢ (𝑢 = ⟨𝑧, 𝑦⟩ → (𝐹‘𝑢) = (𝐹‘⟨𝑧, 𝑦⟩))
38		opelxpi 5660	. . . . . . . . . . . . . . 15 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → ⟨𝑧, 𝑦⟩ ∈ (𝐴 × 𝐵))
39		sneq 4589	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = ⟨𝑧, 𝑦⟩ → {𝑥} = {⟨𝑧, 𝑦⟩})
40	39	cnveqd 5822	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = ⟨𝑧, 𝑦⟩ → ◡{𝑥} = ◡{⟨𝑧, 𝑦⟩})
41	40	unieqd 4874	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = ⟨𝑧, 𝑦⟩ → ∪ ◡{𝑥} = ∪ ◡{⟨𝑧, 𝑦⟩})
42		opswap 6182	. . . . . . . . . . . . . . . . 17 ⊢ ∪ ◡{⟨𝑧, 𝑦⟩} = ⟨𝑦, 𝑧⟩
43	41, 42	eqtrdi 2780	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = ⟨𝑧, 𝑦⟩ → ∪ ◡{𝑥} = ⟨𝑦, 𝑧⟩)
44		opex 5411	. . . . . . . . . . . . . . . 16 ⊢ ⟨𝑦, 𝑧⟩ ∈ V
45	43, 1, 44	fvmpt 6934	. . . . . . . . . . . . . . 15 ⊢ (⟨𝑧, 𝑦⟩ ∈ (𝐴 × 𝐵) → (𝐹‘⟨𝑧, 𝑦⟩) = ⟨𝑦, 𝑧⟩)
46	38, 45	syl 17	. . . . . . . . . . . . . 14 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (𝐹‘⟨𝑧, 𝑦⟩) = ⟨𝑦, 𝑧⟩)
47	37, 46	sylan9eq 2784	. . . . . . . . . . . . 13 ⊢ ((𝑢 = ⟨𝑧, 𝑦⟩ ∧ (𝑧 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (𝐹‘𝑢) = ⟨𝑦, 𝑧⟩)
48	47	breq1d 5105	. . . . . . . . . . . 12 ⊢ ((𝑢 = ⟨𝑧, 𝑦⟩ ∧ (𝑧 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → ((𝐹‘𝑢)𝐺𝑣 ↔ ⟨𝑦, 𝑧⟩𝐺𝑣))
49		df-br 5096	. . . . . . . . . . . . . . . 16 ⊢ (⟨𝑦, 𝑧⟩𝐺𝑣 ↔ ⟨⟨𝑦, 𝑧⟩, 𝑣⟩ ∈ 𝐺)
50		df-mpo 7358	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑦 ∈ 𝐵, 𝑧 ∈ 𝐴 ↦ 𝐶) = {⟨⟨𝑦, 𝑧⟩, 𝑣⟩ ∣ ((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐴) ∧ 𝑣 = 𝐶)}
51	24, 50	eqtri 2752	. . . . . . . . . . . . . . . . 17 ⊢ 𝐺 = {⟨⟨𝑦, 𝑧⟩, 𝑣⟩ ∣ ((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐴) ∧ 𝑣 = 𝐶)}
52	51	eleq2i 2820	. . . . . . . . . . . . . . . 16 ⊢ (⟨⟨𝑦, 𝑧⟩, 𝑣⟩ ∈ 𝐺 ↔ ⟨⟨𝑦, 𝑧⟩, 𝑣⟩ ∈ {⟨⟨𝑦, 𝑧⟩, 𝑣⟩ ∣ ((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐴) ∧ 𝑣 = 𝐶)})
53		oprabidw 7384	. . . . . . . . . . . . . . . 16 ⊢ (⟨⟨𝑦, 𝑧⟩, 𝑣⟩ ∈ {⟨⟨𝑦, 𝑧⟩, 𝑣⟩ ∣ ((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐴) ∧ 𝑣 = 𝐶)} ↔ ((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐴) ∧ 𝑣 = 𝐶))
54	49, 52, 53	3bitri 297	. . . . . . . . . . . . . . 15 ⊢ (⟨𝑦, 𝑧⟩𝐺𝑣 ↔ ((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐴) ∧ 𝑣 = 𝐶))
55	54	baib 535	. . . . . . . . . . . . . 14 ⊢ ((𝑦 ∈ 𝐵 ∧ 𝑧 ∈ 𝐴) → (⟨𝑦, 𝑧⟩𝐺𝑣 ↔ 𝑣 = 𝐶))
56	55	ancoms 458	. . . . . . . . . . . . 13 ⊢ ((𝑧 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) → (⟨𝑦, 𝑧⟩𝐺𝑣 ↔ 𝑣 = 𝐶))
57	56	adantl 481	. . . . . . . . . . . 12 ⊢ ((𝑢 = ⟨𝑧, 𝑦⟩ ∧ (𝑧 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → (⟨𝑦, 𝑧⟩𝐺𝑣 ↔ 𝑣 = 𝐶))
58	48, 57	bitrd 279	. . . . . . . . . . 11 ⊢ ((𝑢 = ⟨𝑧, 𝑦⟩ ∧ (𝑧 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) → ((𝐹‘𝑢)𝐺𝑣 ↔ 𝑣 = 𝐶))
59	58	pm5.32da 579	. . . . . . . . . 10 ⊢ (𝑢 = ⟨𝑧, 𝑦⟩ → (((𝑧 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝐹‘𝑢)𝐺𝑣) ↔ ((𝑧 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑣 = 𝐶)))
60	59	pm5.32i 574	. . . . . . . . 9 ⊢ ((𝑢 = ⟨𝑧, 𝑦⟩ ∧ ((𝑧 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ (𝐹‘𝑢)𝐺𝑣)) ↔ (𝑢 = ⟨𝑧, 𝑦⟩ ∧ ((𝑧 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑣 = 𝐶)))
61	36, 60	bitri 275	. . . . . . . 8 ⊢ (((𝑢 = ⟨𝑧, 𝑦⟩ ∧ (𝑧 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) ∧ (𝐹‘𝑢)𝐺𝑣) ↔ (𝑢 = ⟨𝑧, 𝑦⟩ ∧ ((𝑧 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑣 = 𝐶)))
62	61	exbii 1848	. . . . . . 7 ⊢ (∃𝑦((𝑢 = ⟨𝑧, 𝑦⟩ ∧ (𝑧 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) ∧ (𝐹‘𝑢)𝐺𝑣) ↔ ∃𝑦(𝑢 = ⟨𝑧, 𝑦⟩ ∧ ((𝑧 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑣 = 𝐶)))
63	35, 62	bitr3i 277	. . . . . 6 ⊢ ((∃𝑦(𝑢 = ⟨𝑧, 𝑦⟩ ∧ (𝑧 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) ∧ (𝐹‘𝑢)𝐺𝑣) ↔ ∃𝑦(𝑢 = ⟨𝑧, 𝑦⟩ ∧ ((𝑧 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑣 = 𝐶)))
64	63	exbii 1848	. . . . 5 ⊢ (∃𝑧(∃𝑦(𝑢 = ⟨𝑧, 𝑦⟩ ∧ (𝑧 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵)) ∧ (𝐹‘𝑢)𝐺𝑣) ↔ ∃𝑧∃𝑦(𝑢 = ⟨𝑧, 𝑦⟩ ∧ ((𝑧 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑣 = 𝐶)))
65	22, 29, 64	3bitr2i 299	. . . 4 ⊢ ((𝑢 ∈ (𝐴 × 𝐵) ∧ (𝐹‘𝑢)𝐺𝑣) ↔ ∃𝑧∃𝑦(𝑢 = ⟨𝑧, 𝑦⟩ ∧ ((𝑧 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑣 = 𝐶)))
66	16, 20, 65	3bitri 297	. . 3 ⊢ (∃𝑤(𝑢𝐹𝑤 ∧ 𝑤𝐺𝑣) ↔ ∃𝑧∃𝑦(𝑢 = ⟨𝑧, 𝑦⟩ ∧ ((𝑧 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑣 = 𝐶)))
67	66	opabbii 5162	. 2 ⊢ {⟨𝑢, 𝑣⟩ ∣ ∃𝑤(𝑢𝐹𝑤 ∧ 𝑤𝐺𝑣)} = {⟨𝑢, 𝑣⟩ ∣ ∃𝑧∃𝑦(𝑢 = ⟨𝑧, 𝑦⟩ ∧ ((𝑧 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑣 = 𝐶))}
68		df-co 5632	. 2 ⊢ (𝐺 ∘ 𝐹) = {⟨𝑢, 𝑣⟩ ∣ ∃𝑤(𝑢𝐹𝑤 ∧ 𝑤𝐺𝑣)}
69		df-mpo 7358	. . 3 ⊢ (𝑧 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨⟨𝑧, 𝑦⟩, 𝑣⟩ ∣ ((𝑧 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑣 = 𝐶)}
70		dfoprab2 7411	. . 3 ⊢ {⟨⟨𝑧, 𝑦⟩, 𝑣⟩ ∣ ((𝑧 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑣 = 𝐶)} = {⟨𝑢, 𝑣⟩ ∣ ∃𝑧∃𝑦(𝑢 = ⟨𝑧, 𝑦⟩ ∧ ((𝑧 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑣 = 𝐶))}
71	69, 70	eqtri 2752	. 2 ⊢ (𝑧 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶) = {⟨𝑢, 𝑣⟩ ∣ ∃𝑧∃𝑦(𝑢 = ⟨𝑧, 𝑦⟩ ∧ ((𝑧 ∈ 𝐴 ∧ 𝑦 ∈ 𝐵) ∧ 𝑣 = 𝐶))}
72	67, 68, 71	3eqtr4i 2762	1 ⊢ (𝐺 ∘ 𝐹) = (𝑧 ∈ 𝐴, 𝑦 ∈ 𝐵 ↦ 𝐶)

Syntax hints:   ↔ wb 206   ∧ wa 395   = wceq 1540  ∃wex 1779   ∈ wcel 2109  {csn 4579  ⟨cop 4585  ∪ cuni 4861   class class class wbr 5095  {copab 5157   ↦ cmpt 5176   × cxp 5621  ^◡ccnv 5622  dom cdm 5623   ∘ ccom 5627  Fun wfun 6480  –1-1-onto→wf1o 6485  ‘cfv 6486  {coprab 7354   ∈ cmpo 7355

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7675

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4862  df-br 5096  df-opab 5158  df-mpt 5177  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-oprab 7357  df-mpo 7358  df-1st 7931  df-2nd 7932

This theorem is referenced by:  omf1o  9004