1arymaptf1 - Mathbox for Alexander van der Vekens

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Theorem 1arymaptf1 48999

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 48998	. 2 ⊢ (𝑋 ∈ 𝑉 → 𝐻:(1-aryF 𝑋)⟶(𝑋 ↑_m 𝑋))
3	1	1arymaptfv 48997	. . . . . 6 ⊢ (𝑓 ∈ (1-aryF 𝑋) → (𝐻‘𝑓) = (𝑥 ∈ 𝑋 ↦ (𝑓‘{⟨0, 𝑥⟩})))
4	3	ad2antrl 729	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝑓 ∈ (1-aryF 𝑋) ∧ 𝑔 ∈ (1-aryF 𝑋))) → (𝐻‘𝑓) = (𝑥 ∈ 𝑋 ↦ (𝑓‘{⟨0, 𝑥⟩})))
5	1	1arymaptfv 48997	. . . . . 6 ⊢ (𝑔 ∈ (1-aryF 𝑋) → (𝐻‘𝑔) = (𝑥 ∈ 𝑋 ↦ (𝑔‘{⟨0, 𝑥⟩})))
6	5	ad2antll 730	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝑓 ∈ (1-aryF 𝑋) ∧ 𝑔 ∈ (1-aryF 𝑋))) → (𝐻‘𝑔) = (𝑥 ∈ 𝑋 ↦ (𝑔‘{⟨0, 𝑥⟩})))
7	4, 6	eqeq12d 2753	. . . 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 3056	. . . . . 6 ⊢ ∀𝑥 ∈ 𝑋 (𝑓‘{⟨0, 𝑥⟩}) ∈ V
10		mpteqb 6969	. . . . . 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 48993	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑉 → (𝑓 ∈ (1-aryF 𝑋) ↔ 𝑓:(𝑋 ↑_m {0})⟶𝑋))
13		1aryfvalel 48993	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑉 → (𝑔 ∈ (1-aryF 𝑋) ↔ 𝑔:(𝑋 ↑_m {0})⟶𝑋))
14	12, 13	anbi12d 633	. . . . . . 7 ⊢ (𝑋 ∈ 𝑉 → ((𝑓 ∈ (1-aryF 𝑋) ∧ 𝑔 ∈ (1-aryF 𝑋)) ↔ (𝑓:(𝑋 ↑_m {0})⟶𝑋 ∧ 𝑔:(𝑋 ↑_m {0})⟶𝑋)))
15		ffn 6670	. . . . . . . . . . 11 ⊢ (𝑓:(𝑋 ↑_m {0})⟶𝑋 → 𝑓 Fn (𝑋 ↑_m {0}))
16	15	adantr 480	. . . . . . . . . 10 ⊢ ((𝑓:(𝑋 ↑_m {0})⟶𝑋 ∧ 𝑔:(𝑋 ↑_m {0})⟶𝑋) → 𝑓 Fn (𝑋 ↑_m {0}))
17	16	3ad2ant2 1135	. . . . . . . . 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 6670	. . . . . . . . . . 11 ⊢ (𝑔:(𝑋 ↑_m {0})⟶𝑋 → 𝑔 Fn (𝑋 ↑_m {0}))
19	18	adantl 481	. . . . . . . . . 10 ⊢ ((𝑓:(𝑋 ↑_m {0})⟶𝑋 ∧ 𝑔:(𝑋 ↑_m {0})⟶𝑋) → 𝑔 Fn (𝑋 ↑_m {0}))
20	19	3ad2ant2 1135	. . . . . . . . 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 8798	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (𝑋 ↑_m {0}) → 𝑦:{0}⟶𝑋)
22		c0ex 11138	. . . . . . . . . . . . 13 ⊢ 0 ∈ V
23	22	fsn2 7091	. . . . . . . . . . . 12 ⊢ (𝑦:{0}⟶𝑋 ↔ ((𝑦‘0) ∈ 𝑋 ∧ 𝑦 = {⟨0, (𝑦‘0)⟩}))
24	21, 23	sylib 218	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (𝑋 ↑_m {0}) → ((𝑦‘0) ∈ 𝑋 ∧ 𝑦 = {⟨0, (𝑦‘0)⟩}))
25		opeq2 4832	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = (𝑦‘0) → ⟨0, 𝑥⟩ = ⟨0, (𝑦‘0)⟩)
26	25	sneqd 4594	. . . . . . . . . . . . . . . . . 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 2753	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (𝑦‘0) → ((𝑓‘{⟨0, 𝑥⟩}) = (𝑔‘{⟨0, 𝑥⟩}) ↔ (𝑓‘{⟨0, (𝑦‘0)⟩}) = (𝑔‘{⟨0, (𝑦‘0)⟩})))
30	29	rspccv 3575	. . . . . . . . . . . . . . 15 ⊢ (∀𝑥 ∈ 𝑋 (𝑓‘{⟨0, 𝑥⟩}) = (𝑔‘{⟨0, 𝑥⟩}) → ((𝑦‘0) ∈ 𝑋 → (𝑓‘{⟨0, (𝑦‘0)⟩}) = (𝑔‘{⟨0, (𝑦‘0)⟩})))
31	30	3ad2ant3 1136	. . . . . . . . . . . . . 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 480	. . . . . . . . . . . 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 2753	. . . . . . . . . . . . 13 ⊢ (𝑦 = {⟨0, (𝑦‘0)⟩} → ((𝑓‘𝑦) = (𝑔‘𝑦) ↔ (𝑓‘{⟨0, (𝑦‘0)⟩}) = (𝑔‘{⟨0, (𝑦‘0)⟩})))
37	36	adantl 481	. . . . . . . . . . . 12 ⊢ (((𝑦‘0) ∈ 𝑋 ∧ 𝑦 = {⟨0, (𝑦‘0)⟩}) → ((𝑓‘𝑦) = (𝑔‘𝑦) ↔ (𝑓‘{⟨0, (𝑦‘0)⟩}) = (𝑔‘{⟨0, (𝑦‘0)⟩})))
38	33, 37	sylibrd 259	. . . . . . . . . . 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 407	. . . . . . . . 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 6988	. . . . . . . 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 1120	. . . . . . 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 240	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 → ((𝑓 ∈ (1-aryF 𝑋) ∧ 𝑔 ∈ (1-aryF 𝑋)) → (∀𝑥 ∈ 𝑋 (𝑓‘{⟨0, 𝑥⟩}) = (𝑔‘{⟨0, 𝑥⟩}) → 𝑓 = 𝑔)))
44	43	imp 406	. . . . 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 240	. . . 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 240	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝑓 ∈ (1-aryF 𝑋) ∧ 𝑔 ∈ (1-aryF 𝑋))) → ((𝐻‘𝑓) = (𝐻‘𝑔) → 𝑓 = 𝑔))
47	46	ralrimivva 3181	. 2 ⊢ (𝑋 ∈ 𝑉 → ∀𝑓 ∈ (1-aryF 𝑋)∀𝑔 ∈ (1-aryF 𝑋)((𝐻‘𝑓) = (𝐻‘𝑔) → 𝑓 = 𝑔))
48		dff13 7210	. 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 584	1 ⊢ (𝑋 ∈ 𝑉 → 𝐻:(1-aryF 𝑋)–1-1→(𝑋 ↑_m 𝑋))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∧ w3a 1087 = wceq 1542 ∈ wcel 2114 ∀wral 3052 Vcvv 3442 {csn 4582 ⟨cop 4588 ↦ cmpt 5181 Fn wfn 6495 ⟶wf 6496 –1-1→wf1 6497 ‘cfv 6500 (class class class)co 7368 ↑_m cmap 8775 0cc0 11038 1c1 11039 -aryF cnaryf 48983

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-rep 5226 ax-sep 5243 ax-nul 5253 ax-pow 5312 ax-pr 5379 ax-un 7690 ax-cnex 11094 ax-resscn 11095 ax-1cn 11096 ax-icn 11097 ax-addcl 11098 ax-addrcl 11099 ax-mulcl 11100 ax-mulrcl 11101 ax-mulcom 11102 ax-addass 11103 ax-mulass 11104 ax-distr 11105 ax-i2m1 11106 ax-1ne0 11107 ax-1rid 11108 ax-rnegex 11109 ax-rrecex 11110 ax-cnre 11111 ax-pre-lttri 11112 ax-pre-lttrn 11113 ax-pre-ltadd 11114 ax-pre-mulgt0 11115

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-reu 3353 df-rab 3402 df-v 3444 df-sbc 3743 df-csb 3852 df-dif 3906 df-un 3908 df-in 3910 df-ss 3920 df-pss 3923 df-nul 4288 df-if 4482 df-pw 4558 df-sn 4583 df-pr 4585 df-op 4589 df-uni 4866 df-iun 4950 df-br 5101 df-opab 5163 df-mpt 5182 df-tr 5208 df-id 5527 df-eprel 5532 df-po 5540 df-so 5541 df-fr 5585 df-we 5587 df-xp 5638 df-rel 5639 df-cnv 5640 df-co 5641 df-dm 5642 df-rn 5643 df-res 5644 df-ima 5645 df-pred 6267 df-ord 6328 df-on 6329 df-lim 6330 df-suc 6331 df-iota 6456 df-fun 6502 df-fn 6503 df-f 6504 df-f1 6505 df-fo 6506 df-f1o 6507 df-fv 6508 df-riota 7325 df-ov 7371 df-oprab 7372 df-mpo 7373 df-om 7819 df-1st 7943 df-2nd 7944 df-frecs 8233 df-wrecs 8264 df-recs 8313 df-rdg 8351 df-er 8645 df-map 8777 df-en 8896 df-dom 8897 df-sdom 8898 df-pnf 11180 df-mnf 11181 df-xr 11182 df-ltxr 11183 df-le 11184 df-sub 11378 df-neg 11379 df-nn 12158 df-n0 12414 df-z 12501 df-uz 12764 df-fz 13436 df-fzo 13583 df-naryf 48984

This theorem is referenced by: 1arymaptf1o 49001

W3C validator