Theorem f1ofvswap 7300

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 6865	. . . . . 6 ⊢ ( I ↾ (𝐴 ∖ {𝑋, 𝑌})):(𝐴 ∖ {𝑋, 𝑌})–1-1-onto→(𝐴 ∖ {𝑋, 𝑌})
2		f1oprswap 6871	. . . . . 6 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → {⟨𝑋, 𝑌⟩, ⟨𝑌, 𝑋⟩}:{𝑋, 𝑌}–1-1-onto→{𝑋, 𝑌})
3		disjdifr 4467	. . . . . . 7 ⊢ ((𝐴 ∖ {𝑋, 𝑌}) ∩ {𝑋, 𝑌}) = ∅
4		f1oun 6846	. . . . . . 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 702	. . . . . 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 586	. . . . 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 4819	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → {𝑋, 𝑌} ⊆ 𝐴)
8		undifr 4477	. . . . . . 7 ⊢ ({𝑋, 𝑌} ⊆ 𝐴 ↔ ((𝐴 ∖ {𝑋, 𝑌}) ∪ {𝑋, 𝑌}) = 𝐴)
9	7, 8	sylib 217	. . . . . 6 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → ((𝐴 ∖ {𝑋, 𝑌}) ∪ {𝑋, 𝑌}) = 𝐴)
10		f1oeq23 6818	. . . . . 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 583	. . . . 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 231	. . . 4 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (( I ↾ (𝐴 ∖ {𝑋, 𝑌})) ∪ {⟨𝑋, 𝑌⟩, ⟨𝑌, 𝑋⟩}):𝐴–1-1-onto→𝐴)
13		f1oco 6850	. . . 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 592	. . 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 1112	. 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 6828	. . . 4 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹 Fn 𝐴)
17		coundi 6240	. . . . . 6 ⊢ (𝐹 ∘ (( I ↾ (𝐴 ∖ {𝑋, 𝑌})) ∪ {⟨𝑋, 𝑌⟩, ⟨𝑌, 𝑋⟩})) = ((𝐹 ∘ ( I ↾ (𝐴 ∖ {𝑋, 𝑌}))) ∪ (𝐹 ∘ {⟨𝑋, 𝑌⟩, ⟨𝑌, 𝑋⟩}))
18		fcoconst 7128	. . . . . . . . . . 11 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝐹 ∘ ({𝑋} × {𝑌})) = ({𝑋} × {(𝐹‘𝑌)}))
19	18	3adant2 1128	. . . . . . . . . 10 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝐹 ∘ ({𝑋} × {𝑌})) = ({𝑋} × {(𝐹‘𝑌)}))
20		xpsng 7133	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → ({𝑋} × {𝑌}) = {⟨𝑋, 𝑌⟩})
21	20	coeq2d 5856	. . . . . . . . . . 11 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝐹 ∘ ({𝑋} × {𝑌})) = (𝐹 ∘ {⟨𝑋, 𝑌⟩}))
22	21	3adant1 1127	. . . . . . . . . 10 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝐹 ∘ ({𝑋} × {𝑌})) = (𝐹 ∘ {⟨𝑋, 𝑌⟩}))
23		fvex 6898	. . . . . . . . . . . 12 ⊢ (𝐹‘𝑌) ∈ V
24		xpsng 7133	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ 𝐴 ∧ (𝐹‘𝑌) ∈ V) → ({𝑋} × {(𝐹‘𝑌)}) = {⟨𝑋, (𝐹‘𝑌)⟩})
25	23, 24	mpan2 688	. . . . . . . . . . 11 ⊢ (𝑋 ∈ 𝐴 → ({𝑋} × {(𝐹‘𝑌)}) = {⟨𝑋, (𝐹‘𝑌)⟩})
26	25	3ad2ant2 1131	. . . . . . . . . 10 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → ({𝑋} × {(𝐹‘𝑌)}) = {⟨𝑋, (𝐹‘𝑌)⟩})
27	19, 22, 26	3eqtr3d 2774	. . . . . . . . 9 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝐹 ∘ {⟨𝑋, 𝑌⟩}) = {⟨𝑋, (𝐹‘𝑌)⟩})
28		fcoconst 7128	. . . . . . . . . . 11 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴) → (𝐹 ∘ ({𝑌} × {𝑋})) = ({𝑌} × {(𝐹‘𝑋)}))
29	28	3adant3 1129	. . . . . . . . . 10 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝐹 ∘ ({𝑌} × {𝑋})) = ({𝑌} × {(𝐹‘𝑋)}))
30		xpsng 7133	. . . . . . . . . . . . 13 ⊢ ((𝑌 ∈ 𝐴 ∧ 𝑋 ∈ 𝐴) → ({𝑌} × {𝑋}) = {⟨𝑌, 𝑋⟩})
31	30	coeq2d 5856	. . . . . . . . . . . 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 1127	. . . . . . . . . 10 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝐹 ∘ ({𝑌} × {𝑋})) = (𝐹 ∘ {⟨𝑌, 𝑋⟩}))
34		fvex 6898	. . . . . . . . . . . 12 ⊢ (𝐹‘𝑋) ∈ V
35		xpsng 7133	. . . . . . . . . . . 12 ⊢ ((𝑌 ∈ 𝐴 ∧ (𝐹‘𝑋) ∈ V) → ({𝑌} × {(𝐹‘𝑋)}) = {⟨𝑌, (𝐹‘𝑋)⟩})
36	34, 35	mpan2 688	. . . . . . . . . . 11 ⊢ (𝑌 ∈ 𝐴 → ({𝑌} × {(𝐹‘𝑋)}) = {⟨𝑌, (𝐹‘𝑋)⟩})
37	36	3ad2ant3 1132	. . . . . . . . . 10 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → ({𝑌} × {(𝐹‘𝑋)}) = {⟨𝑌, (𝐹‘𝑋)⟩})
38	29, 33, 37	3eqtr3d 2774	. . . . . . . . 9 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝐹 ∘ {⟨𝑌, 𝑋⟩}) = {⟨𝑌, (𝐹‘𝑋)⟩})
39	27, 38	uneq12d 4159	. . . . . . . 8 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → ((𝐹 ∘ {⟨𝑋, 𝑌⟩}) ∪ (𝐹 ∘ {⟨𝑌, 𝑋⟩})) = ({⟨𝑋, (𝐹‘𝑌)⟩} ∪ {⟨𝑌, (𝐹‘𝑋)⟩}))
40		df-pr 4626	. . . . . . . . . 10 ⊢ {⟨𝑋, 𝑌⟩, ⟨𝑌, 𝑋⟩} = ({⟨𝑋, 𝑌⟩} ∪ {⟨𝑌, 𝑋⟩})
41	40	coeq2i 5854	. . . . . . . . 9 ⊢ (𝐹 ∘ {⟨𝑋, 𝑌⟩, ⟨𝑌, 𝑋⟩}) = (𝐹 ∘ ({⟨𝑋, 𝑌⟩} ∪ {⟨𝑌, 𝑋⟩}))
42		coundi 6240	. . . . . . . . 9 ⊢ (𝐹 ∘ ({⟨𝑋, 𝑌⟩} ∪ {⟨𝑌, 𝑋⟩})) = ((𝐹 ∘ {⟨𝑋, 𝑌⟩}) ∪ (𝐹 ∘ {⟨𝑌, 𝑋⟩}))
43	41, 42	eqtri 2754	. . . . . . . 8 ⊢ (𝐹 ∘ {⟨𝑋, 𝑌⟩, ⟨𝑌, 𝑋⟩}) = ((𝐹 ∘ {⟨𝑋, 𝑌⟩}) ∪ (𝐹 ∘ {⟨𝑌, 𝑋⟩}))
44		df-pr 4626	. . . . . . . 8 ⊢ {⟨𝑋, (𝐹‘𝑌)⟩, ⟨𝑌, (𝐹‘𝑋)⟩} = ({⟨𝑋, (𝐹‘𝑌)⟩} ∪ {⟨𝑌, (𝐹‘𝑋)⟩})
45	39, 43, 44	3eqtr4g 2791	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝐹 ∘ {⟨𝑋, 𝑌⟩, ⟨𝑌, 𝑋⟩}) = {⟨𝑋, (𝐹‘𝑌)⟩, ⟨𝑌, (𝐹‘𝑋)⟩})
46	45	uneq2d 4158	. . . . . 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 2778	. . . . 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 6257	. . . . . 6 ⊢ (𝐹 ∘ ( I ↾ (𝐴 ∖ {𝑋, 𝑌}))) = (𝐹 ↾ (𝐴 ∖ {𝑋, 𝑌}))
49	48	uneq1i 4154	. . . . 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 2782	. . . 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 1160	. . 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 6822	. 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 231	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 205   ∧ wa 395   ∧ w3a 1084   = wceq 1533   ∈ wcel 2098  Vcvv 3468   ∖ cdif 3940   ∪ cun 3941   ∩ cin 3942   ⊆ wss 3943  ∅c0 4317  {csn 4623  {cpr 4625  ⟨cop 4629   I cid 5566   × cxp 5667   ↾ cres 5671   ∘ ccom 5673   Fn wfn 6532  –1-1-onto→wf1o 6536  ‘cfv 6537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-mpt 5225  df-id 5567  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545

This theorem is referenced by:  dif1en  9162  dif1enOLD  9164