Theorem f1ofvswap 7284

Description: Swapping two values in a bijection between two classes yields another bijection between those classes. (Contributed by BTernaryTau, 31-Aug-2024.)

Assertion

Ref	Expression
f1ofvswap	⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → ((𝐹 ↾ (𝐴 ∖ {𝑋, 𝑌})) ∪ {⟨𝑋, (𝐹‘𝑌)⟩, ⟨𝑌, (𝐹‘𝑋)⟩}):𝐴–1-1-onto→𝐵)

Proof of Theorem f1ofvswap

Step	Hyp	Ref	Expression
1		f1oi 6841	. . . . . 6 ⊢ ( I ↾ (𝐴 ∖ {𝑋, 𝑌})):(𝐴 ∖ {𝑋, 𝑌})–1-1-onto→(𝐴 ∖ {𝑋, 𝑌})
2		f1oprswap 6847	. . . . . 6 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → {⟨𝑋, 𝑌⟩, ⟨𝑌, 𝑋⟩}:{𝑋, 𝑌}–1-1-onto→{𝑋, 𝑌})
3		disjdifr 4439	. . . . . . 7 ⊢ ((𝐴 ∖ {𝑋, 𝑌}) ∩ {𝑋, 𝑌}) = ∅
4		f1oun 6822	. . . . . . 7 ⊢ (((( I ↾ (𝐴 ∖ {𝑋, 𝑌})):(𝐴 ∖ {𝑋, 𝑌})–1-1-onto→(𝐴 ∖ {𝑋, 𝑌}) ∧ {⟨𝑋, 𝑌⟩, ⟨𝑌, 𝑋⟩}:{𝑋, 𝑌}–1-1-onto→{𝑋, 𝑌}) ∧ (((𝐴 ∖ {𝑋, 𝑌}) ∩ {𝑋, 𝑌}) = ∅ ∧ ((𝐴 ∖ {𝑋, 𝑌}) ∩ {𝑋, 𝑌}) = ∅)) → (( I ↾ (𝐴 ∖ {𝑋, 𝑌})) ∪ {⟨𝑋, 𝑌⟩, ⟨𝑌, 𝑋⟩}):((𝐴 ∖ {𝑋, 𝑌}) ∪ {𝑋, 𝑌})–1-1-onto→((𝐴 ∖ {𝑋, 𝑌}) ∪ {𝑋, 𝑌}))
5	3, 3, 4	mpanr12 705	. . . . . 6 ⊢ ((( I ↾ (𝐴 ∖ {𝑋, 𝑌})):(𝐴 ∖ {𝑋, 𝑌})–1-1-onto→(𝐴 ∖ {𝑋, 𝑌}) ∧ {⟨𝑋, 𝑌⟩, ⟨𝑌, 𝑋⟩}:{𝑋, 𝑌}–1-1-onto→{𝑋, 𝑌}) → (( I ↾ (𝐴 ∖ {𝑋, 𝑌})) ∪ {⟨𝑋, 𝑌⟩, ⟨𝑌, 𝑋⟩}):((𝐴 ∖ {𝑋, 𝑌}) ∪ {𝑋, 𝑌})–1-1-onto→((𝐴 ∖ {𝑋, 𝑌}) ∪ {𝑋, 𝑌}))
6	1, 2, 5	sylancr 587	. . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (( I ↾ (𝐴 ∖ {𝑋, 𝑌})) ∪ {⟨𝑋, 𝑌⟩, ⟨𝑌, 𝑋⟩}):((𝐴 ∖ {𝑋, 𝑌}) ∪ {𝑋, 𝑌})–1-1-onto→((𝐴 ∖ {𝑋, 𝑌}) ∪ {𝑋, 𝑌}))
7		prssi 4788	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → {𝑋, 𝑌} ⊆ 𝐴)
8		undifr 4449	. . . . . . 7 ⊢ ({𝑋, 𝑌} ⊆ 𝐴 ↔ ((𝐴 ∖ {𝑋, 𝑌}) ∪ {𝑋, 𝑌}) = 𝐴)
9	7, 8	sylib 218	. . . . . 6 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → ((𝐴 ∖ {𝑋, 𝑌}) ∪ {𝑋, 𝑌}) = 𝐴)
10		f1oeq23 6794	. . . . . 6 ⊢ ((((𝐴 ∖ {𝑋, 𝑌}) ∪ {𝑋, 𝑌}) = 𝐴 ∧ ((𝐴 ∖ {𝑋, 𝑌}) ∪ {𝑋, 𝑌}) = 𝐴) → ((( I ↾ (𝐴 ∖ {𝑋, 𝑌})) ∪ {⟨𝑋, 𝑌⟩, ⟨𝑌, 𝑋⟩}):((𝐴 ∖ {𝑋, 𝑌}) ∪ {𝑋, 𝑌})–1-1-onto→((𝐴 ∖ {𝑋, 𝑌}) ∪ {𝑋, 𝑌}) ↔ (( I ↾ (𝐴 ∖ {𝑋, 𝑌})) ∪ {⟨𝑋, 𝑌⟩, ⟨𝑌, 𝑋⟩}):𝐴–1-1-onto→𝐴))
11	9, 9, 10	syl2anc 584	. . . . 5 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → ((( I ↾ (𝐴 ∖ {𝑋, 𝑌})) ∪ {⟨𝑋, 𝑌⟩, ⟨𝑌, 𝑋⟩}):((𝐴 ∖ {𝑋, 𝑌}) ∪ {𝑋, 𝑌})–1-1-onto→((𝐴 ∖ {𝑋, 𝑌}) ∪ {𝑋, 𝑌}) ↔ (( I ↾ (𝐴 ∖ {𝑋, 𝑌})) ∪ {⟨𝑋, 𝑌⟩, ⟨𝑌, 𝑋⟩}):𝐴–1-1-onto→𝐴))
12	6, 11	mpbid 232	. . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (( I ↾ (𝐴 ∖ {𝑋, 𝑌})) ∪ {⟨𝑋, 𝑌⟩, ⟨𝑌, 𝑋⟩}):𝐴–1-1-onto→𝐴)
13		f1oco 6826	. . . 4 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ (( I ↾ (𝐴 ∖ {𝑋, 𝑌})) ∪ {⟨𝑋, 𝑌⟩, ⟨𝑌, 𝑋⟩}):𝐴–1-1-onto→𝐴) → (𝐹 ∘ (( I ↾ (𝐴 ∖ {𝑋, 𝑌})) ∪ {⟨𝑋, 𝑌⟩, ⟨𝑌, 𝑋⟩})):𝐴–1-1-onto→𝐵)
14	12, 13	sylan2 593	. . 3 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴)) → (𝐹 ∘ (( I ↾ (𝐴 ∖ {𝑋, 𝑌})) ∪ {⟨𝑋, 𝑌⟩, ⟨𝑌, 𝑋⟩})):𝐴–1-1-onto→𝐵)
15	14	3impb 1114	. 2 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝐹 ∘ (( I ↾ (𝐴 ∖ {𝑋, 𝑌})) ∪ {⟨𝑋, 𝑌⟩, ⟨𝑌, 𝑋⟩})):𝐴–1-1-onto→𝐵)
16		f1ofn 6804	. . . 4 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹 Fn 𝐴)
17		coundi 6223	. . . . . 6 ⊢ (𝐹 ∘ (( I ↾ (𝐴 ∖ {𝑋, 𝑌})) ∪ {⟨𝑋, 𝑌⟩, ⟨𝑌, 𝑋⟩})) = ((𝐹 ∘ ( I ↾ (𝐴 ∖ {𝑋, 𝑌}))) ∪ (𝐹 ∘ {⟨𝑋, 𝑌⟩, ⟨𝑌, 𝑋⟩}))
18		fcoconst 7109	. . . . . . . . . . 11 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝐹 ∘ ({𝑋} × {𝑌})) = ({𝑋} × {(𝐹‘𝑌)}))
19	18	3adant2 1131	. . . . . . . . . 10 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝐹 ∘ ({𝑋} × {𝑌})) = ({𝑋} × {(𝐹‘𝑌)}))
20		xpsng 7114	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → ({𝑋} × {𝑌}) = {⟨𝑋, 𝑌⟩})
21	20	coeq2d 5829	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝐹 ∘ ({𝑋} × {𝑌})) = (𝐹 ∘ {⟨𝑋, 𝑌⟩}))
22	21	3adant1 1130	. . . . . . . . . 10 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝐹 ∘ ({𝑋} × {𝑌})) = (𝐹 ∘ {⟨𝑋, 𝑌⟩}))
23		fvex 6874	. . . . . . . . . . . 12 ⊢ (𝐹‘𝑌) ∈ V
24		xpsng 7114	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ 𝐴 ∧ (𝐹‘𝑌) ∈ V) → ({𝑋} × {(𝐹‘𝑌)}) = {⟨𝑋, (𝐹‘𝑌)⟩})
25	23, 24	mpan2 691	. . . . . . . . . . 11 ⊢ (𝑋 ∈ 𝐴 → ({𝑋} × {(𝐹‘𝑌)}) = {⟨𝑋, (𝐹‘𝑌)⟩})
26	25	3ad2ant2 1134	. . . . . . . . . 10 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → ({𝑋} × {(𝐹‘𝑌)}) = {⟨𝑋, (𝐹‘𝑌)⟩})
27	19, 22, 26	3eqtr3d 2773	. . . . . . . . 9 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝐹 ∘ {⟨𝑋, 𝑌⟩}) = {⟨𝑋, (𝐹‘𝑌)⟩})
28		fcoconst 7109	. . . . . . . . . . 11 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝐹 ∘ ({𝑌} × {𝑋})) = ({𝑌} × {(𝐹‘𝑋)}))
29	28	3adant3 1132	. . . . . . . . . 10 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝐹 ∘ ({𝑌} × {𝑋})) = ({𝑌} × {(𝐹‘𝑋)}))
30		xpsng 7114	. . . . . . . . . . . . 13 ⊢ ((𝑌 ∈ 𝐴 ∧ 𝑋 ∈ 𝐴) → ({𝑌} × {𝑋}) = {⟨𝑌, 𝑋⟩})
31	30	coeq2d 5829	. . . . . . . . . . . 12 ⊢ ((𝑌 ∈ 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝐹 ∘ ({𝑌} × {𝑋})) = (𝐹 ∘ {⟨𝑌, 𝑋⟩}))
32	31	ancoms 458	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝐹 ∘ ({𝑌} × {𝑋})) = (𝐹 ∘ {⟨𝑌, 𝑋⟩}))
33	32	3adant1 1130	. . . . . . . . . 10 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝐹 ∘ ({𝑌} × {𝑋})) = (𝐹 ∘ {⟨𝑌, 𝑋⟩}))
34		fvex 6874	. . . . . . . . . . . 12 ⊢ (𝐹‘𝑋) ∈ V
35		xpsng 7114	. . . . . . . . . . . 12 ⊢ ((𝑌 ∈ 𝐴 ∧ (𝐹‘𝑋) ∈ V) → ({𝑌} × {(𝐹‘𝑋)}) = {⟨𝑌, (𝐹‘𝑋)⟩})
36	34, 35	mpan2 691	. . . . . . . . . . 11 ⊢ (𝑌 ∈ 𝐴 → ({𝑌} × {(𝐹‘𝑋)}) = {⟨𝑌, (𝐹‘𝑋)⟩})
37	36	3ad2ant3 1135	. . . . . . . . . 10 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → ({𝑌} × {(𝐹‘𝑋)}) = {⟨𝑌, (𝐹‘𝑋)⟩})
38	29, 33, 37	3eqtr3d 2773	. . . . . . . . 9 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝐹 ∘ {⟨𝑌, 𝑋⟩}) = {⟨𝑌, (𝐹‘𝑋)⟩})
39	27, 38	uneq12d 4135	. . . . . . . 8 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → ((𝐹 ∘ {⟨𝑋, 𝑌⟩}) ∪ (𝐹 ∘ {⟨𝑌, 𝑋⟩})) = ({⟨𝑋, (𝐹‘𝑌)⟩} ∪ {⟨𝑌, (𝐹‘𝑋)⟩}))
40		df-pr 4595	. . . . . . . . . 10 ⊢ {⟨𝑋, 𝑌⟩, ⟨𝑌, 𝑋⟩} = ({⟨𝑋, 𝑌⟩} ∪ {⟨𝑌, 𝑋⟩})
41	40	coeq2i 5827	. . . . . . . . 9 ⊢ (𝐹 ∘ {⟨𝑋, 𝑌⟩, ⟨𝑌, 𝑋⟩}) = (𝐹 ∘ ({⟨𝑋, 𝑌⟩} ∪ {⟨𝑌, 𝑋⟩}))
42		coundi 6223	. . . . . . . . 9 ⊢ (𝐹 ∘ ({⟨𝑋, 𝑌⟩} ∪ {⟨𝑌, 𝑋⟩})) = ((𝐹 ∘ {⟨𝑋, 𝑌⟩}) ∪ (𝐹 ∘ {⟨𝑌, 𝑋⟩}))
43	41, 42	eqtri 2753	. . . . . . . 8 ⊢ (𝐹 ∘ {⟨𝑋, 𝑌⟩, ⟨𝑌, 𝑋⟩}) = ((𝐹 ∘ {⟨𝑋, 𝑌⟩}) ∪ (𝐹 ∘ {⟨𝑌, 𝑋⟩}))
44		df-pr 4595	. . . . . . . 8 ⊢ {⟨𝑋, (𝐹‘𝑌)⟩, ⟨𝑌, (𝐹‘𝑋)⟩} = ({⟨𝑋, (𝐹‘𝑌)⟩} ∪ {⟨𝑌, (𝐹‘𝑋)⟩})
45	39, 43, 44	3eqtr4g 2790	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝐹 ∘ {⟨𝑋, 𝑌⟩, ⟨𝑌, 𝑋⟩}) = {⟨𝑋, (𝐹‘𝑌)⟩, ⟨𝑌, (𝐹‘𝑋)⟩})
46	45	uneq2d 4134	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → ((𝐹 ∘ ( I ↾ (𝐴 ∖ {𝑋, 𝑌}))) ∪ (𝐹 ∘ {⟨𝑋, 𝑌⟩, ⟨𝑌, 𝑋⟩})) = ((𝐹 ∘ ( I ↾ (𝐴 ∖ {𝑋, 𝑌}))) ∪ {⟨𝑋, (𝐹‘𝑌)⟩, ⟨𝑌, (𝐹‘𝑋)⟩}))
47	17, 46	eqtrid 2777	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝐹 ∘ (( I ↾ (𝐴 ∖ {𝑋, 𝑌})) ∪ {⟨𝑋, 𝑌⟩, ⟨𝑌, 𝑋⟩})) = ((𝐹 ∘ ( I ↾ (𝐴 ∖ {𝑋, 𝑌}))) ∪ {⟨𝑋, (𝐹‘𝑌)⟩, ⟨𝑌, (𝐹‘𝑋)⟩}))
48		coires1 6240	. . . . . 6 ⊢ (𝐹 ∘ ( I ↾ (𝐴 ∖ {𝑋, 𝑌}))) = (𝐹 ↾ (𝐴 ∖ {𝑋, 𝑌}))
49	48	uneq1i 4130	. . . . 5 ⊢ ((𝐹 ∘ ( I ↾ (𝐴 ∖ {𝑋, 𝑌}))) ∪ {⟨𝑋, (𝐹‘𝑌)⟩, ⟨𝑌, (𝐹‘𝑋)⟩}) = ((𝐹 ↾ (𝐴 ∖ {𝑋, 𝑌})) ∪ {⟨𝑋, (𝐹‘𝑌)⟩, ⟨𝑌, (𝐹‘𝑋)⟩})
50	47, 49	eqtrdi 2781	. . . 4 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝐹 ∘ (( I ↾ (𝐴 ∖ {𝑋, 𝑌})) ∪ {⟨𝑋, 𝑌⟩, ⟨𝑌, 𝑋⟩})) = ((𝐹 ↾ (𝐴 ∖ {𝑋, 𝑌})) ∪ {⟨𝑋, (𝐹‘𝑌)⟩, ⟨𝑌, (𝐹‘𝑋)⟩}))
51	16, 50	syl3an1 1163	. . 3 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝐹 ∘ (( I ↾ (𝐴 ∖ {𝑋, 𝑌})) ∪ {⟨𝑋, 𝑌⟩, ⟨𝑌, 𝑋⟩})) = ((𝐹 ↾ (𝐴 ∖ {𝑋, 𝑌})) ∪ {⟨𝑋, (𝐹‘𝑌)⟩, ⟨𝑌, (𝐹‘𝑋)⟩}))
52	51	f1oeq1d 6798	. 2 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → ((𝐹 ∘ (( I ↾ (𝐴 ∖ {𝑋, 𝑌})) ∪ {⟨𝑋, 𝑌⟩, ⟨𝑌, 𝑋⟩})):𝐴–1-1-onto→𝐵 ↔ ((𝐹 ↾ (𝐴 ∖ {𝑋, 𝑌})) ∪ {⟨𝑋, (𝐹‘𝑌)⟩, ⟨𝑌, (𝐹‘𝑋)⟩}):𝐴–1-1-onto→𝐵))
53	15, 52	mpbid 232	1 ⊢ ((𝐹:𝐴–1-1-onto→𝐵 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → ((𝐹 ↾ (𝐴 ∖ {𝑋, 𝑌})) ∪ {⟨𝑋, (𝐹‘𝑌)⟩, ⟨𝑌, (𝐹‘𝑋)⟩}):𝐴–1-1-onto→𝐵)

Syntax hints:   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109  Vcvv 3450   ∖ cdif 3914   ∪ cun 3915   ∩ cin 3916   ⊆ wss 3917  ∅c0 4299  {csn 4592  {cpr 4594  ⟨cop 4598   I cid 5535   × cxp 5639   ↾ cres 5643   ∘ ccom 5645   Fn wfn 6509  –1-1-onto→wf1o 6513  ‘cfv 6514

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 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

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 2534  df-eu 2563  df-clab 2709  df-cleq 2722  df-clel 2804  df-nfc 2879  df-ne 2927  df-ral 3046  df-rex 3055  df-reu 3357  df-rab 3409  df-v 3452  df-sbc 3757  df-csb 3866  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-uni 4875  df-br 5111  df-opab 5173  df-mpt 5192  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-res 5653  df-ima 5654  df-iota 6467  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-fv 6522

This theorem is referenced by:  dif1en  9130  dif1enOLD  9132  nregmodelf1o  45012