1arymaptf1 - Mathbox for Alexander van der Vekens

	Mathbox for Alexander van der Vekens		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > Mathboxes > 1arymaptf1		Structured version Visualization version GIF version

Theorem 1arymaptf1 45988

Description: The mapping of unary (endo)functions is a one-to-one function into the set of endofunctions. (Contributed by AV, 19-May-2024.)

**Hypothesis**
Ref	Expression
1arymaptfv.h	⊢ 𝐻 = (ℎ ∈ (1-aryF 𝑋) ↦ (𝑥 ∈ 𝑋 ↦ (ℎ‘{⟨0, 𝑥⟩})))

**Assertion**
Ref	Expression
1arymaptf1	⊢ (𝑋 ∈ 𝑉 → 𝐻:(1-aryF 𝑋)–1-1→(𝑋 ↑_m 𝑋))

Distinct variable groups: 𝑥,ℎ,𝑋 ℎ,𝑉,𝑥 Allowed substitution hints: 𝐻(𝑥,ℎ)

Proof of Theorem 1arymaptf1 Dummy variables 𝑓 𝑔 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		1arymaptfv.h	. . 3 ⊢ 𝐻 = (ℎ ∈ (1-aryF 𝑋) ↦ (𝑥 ∈ 𝑋 ↦ (ℎ‘{⟨0, 𝑥⟩})))
2	1	1arymaptf 45987	. 2 ⊢ (𝑋 ∈ 𝑉 → 𝐻:(1-aryF 𝑋)⟶(𝑋 ↑_m 𝑋))
3	1	1arymaptfv 45986	. . . . . 6 ⊢ (𝑓 ∈ (1-aryF 𝑋) → (𝐻‘𝑓) = (𝑥 ∈ 𝑋 ↦ (𝑓‘{⟨0, 𝑥⟩})))
4	3	ad2antrl 725	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝑓 ∈ (1-aryF 𝑋) ∧ 𝑔 ∈ (1-aryF 𝑋))) → (𝐻‘𝑓) = (𝑥 ∈ 𝑋 ↦ (𝑓‘{⟨0, 𝑥⟩})))
5	1	1arymaptfv 45986	. . . . . 6 ⊢ (𝑔 ∈ (1-aryF 𝑋) → (𝐻‘𝑔) = (𝑥 ∈ 𝑋 ↦ (𝑔‘{⟨0, 𝑥⟩})))
6	5	ad2antll 726	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝑓 ∈ (1-aryF 𝑋) ∧ 𝑔 ∈ (1-aryF 𝑋))) → (𝐻‘𝑔) = (𝑥 ∈ 𝑋 ↦ (𝑔‘{⟨0, 𝑥⟩})))
7	4, 6	eqeq12d 2754	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝑓 ∈ (1-aryF 𝑋) ∧ 𝑔 ∈ (1-aryF 𝑋))) → ((𝐻‘𝑓) = (𝐻‘𝑔) ↔ (𝑥 ∈ 𝑋 ↦ (𝑓‘{⟨0, 𝑥⟩})) = (𝑥 ∈ 𝑋 ↦ (𝑔‘{⟨0, 𝑥⟩}))))
8		fvex 6787	. . . . . . 7 ⊢ (𝑓‘{⟨0, 𝑥⟩}) ∈ V
9	8	rgenw 3076	. . . . . 6 ⊢ ∀𝑥 ∈ 𝑋 (𝑓‘{⟨0, 𝑥⟩}) ∈ V
10		mpteqb 6894	. . . . . 6 ⊢ (∀𝑥 ∈ 𝑋 (𝑓‘{⟨0, 𝑥⟩}) ∈ V → ((𝑥 ∈ 𝑋 ↦ (𝑓‘{⟨0, 𝑥⟩})) = (𝑥 ∈ 𝑋 ↦ (𝑔‘{⟨0, 𝑥⟩})) ↔ ∀𝑥 ∈ 𝑋 (𝑓‘{⟨0, 𝑥⟩}) = (𝑔‘{⟨0, 𝑥⟩})))
11	9, 10	mp1i 13	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝑓 ∈ (1-aryF 𝑋) ∧ 𝑔 ∈ (1-aryF 𝑋))) → ((𝑥 ∈ 𝑋 ↦ (𝑓‘{⟨0, 𝑥⟩})) = (𝑥 ∈ 𝑋 ↦ (𝑔‘{⟨0, 𝑥⟩})) ↔ ∀𝑥 ∈ 𝑋 (𝑓‘{⟨0, 𝑥⟩}) = (𝑔‘{⟨0, 𝑥⟩})))
12		1aryfvalel 45982	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑉 → (𝑓 ∈ (1-aryF 𝑋) ↔ 𝑓:(𝑋 ↑_m {0})⟶𝑋))
13		1aryfvalel 45982	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑉 → (𝑔 ∈ (1-aryF 𝑋) ↔ 𝑔:(𝑋 ↑_m {0})⟶𝑋))
14	12, 13	anbi12d 631	. . . . . . 7 ⊢ (𝑋 ∈ 𝑉 → ((𝑓 ∈ (1-aryF 𝑋) ∧ 𝑔 ∈ (1-aryF 𝑋)) ↔ (𝑓:(𝑋 ↑_m {0})⟶𝑋 ∧ 𝑔:(𝑋 ↑_m {0})⟶𝑋)))
15		ffn 6600	. . . . . . . . . . 11 ⊢ (𝑓:(𝑋 ↑_m {0})⟶𝑋 → 𝑓 Fn (𝑋 ↑_m {0}))
16	15	adantr 481	. . . . . . . . . 10 ⊢ ((𝑓:(𝑋 ↑_m {0})⟶𝑋 ∧ 𝑔:(𝑋 ↑_m {0})⟶𝑋) → 𝑓 Fn (𝑋 ↑_m {0}))
17	16	3ad2ant2 1133	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝑓:(𝑋 ↑_m {0})⟶𝑋 ∧ 𝑔:(𝑋 ↑_m {0})⟶𝑋) ∧ ∀𝑥 ∈ 𝑋 (𝑓‘{⟨0, 𝑥⟩}) = (𝑔‘{⟨0, 𝑥⟩})) → 𝑓 Fn (𝑋 ↑_m {0}))
18		ffn 6600	. . . . . . . . . . 11 ⊢ (𝑔:(𝑋 ↑_m {0})⟶𝑋 → 𝑔 Fn (𝑋 ↑_m {0}))
19	18	adantl 482	. . . . . . . . . 10 ⊢ ((𝑓:(𝑋 ↑_m {0})⟶𝑋 ∧ 𝑔:(𝑋 ↑_m {0})⟶𝑋) → 𝑔 Fn (𝑋 ↑_m {0}))
20	19	3ad2ant2 1133	. . . . . . . . 9 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝑓:(𝑋 ↑_m {0})⟶𝑋 ∧ 𝑔:(𝑋 ↑_m {0})⟶𝑋) ∧ ∀𝑥 ∈ 𝑋 (𝑓‘{⟨0, 𝑥⟩}) = (𝑔‘{⟨0, 𝑥⟩})) → 𝑔 Fn (𝑋 ↑_m {0}))
21		elmapi 8637	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (𝑋 ↑_m {0}) → 𝑦:{0}⟶𝑋)
22		c0ex 10969	. . . . . . . . . . . . 13 ⊢ 0 ∈ V
23	22	fsn2 7008	. . . . . . . . . . . 12 ⊢ (𝑦:{0}⟶𝑋 ↔ ((𝑦‘0) ∈ 𝑋 ∧ 𝑦 = {⟨0, (𝑦‘0)⟩}))
24	21, 23	sylib 217	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (𝑋 ↑_m {0}) → ((𝑦‘0) ∈ 𝑋 ∧ 𝑦 = {⟨0, (𝑦‘0)⟩}))
25		opeq2 4805	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = (𝑦‘0) → ⟨0, 𝑥⟩ = ⟨0, (𝑦‘0)⟩)
26	25	sneqd 4573	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = (𝑦‘0) → {⟨0, 𝑥⟩} = {⟨0, (𝑦‘0)⟩})
27	26	fveq2d 6778	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = (𝑦‘0) → (𝑓‘{⟨0, 𝑥⟩}) = (𝑓‘{⟨0, (𝑦‘0)⟩}))
28	26	fveq2d 6778	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = (𝑦‘0) → (𝑔‘{⟨0, 𝑥⟩}) = (𝑔‘{⟨0, (𝑦‘0)⟩}))
29	27, 28	eqeq12d 2754	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (𝑦‘0) → ((𝑓‘{⟨0, 𝑥⟩}) = (𝑔‘{⟨0, 𝑥⟩}) ↔ (𝑓‘{⟨0, (𝑦‘0)⟩}) = (𝑔‘{⟨0, (𝑦‘0)⟩})))
30	29	rspccv 3558	. . . . . . . . . . . . . . 15 ⊢ (∀𝑥 ∈ 𝑋 (𝑓‘{⟨0, 𝑥⟩}) = (𝑔‘{⟨0, 𝑥⟩}) → ((𝑦‘0) ∈ 𝑋 → (𝑓‘{⟨0, (𝑦‘0)⟩}) = (𝑔‘{⟨0, (𝑦‘0)⟩})))
31	30	3ad2ant3 1134	. . . . . . . . . . . . . 14 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝑓:(𝑋 ↑_m {0})⟶𝑋 ∧ 𝑔:(𝑋 ↑_m {0})⟶𝑋) ∧ ∀𝑥 ∈ 𝑋 (𝑓‘{⟨0, 𝑥⟩}) = (𝑔‘{⟨0, 𝑥⟩})) → ((𝑦‘0) ∈ 𝑋 → (𝑓‘{⟨0, (𝑦‘0)⟩}) = (𝑔‘{⟨0, (𝑦‘0)⟩})))
32	31	com12 32	. . . . . . . . . . . . 13 ⊢ ((𝑦‘0) ∈ 𝑋 → ((𝑋 ∈ 𝑉 ∧ (𝑓:(𝑋 ↑_m {0})⟶𝑋 ∧ 𝑔:(𝑋 ↑_m {0})⟶𝑋) ∧ ∀𝑥 ∈ 𝑋 (𝑓‘{⟨0, 𝑥⟩}) = (𝑔‘{⟨0, 𝑥⟩})) → (𝑓‘{⟨0, (𝑦‘0)⟩}) = (𝑔‘{⟨0, (𝑦‘0)⟩})))
33	32	adantr 481	. . . . . . . . . . . 12 ⊢ (((𝑦‘0) ∈ 𝑋 ∧ 𝑦 = {⟨0, (𝑦‘0)⟩}) → ((𝑋 ∈ 𝑉 ∧ (𝑓:(𝑋 ↑_m {0})⟶𝑋 ∧ 𝑔:(𝑋 ↑_m {0})⟶𝑋) ∧ ∀𝑥 ∈ 𝑋 (𝑓‘{⟨0, 𝑥⟩}) = (𝑔‘{⟨0, 𝑥⟩})) → (𝑓‘{⟨0, (𝑦‘0)⟩}) = (𝑔‘{⟨0, (𝑦‘0)⟩})))
34		fveq2 6774	. . . . . . . . . . . . . 14 ⊢ (𝑦 = {⟨0, (𝑦‘0)⟩} → (𝑓‘𝑦) = (𝑓‘{⟨0, (𝑦‘0)⟩}))
35		fveq2 6774	. . . . . . . . . . . . . 14 ⊢ (𝑦 = {⟨0, (𝑦‘0)⟩} → (𝑔‘𝑦) = (𝑔‘{⟨0, (𝑦‘0)⟩}))
36	34, 35	eqeq12d 2754	. . . . . . . . . . . . 13 ⊢ (𝑦 = {⟨0, (𝑦‘0)⟩} → ((𝑓‘𝑦) = (𝑔‘𝑦) ↔ (𝑓‘{⟨0, (𝑦‘0)⟩}) = (𝑔‘{⟨0, (𝑦‘0)⟩})))
37	36	adantl 482	. . . . . . . . . . . 12 ⊢ (((𝑦‘0) ∈ 𝑋 ∧ 𝑦 = {⟨0, (𝑦‘0)⟩}) → ((𝑓‘𝑦) = (𝑔‘𝑦) ↔ (𝑓‘{⟨0, (𝑦‘0)⟩}) = (𝑔‘{⟨0, (𝑦‘0)⟩})))
38	33, 37	sylibrd 258	. . . . . . . . . . 11 ⊢ (((𝑦‘0) ∈ 𝑋 ∧ 𝑦 = {⟨0, (𝑦‘0)⟩}) → ((𝑋 ∈ 𝑉 ∧ (𝑓:(𝑋 ↑_m {0})⟶𝑋 ∧ 𝑔:(𝑋 ↑_m {0})⟶𝑋) ∧ ∀𝑥 ∈ 𝑋 (𝑓‘{⟨0, 𝑥⟩}) = (𝑔‘{⟨0, 𝑥⟩})) → (𝑓‘𝑦) = (𝑔‘𝑦)))
39	24, 38	syl 17	. . . . . . . . . 10 ⊢ (𝑦 ∈ (𝑋 ↑_m {0}) → ((𝑋 ∈ 𝑉 ∧ (𝑓:(𝑋 ↑_m {0})⟶𝑋 ∧ 𝑔:(𝑋 ↑_m {0})⟶𝑋) ∧ ∀𝑥 ∈ 𝑋 (𝑓‘{⟨0, 𝑥⟩}) = (𝑔‘{⟨0, 𝑥⟩})) → (𝑓‘𝑦) = (𝑔‘𝑦)))
40	39	impcom 408	. . . . . . . . 9 ⊢ (((𝑋 ∈ 𝑉 ∧ (𝑓:(𝑋 ↑_m {0})⟶𝑋 ∧ 𝑔:(𝑋 ↑_m {0})⟶𝑋) ∧ ∀𝑥 ∈ 𝑋 (𝑓‘{⟨0, 𝑥⟩}) = (𝑔‘{⟨0, 𝑥⟩})) ∧ 𝑦 ∈ (𝑋 ↑_m {0})) → (𝑓‘𝑦) = (𝑔‘𝑦))
41	17, 20, 40	eqfnfvd 6912	. . . . . . . 8 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝑓:(𝑋 ↑_m {0})⟶𝑋 ∧ 𝑔:(𝑋 ↑_m {0})⟶𝑋) ∧ ∀𝑥 ∈ 𝑋 (𝑓‘{⟨0, 𝑥⟩}) = (𝑔‘{⟨0, 𝑥⟩})) → 𝑓 = 𝑔)
42	41	3exp 1118	. . . . . . 7 ⊢ (𝑋 ∈ 𝑉 → ((𝑓:(𝑋 ↑_m {0})⟶𝑋 ∧ 𝑔:(𝑋 ↑_m {0})⟶𝑋) → (∀𝑥 ∈ 𝑋 (𝑓‘{⟨0, 𝑥⟩}) = (𝑔‘{⟨0, 𝑥⟩}) → 𝑓 = 𝑔)))
43	14, 42	sylbid 239	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 → ((𝑓 ∈ (1-aryF 𝑋) ∧ 𝑔 ∈ (1-aryF 𝑋)) → (∀𝑥 ∈ 𝑋 (𝑓‘{⟨0, 𝑥⟩}) = (𝑔‘{⟨0, 𝑥⟩}) → 𝑓 = 𝑔)))
44	43	imp 407	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝑓 ∈ (1-aryF 𝑋) ∧ 𝑔 ∈ (1-aryF 𝑋))) → (∀𝑥 ∈ 𝑋 (𝑓‘{⟨0, 𝑥⟩}) = (𝑔‘{⟨0, 𝑥⟩}) → 𝑓 = 𝑔))
45	11, 44	sylbid 239	. . . 4 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝑓 ∈ (1-aryF 𝑋) ∧ 𝑔 ∈ (1-aryF 𝑋))) → ((𝑥 ∈ 𝑋 ↦ (𝑓‘{⟨0, 𝑥⟩})) = (𝑥 ∈ 𝑋 ↦ (𝑔‘{⟨0, 𝑥⟩})) → 𝑓 = 𝑔))
46	7, 45	sylbid 239	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝑓 ∈ (1-aryF 𝑋) ∧ 𝑔 ∈ (1-aryF 𝑋))) → ((𝐻‘𝑓) = (𝐻‘𝑔) → 𝑓 = 𝑔))
47	46	ralrimivva 3123	. 2 ⊢ (𝑋 ∈ 𝑉 → ∀𝑓 ∈ (1-aryF 𝑋)∀𝑔 ∈ (1-aryF 𝑋)((𝐻‘𝑓) = (𝐻‘𝑔) → 𝑓 = 𝑔))
48		dff13 7128	. 2 ⊢ (𝐻:(1-aryF 𝑋)–1-1→(𝑋 ↑_m 𝑋) ↔ (𝐻:(1-aryF 𝑋)⟶(𝑋 ↑_m 𝑋) ∧ ∀𝑓 ∈ (1-aryF 𝑋)∀𝑔 ∈ (1-aryF 𝑋)((𝐻‘𝑓) = (𝐻‘𝑔) → 𝑓 = 𝑔)))
49	2, 47, 48	sylanbrc 583	1 ⊢ (𝑋 ∈ 𝑉 → 𝐻:(1-aryF 𝑋)–1-1→(𝑋 ↑_m 𝑋))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∈ wcel 2106 ∀wral 3064 Vcvv 3432 {csn 4561 ⟨cop 4567 ↦ cmpt 5157 Fn wfn 6428 ⟶wf 6429 –1-1→wf1 6430 ‘cfv 6433 (class class class)co 7275 ↑_m cmap 8615 0cc0 10871 1c1 10872 -aryF cnaryf 45972

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588 ax-cnex 10927 ax-resscn 10928 ax-1cn 10929 ax-icn 10930 ax-addcl 10931 ax-addrcl 10932 ax-mulcl 10933 ax-mulrcl 10934 ax-mulcom 10935 ax-addass 10936 ax-mulass 10937 ax-distr 10938 ax-i2m1 10939 ax-1ne0 10940 ax-1rid 10941 ax-rnegex 10942 ax-rrecex 10943 ax-cnre 10944 ax-pre-lttri 10945 ax-pre-lttrn 10946 ax-pre-ltadd 10947 ax-pre-mulgt0 10948

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3or 1087 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-pss 3906 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-tr 5192 df-id 5489 df-eprel 5495 df-po 5503 df-so 5504 df-fr 5544 df-we 5546 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-ord 6269 df-on 6270 df-lim 6271 df-suc 6272 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-riota 7232 df-ov 7278 df-oprab 7279 df-mpo 7280 df-om 7713 df-1st 7831 df-2nd 7832 df-frecs 8097 df-wrecs 8128 df-recs 8202 df-rdg 8241 df-er 8498 df-map 8617 df-en 8734 df-dom 8735 df-sdom 8736 df-pnf 11011 df-mnf 11012 df-xr 11013 df-ltxr 11014 df-le 11015 df-sub 11207 df-neg 11208 df-nn 11974 df-n0 12234 df-z 12320 df-uz 12583 df-fz 13240 df-fzo 13383 df-naryf 45973

This theorem is referenced by: 1arymaptf1o 45990

W3C validator