1arymaptf1 - Mathbox for Alexander van der Vekens

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Theorem 1arymaptf1 46718

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 46717	. 2 ⊢ (𝑋 ∈ 𝑉 → 𝐻:(1-aryF 𝑋)⟶(𝑋 ↑_m 𝑋))
3	1	1arymaptfv 46716	. . . . . 6 ⊢ (𝑓 ∈ (1-aryF 𝑋) → (𝐻‘𝑓) = (𝑥 ∈ 𝑋 ↦ (𝑓‘{⟨0, 𝑥⟩})))
4	3	ad2antrl 726	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝑓 ∈ (1-aryF 𝑋) ∧ 𝑔 ∈ (1-aryF 𝑋))) → (𝐻‘𝑓) = (𝑥 ∈ 𝑋 ↦ (𝑓‘{⟨0, 𝑥⟩})))
5	1	1arymaptfv 46716	. . . . . 6 ⊢ (𝑔 ∈ (1-aryF 𝑋) → (𝐻‘𝑔) = (𝑥 ∈ 𝑋 ↦ (𝑔‘{⟨0, 𝑥⟩})))
6	5	ad2antll 727	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝑓 ∈ (1-aryF 𝑋) ∧ 𝑔 ∈ (1-aryF 𝑋))) → (𝐻‘𝑔) = (𝑥 ∈ 𝑋 ↦ (𝑔‘{⟨0, 𝑥⟩})))
7	4, 6	eqeq12d 2752	. . . 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 6855	. . . . . . 7 ⊢ (𝑓‘{⟨0, 𝑥⟩}) ∈ V
9	8	rgenw 3068	. . . . . 6 ⊢ ∀𝑥 ∈ 𝑋 (𝑓‘{⟨0, 𝑥⟩}) ∈ V
10		mpteqb 6967	. . . . . 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 46712	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑉 → (𝑓 ∈ (1-aryF 𝑋) ↔ 𝑓:(𝑋 ↑_m {0})⟶𝑋))
13		1aryfvalel 46712	. . . . . . . 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 6668	. . . . . . . . . . 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 1134	. . . . . . . . 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 6668	. . . . . . . . . . 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 1134	. . . . . . . . 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 8787	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (𝑋 ↑_m {0}) → 𝑦:{0}⟶𝑋)
22		c0ex 11149	. . . . . . . . . . . . 13 ⊢ 0 ∈ V
23	22	fsn2 7082	. . . . . . . . . . . 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 4831	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = (𝑦‘0) → ⟨0, 𝑥⟩ = ⟨0, (𝑦‘0)⟩)
26	25	sneqd 4598	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = (𝑦‘0) → {⟨0, 𝑥⟩} = {⟨0, (𝑦‘0)⟩})
27	26	fveq2d 6846	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = (𝑦‘0) → (𝑓‘{⟨0, 𝑥⟩}) = (𝑓‘{⟨0, (𝑦‘0)⟩}))
28	26	fveq2d 6846	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = (𝑦‘0) → (𝑔‘{⟨0, 𝑥⟩}) = (𝑔‘{⟨0, (𝑦‘0)⟩}))
29	27, 28	eqeq12d 2752	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (𝑦‘0) → ((𝑓‘{⟨0, 𝑥⟩}) = (𝑔‘{⟨0, 𝑥⟩}) ↔ (𝑓‘{⟨0, (𝑦‘0)⟩}) = (𝑔‘{⟨0, (𝑦‘0)⟩})))
30	29	rspccv 3578	. . . . . . . . . . . . . . 15 ⊢ (∀𝑥 ∈ 𝑋 (𝑓‘{⟨0, 𝑥⟩}) = (𝑔‘{⟨0, 𝑥⟩}) → ((𝑦‘0) ∈ 𝑋 → (𝑓‘{⟨0, (𝑦‘0)⟩}) = (𝑔‘{⟨0, (𝑦‘0)⟩})))
31	30	3ad2ant3 1135	. . . . . . . . . . . . . 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 6842	. . . . . . . . . . . . . 14 ⊢ (𝑦 = {⟨0, (𝑦‘0)⟩} → (𝑓‘𝑦) = (𝑓‘{⟨0, (𝑦‘0)⟩}))
35		fveq2 6842	. . . . . . . . . . . . . 14 ⊢ (𝑦 = {⟨0, (𝑦‘0)⟩} → (𝑔‘𝑦) = (𝑔‘{⟨0, (𝑦‘0)⟩}))
36	34, 35	eqeq12d 2752	. . . . . . . . . . . . 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 6985	. . . . . . . 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 1119	. . . . . . 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 3197	. 2 ⊢ (𝑋 ∈ 𝑉 → ∀𝑓 ∈ (1-aryF 𝑋)∀𝑔 ∈ (1-aryF 𝑋)((𝐻‘𝑓) = (𝐻‘𝑔) → 𝑓 = 𝑔))
48		dff13 7202	. 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 1087 = wceq 1541 ∈ wcel 2106 ∀wral 3064 Vcvv 3445 {csn 4586 ⟨cop 4592 ↦ cmpt 5188 Fn wfn 6491 ⟶wf 6492 –1-1→wf1 6493 ‘cfv 6496 (class class class)co 7357 ↑_m cmap 8765 0cc0 11051 1c1 11052 -aryF cnaryf 46702

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 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 2707 ax-rep 5242 ax-sep 5256 ax-nul 5263 ax-pow 5320 ax-pr 5384 ax-un 7672 ax-cnex 11107 ax-resscn 11108 ax-1cn 11109 ax-icn 11110 ax-addcl 11111 ax-addrcl 11112 ax-mulcl 11113 ax-mulrcl 11114 ax-mulcom 11115 ax-addass 11116 ax-mulass 11117 ax-distr 11118 ax-i2m1 11119 ax-1ne0 11120 ax-1rid 11121 ax-rnegex 11122 ax-rrecex 11123 ax-cnre 11124 ax-pre-lttri 11125 ax-pre-lttrn 11126 ax-pre-ltadd 11127 ax-pre-mulgt0 11128

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-nel 3050 df-ral 3065 df-rex 3074 df-reu 3354 df-rab 3408 df-v 3447 df-sbc 3740 df-csb 3856 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-pss 3929 df-nul 4283 df-if 4487 df-pw 4562 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-iun 4956 df-br 5106 df-opab 5168 df-mpt 5189 df-tr 5223 df-id 5531 df-eprel 5537 df-po 5545 df-so 5546 df-fr 5588 df-we 5590 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-pred 6253 df-ord 6320 df-on 6321 df-lim 6322 df-suc 6323 df-iota 6448 df-fun 6498 df-fn 6499 df-f 6500 df-f1 6501 df-fo 6502 df-f1o 6503 df-fv 6504 df-riota 7313 df-ov 7360 df-oprab 7361 df-mpo 7362 df-om 7803 df-1st 7921 df-2nd 7922 df-frecs 8212 df-wrecs 8243 df-recs 8317 df-rdg 8356 df-er 8648 df-map 8767 df-en 8884 df-dom 8885 df-sdom 8886 df-pnf 11191 df-mnf 11192 df-xr 11193 df-ltxr 11194 df-le 11195 df-sub 11387 df-neg 11388 df-nn 12154 df-n0 12414 df-z 12500 df-uz 12764 df-fz 13425 df-fzo 13568 df-naryf 46703

This theorem is referenced by: 1arymaptf1o 46720

W3C validator