1arymaptf1 - Mathbox for Alexander van der Vekens

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Theorem 1arymaptf1 49225

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 49224	. 2 ⊢ (𝑋 ∈ 𝑉 → 𝐻:(1-aryF 𝑋)⟶(𝑋 ↑_m 𝑋))
3	1	1arymaptfv 49223	. . . . . 6 ⊢ (𝑓 ∈ (1-aryF 𝑋) → (𝐻‘𝑓) = (𝑥 ∈ 𝑋 ↦ (𝑓‘{⟨0, 𝑥⟩})))
4	3	ad2antrl 738	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝑓 ∈ (1-aryF 𝑋) ∧ 𝑔 ∈ (1-aryF 𝑋))) → (𝐻‘𝑓) = (𝑥 ∈ 𝑋 ↦ (𝑓‘{⟨0, 𝑥⟩})))
5	1	1arymaptfv 49223	. . . . . 6 ⊢ (𝑔 ∈ (1-aryF 𝑋) → (𝐻‘𝑔) = (𝑥 ∈ 𝑋 ↦ (𝑔‘{⟨0, 𝑥⟩})))
6	5	ad2antll 739	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝑓 ∈ (1-aryF 𝑋) ∧ 𝑔 ∈ (1-aryF 𝑋))) → (𝐻‘𝑔) = (𝑥 ∈ 𝑋 ↦ (𝑔‘{⟨0, 𝑥⟩})))
7	4, 6	eqeq12d 2777	. . . 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 6875	. . . . . . 7 ⊢ (𝑓‘{⟨0, 𝑥⟩}) ∈ V
9	8	rgenw 3079	. . . . . 6 ⊢ ∀𝑥 ∈ 𝑋 (𝑓‘{⟨0, 𝑥⟩}) ∈ V
10		mpteqb 6990	. . . . . 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 49219	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑉 → (𝑓 ∈ (1-aryF 𝑋) ↔ 𝑓:(𝑋 ↑_m {0})⟶𝑋))
13		1aryfvalel 49219	. . . . . . . 8 ⊢ (𝑋 ∈ 𝑉 → (𝑔 ∈ (1-aryF 𝑋) ↔ 𝑔:(𝑋 ↑_m {0})⟶𝑋))
14	12, 13	anbi12d 641	. . . . . . 7 ⊢ (𝑋 ∈ 𝑉 → ((𝑓 ∈ (1-aryF 𝑋) ∧ 𝑔 ∈ (1-aryF 𝑋)) ↔ (𝑓:(𝑋 ↑_m {0})⟶𝑋 ∧ 𝑔:(𝑋 ↑_m {0})⟶𝑋)))
15		ffn 6686	. . . . . . . . . . 11 ⊢ (𝑓:(𝑋 ↑_m {0})⟶𝑋 → 𝑓 Fn (𝑋 ↑_m {0}))
16	15	adantr 484	. . . . . . . . . 10 ⊢ ((𝑓:(𝑋 ↑_m {0})⟶𝑋 ∧ 𝑔:(𝑋 ↑_m {0})⟶𝑋) → 𝑓 Fn (𝑋 ↑_m {0}))
17	16	3ad2ant2 1146	. . . . . . . . 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 6686	. . . . . . . . . . 11 ⊢ (𝑔:(𝑋 ↑_m {0})⟶𝑋 → 𝑔 Fn (𝑋 ↑_m {0}))
19	18	adantl 485	. . . . . . . . . 10 ⊢ ((𝑓:(𝑋 ↑_m {0})⟶𝑋 ∧ 𝑔:(𝑋 ↑_m {0})⟶𝑋) → 𝑔 Fn (𝑋 ↑_m {0}))
20	19	3ad2ant2 1146	. . . . . . . . 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 8824	. . . . . . . . . . . 12 ⊢ (𝑦 ∈ (𝑋 ↑_m {0}) → 𝑦:{0}⟶𝑋)
22		c0ex 11167	. . . . . . . . . . . . 13 ⊢ 0 ∈ V
23	22	fsn2 7113	. . . . . . . . . . . 12 ⊢ (𝑦:{0}⟶𝑋 ↔ ((𝑦‘0) ∈ 𝑋 ∧ 𝑦 = {⟨0, (𝑦‘0)⟩}))
24	21, 23	sylib 220	. . . . . . . . . . 11 ⊢ (𝑦 ∈ (𝑋 ↑_m {0}) → ((𝑦‘0) ∈ 𝑋 ∧ 𝑦 = {⟨0, (𝑦‘0)⟩}))
25		opeq2 4829	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑥 = (𝑦‘0) → ⟨0, 𝑥⟩ = ⟨0, (𝑦‘0)⟩)
26	25	sneqd 4591	. . . . . . . . . . . . . . . . . 18 ⊢ (𝑥 = (𝑦‘0) → {⟨0, 𝑥⟩} = {⟨0, (𝑦‘0)⟩})
27	26	fveq2d 6866	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = (𝑦‘0) → (𝑓‘{⟨0, 𝑥⟩}) = (𝑓‘{⟨0, (𝑦‘0)⟩}))
28	26	fveq2d 6866	. . . . . . . . . . . . . . . . 17 ⊢ (𝑥 = (𝑦‘0) → (𝑔‘{⟨0, 𝑥⟩}) = (𝑔‘{⟨0, (𝑦‘0)⟩}))
29	27, 28	eqeq12d 2777	. . . . . . . . . . . . . . . 16 ⊢ (𝑥 = (𝑦‘0) → ((𝑓‘{⟨0, 𝑥⟩}) = (𝑔‘{⟨0, 𝑥⟩}) ↔ (𝑓‘{⟨0, (𝑦‘0)⟩}) = (𝑔‘{⟨0, (𝑦‘0)⟩})))
30	29	rspccv 3577	. . . . . . . . . . . . . . 15 ⊢ (∀𝑥 ∈ 𝑋 (𝑓‘{⟨0, 𝑥⟩}) = (𝑔‘{⟨0, 𝑥⟩}) → ((𝑦‘0) ∈ 𝑋 → (𝑓‘{⟨0, (𝑦‘0)⟩}) = (𝑔‘{⟨0, (𝑦‘0)⟩})))
31	30	3ad2ant3 1147	. . . . . . . . . . . . . 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 484	. . . . . . . . . . . 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 6862	. . . . . . . . . . . . . 14 ⊢ (𝑦 = {⟨0, (𝑦‘0)⟩} → (𝑓‘𝑦) = (𝑓‘{⟨0, (𝑦‘0)⟩}))
35		fveq2 6862	. . . . . . . . . . . . . 14 ⊢ (𝑦 = {⟨0, (𝑦‘0)⟩} → (𝑔‘𝑦) = (𝑔‘{⟨0, (𝑦‘0)⟩}))
36	34, 35	eqeq12d 2777	. . . . . . . . . . . . 13 ⊢ (𝑦 = {⟨0, (𝑦‘0)⟩} → ((𝑓‘𝑦) = (𝑔‘𝑦) ↔ (𝑓‘{⟨0, (𝑦‘0)⟩}) = (𝑔‘{⟨0, (𝑦‘0)⟩})))
37	36	adantl 485	. . . . . . . . . . . 12 ⊢ (((𝑦‘0) ∈ 𝑋 ∧ 𝑦 = {⟨0, (𝑦‘0)⟩}) → ((𝑓‘𝑦) = (𝑔‘𝑦) ↔ (𝑓‘{⟨0, (𝑦‘0)⟩}) = (𝑔‘{⟨0, (𝑦‘0)⟩})))
38	33, 37	sylibrd 261	. . . . . . . . . . 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 411	. . . . . . . . 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 7009	. . . . . . . 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 1131	. . . . . . 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 242	. . . . . 6 ⊢ (𝑋 ∈ 𝑉 → ((𝑓 ∈ (1-aryF 𝑋) ∧ 𝑔 ∈ (1-aryF 𝑋)) → (∀𝑥 ∈ 𝑋 (𝑓‘{⟨0, 𝑥⟩}) = (𝑔‘{⟨0, 𝑥⟩}) → 𝑓 = 𝑔)))
44	43	imp 410	. . . . 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 242	. . . 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 242	. . 3 ⊢ ((𝑋 ∈ 𝑉 ∧ (𝑓 ∈ (1-aryF 𝑋) ∧ 𝑔 ∈ (1-aryF 𝑋))) → ((𝐻‘𝑓) = (𝐻‘𝑔) → 𝑓 = 𝑔))
47	46	ralrimivva 3204	. 2 ⊢ (𝑋 ∈ 𝑉 → ∀𝑓 ∈ (1-aryF 𝑋)∀𝑔 ∈ (1-aryF 𝑋)((𝐻‘𝑓) = (𝐻‘𝑔) → 𝑓 = 𝑔))
48		dff13 7233	. 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 592	1 ⊢ (𝑋 ∈ 𝑉 → 𝐻:(1-aryF 𝑋)–1-1→(𝑋 ↑_m 𝑋))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 399 ∧ w3a 1097 = wceq 1559 ∈ wcel 2141 ∀wral 3075 Vcvv 3453 {csn 4579 ⟨cop 4585 ↦ cmpt 5178 Fn wfn 6511 ⟶wf 6512 –1-1→wf1 6513 ‘cfv 6516 (class class class)co 7391 ↑_m cmap 8802 0cc0 11067 1c1 11068 -aryF cnaryf 49209

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-rep 5224 ax-sep 5243 ax-nul 5253 ax-pow 5319 ax-pr 5387 ax-un 7713 ax-cnex 11123 ax-resscn 11124 ax-1cn 11125 ax-icn 11126 ax-addcl 11127 ax-addrcl 11128 ax-mulcl 11129 ax-mulrcl 11130 ax-mulcom 11131 ax-addass 11132 ax-mulass 11133 ax-distr 11134 ax-i2m1 11135 ax-1ne0 11136 ax-1rid 11137 ax-rnegex 11138 ax-rrecex 11139 ax-cnre 11140 ax-pre-lttri 11141 ax-pre-lttrn 11142 ax-pre-ltadd 11143 ax-pre-mulgt0 11144

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3or 1098 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ne 2957 df-nel 3061 df-ral 3076 df-rex 3086 df-reu 3367 df-rab 3414 df-v 3455 df-sbc 3743 df-csb 3851 df-dif 3905 df-un 3907 df-in 3909 df-ss 3919 df-pss 3922 df-nul 4284 df-if 4478 df-pw 4554 df-sn 4580 df-pr 4582 df-op 4586 df-uni 4863 df-iun 4948 df-br 5098 df-opab 5160 df-mpt 5179 df-tr 5205 df-id 5538 df-eprel 5543 df-po 5551 df-so 5552 df-fr 5596 df-we 5598 df-xp 5649 df-rel 5650 df-cnv 5651 df-co 5652 df-dm 5653 df-rn 5654 df-res 5655 df-ima 5656 df-pred 6283 df-ord 6344 df-on 6345 df-lim 6346 df-suc 6347 df-iota 6472 df-fun 6518 df-fn 6519 df-f 6520 df-f1 6521 df-fo 6522 df-f1o 6523 df-fv 6524 df-riota 7348 df-ov 7394 df-oprab 7395 df-mpo 7396 df-om 7842 df-1st 7965 df-2nd 7966 df-frecs 8256 df-wrecs 8287 df-recs 8336 df-rdg 8375 df-er 8672 df-map 8804 df-en 8922 df-dom 8923 df-sdom 8924 df-pnf 11212 df-mnf 11213 df-xr 11214 df-ltxr 11215 df-le 11216 df-sub 11410 df-neg 11411 df-nn 12205 df-n0 12476 df-z 12563 df-uz 12834 df-fz 13507 df-fzo 13654 df-naryf 49210

This theorem is referenced by: 1arymaptf1o 49227

W3C validator