Theorem f1ofvswap 7240

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 6801	. . . . . 6 ⊢ ( I ↾ (𝐴 ∖ {𝑋, 𝑌})):(𝐴 ∖ {𝑋, 𝑌})–1-1-onto→(𝐴 ∖ {𝑋, 𝑌})
2		f1oprswap 6807	. . . . . 6 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → {⟨𝑋, 𝑌⟩, ⟨𝑌, 𝑋⟩}:{𝑋, 𝑌}–1-1-onto→{𝑋, 𝑌})
3		disjdifr 4420	. . . . . . 7 ⊢ ((𝐴 ∖ {𝑋, 𝑌}) ∩ {𝑋, 𝑌}) = ∅
4		f1oun 6782	. . . . . . 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 4770	. . . . . . 7 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → {𝑋, 𝑌} ⊆ 𝐴)
8		undifr 4430	. . . . . . 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 6754	. . . . . 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 6786	. . . 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 6764	. . . 4 ⊢ (𝐹:𝐴–1-1-onto→𝐵 → 𝐹 Fn 𝐴)
17		coundi 6194	. . . . . 6 ⊢ (𝐹 ∘ (( I ↾ (𝐴 ∖ {𝑋, 𝑌})) ∪ {⟨𝑋, 𝑌⟩, ⟨𝑌, 𝑋⟩})) = ((𝐹 ∘ ( I ↾ (𝐴 ∖ {𝑋, 𝑌}))) ∪ (𝐹 ∘ {⟨𝑋, 𝑌⟩, ⟨𝑌, 𝑋⟩}))
18		fcoconst 7067	. . . . . . . . . . 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 7072	. . . . . . . . . . . 12 ⊢ ((𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → ({𝑋} × {𝑌}) = {⟨𝑋, 𝑌⟩})
21	20	coeq2d 5801	. . . . . . . . . . 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 6835	. . . . . . . . . . . 12 ⊢ (𝐹‘𝑌) ∈ V
24		xpsng 7072	. . . . . . . . . . . 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 2774	. . . . . . . . 9 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝐹 ∘ {⟨𝑋, 𝑌⟩}) = {⟨𝑋, (𝐹‘𝑌)⟩})
28		fcoconst 7067	. . . . . . . . . . 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 7072	. . . . . . . . . . . . 13 ⊢ ((𝑌 ∈ 𝐴 ∧ 𝑋 ∈ 𝐴) → ({𝑌} × {𝑋}) = {⟨𝑌, 𝑋⟩})
31	30	coeq2d 5801	. . . . . . . . . . . 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 6835	. . . . . . . . . . . 12 ⊢ (𝐹‘𝑋) ∈ V
35		xpsng 7072	. . . . . . . . . . . 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 2774	. . . . . . . . 9 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → (𝐹 ∘ {⟨𝑌, 𝑋⟩}) = {⟨𝑌, (𝐹‘𝑋)⟩})
39	27, 38	uneq12d 4116	. . . . . . . 8 ⊢ ((𝐹 Fn 𝐴 ∧ 𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴) → ((𝐹 ∘ {⟨𝑋, 𝑌⟩}) ∪ (𝐹 ∘ {⟨𝑌, 𝑋⟩})) = ({⟨𝑋, (𝐹‘𝑌)⟩} ∪ {⟨𝑌, (𝐹‘𝑋)⟩}))
40		df-pr 4576	. . . . . . . . . 10 ⊢ {⟨𝑋, 𝑌⟩, ⟨𝑌, 𝑋⟩} = ({⟨𝑋, 𝑌⟩} ∪ {⟨𝑌, 𝑋⟩})
41	40	coeq2i 5799	. . . . . . . . 9 ⊢ (𝐹 ∘ {⟨𝑋, 𝑌⟩, ⟨𝑌, 𝑋⟩}) = (𝐹 ∘ ({⟨𝑋, 𝑌⟩} ∪ {⟨𝑌, 𝑋⟩}))
42		coundi 6194	. . . . . . . . 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 4576	. . . . . . . 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 4115	. . . . . 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 6212	. . . . . 6 ⊢ (𝐹 ∘ ( I ↾ (𝐴 ∖ {𝑋, 𝑌}))) = (𝐹 ↾ (𝐴 ∖ {𝑋, 𝑌}))
49	48	uneq1i 4111	. . . . 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 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 6758	. 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 1541   ∈ wcel 2111  Vcvv 3436   ∖ cdif 3894   ∪ cun 3895   ∩ cin 3896   ⊆ wss 3897  ∅c0 4280  {csn 4573  {cpr 4575  ⟨cop 4579   I cid 5508   × cxp 5612   ↾ cres 5616   ∘ ccom 5618   Fn wfn 6476  –1-1-onto→wf1o 6480  ‘cfv 6481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489

This theorem is referenced by:  dif1en  9071  nregmodelf1o  45118