fmptco1f1o - Mathbox for Thierry Arnoux

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Theorem fmptco1f1o 30968

Description: The action of composing (to the right) with a bijection is itself a bijection of functions. (Contributed by Thierry Arnoux, 3-Jan-2021.)

**Hypotheses**
Ref	Expression
fmptco1f1o.a	⊢ 𝐴 = (𝑅 ↑_m 𝐸)
fmptco1f1o.b	⊢ 𝐵 = (𝑅 ↑_m 𝐷)
fmptco1f1o.f	⊢ 𝐹 = (𝑓 ∈ 𝐴 ↦ (𝑓 ∘ 𝑇))
fmptco1f1o.d	⊢ (𝜑 → 𝐷 ∈ 𝑉)
fmptco1f1o.e	⊢ (𝜑 → 𝐸 ∈ 𝑊)
fmptco1f1o.r	⊢ (𝜑 → 𝑅 ∈ 𝑋)
fmptco1f1o.t	⊢ (𝜑 → 𝑇:𝐷–1-1-onto→𝐸)

**Assertion**
Ref	Expression
fmptco1f1o	⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵)

Distinct variable groups: 𝐴,𝑓 𝐵,𝑓 𝑇,𝑓 𝜑,𝑓 Allowed substitution hints: 𝐷(𝑓) 𝑅(𝑓) 𝐸(𝑓) 𝐹(𝑓) 𝑉(𝑓) 𝑊(𝑓) 𝑋(𝑓)

Proof of Theorem fmptco1f1o Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fmptco1f1o.f	. . . 4 ⊢ 𝐹 = (𝑓 ∈ 𝐴 ↦ (𝑓 ∘ 𝑇))
2	1	a1i 11	. . 3 ⊢ (𝜑 → 𝐹 = (𝑓 ∈ 𝐴 ↦ (𝑓 ∘ 𝑇)))
3		fmptco1f1o.r	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ 𝑋)
4	3	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑅 ∈ 𝑋)
5		fmptco1f1o.d	. . . . . 6 ⊢ (𝜑 → 𝐷 ∈ 𝑉)
6	5	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝐷 ∈ 𝑉)
7		simpr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓 ∈ 𝐴)
8		fmptco1f1o.a	. . . . . . . 8 ⊢ 𝐴 = (𝑅 ↑_m 𝐸)
9	7, 8	eleqtrdi 2849	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓 ∈ (𝑅 ↑_m 𝐸))
10		elmapi 8637	. . . . . . 7 ⊢ (𝑓 ∈ (𝑅 ↑_m 𝐸) → 𝑓:𝐸⟶𝑅)
11	9, 10	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓:𝐸⟶𝑅)
12		fmptco1f1o.t	. . . . . . . 8 ⊢ (𝜑 → 𝑇:𝐷–1-1-onto→𝐸)
13		f1of 6716	. . . . . . . 8 ⊢ (𝑇:𝐷–1-1-onto→𝐸 → 𝑇:𝐷⟶𝐸)
14	12, 13	syl 17	. . . . . . 7 ⊢ (𝜑 → 𝑇:𝐷⟶𝐸)
15	14	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑇:𝐷⟶𝐸)
16		fco 6624	. . . . . 6 ⊢ ((𝑓:𝐸⟶𝑅 ∧ 𝑇:𝐷⟶𝐸) → (𝑓 ∘ 𝑇):𝐷⟶𝑅)
17	11, 15, 16	syl2anc 584	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → (𝑓 ∘ 𝑇):𝐷⟶𝑅)
18		elmapg 8628	. . . . . 6 ⊢ ((𝑅 ∈ 𝑋 ∧ 𝐷 ∈ 𝑉) → ((𝑓 ∘ 𝑇) ∈ (𝑅 ↑_m 𝐷) ↔ (𝑓 ∘ 𝑇):𝐷⟶𝑅))
19	18	biimpar 478	. . . . 5 ⊢ (((𝑅 ∈ 𝑋 ∧ 𝐷 ∈ 𝑉) ∧ (𝑓 ∘ 𝑇):𝐷⟶𝑅) → (𝑓 ∘ 𝑇) ∈ (𝑅 ↑_m 𝐷))
20	4, 6, 17, 19	syl21anc 835	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → (𝑓 ∘ 𝑇) ∈ (𝑅 ↑_m 𝐷))
21		fmptco1f1o.b	. . . 4 ⊢ 𝐵 = (𝑅 ↑_m 𝐷)
22	20, 21	eleqtrrdi 2850	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → (𝑓 ∘ 𝑇) ∈ 𝐵)
23	3	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐵) → 𝑅 ∈ 𝑋)
24		fmptco1f1o.e	. . . . . 6 ⊢ (𝜑 → 𝐸 ∈ 𝑊)
25	24	adantr 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐵) → 𝐸 ∈ 𝑊)
26		simpr 485	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐵) → 𝑔 ∈ 𝐵)
27	26, 21	eleqtrdi 2849	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐵) → 𝑔 ∈ (𝑅 ↑_m 𝐷))
28		elmapi 8637	. . . . . . 7 ⊢ (𝑔 ∈ (𝑅 ↑_m 𝐷) → 𝑔:𝐷⟶𝑅)
29	27, 28	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐵) → 𝑔:𝐷⟶𝑅)
30		f1ocnv 6728	. . . . . . . 8 ⊢ (𝑇:𝐷–1-1-onto→𝐸 → ^◡𝑇:𝐸–1-1-onto→𝐷)
31		f1of 6716	. . . . . . . 8 ⊢ (^◡𝑇:𝐸–1-1-onto→𝐷 → ^◡𝑇:𝐸⟶𝐷)
32	12, 30, 31	3syl 18	. . . . . . 7 ⊢ (𝜑 → ^◡𝑇:𝐸⟶𝐷)
33	32	adantr 481	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐵) → ^◡𝑇:𝐸⟶𝐷)
34		fco 6624	. . . . . 6 ⊢ ((𝑔:𝐷⟶𝑅 ∧ ^◡𝑇:𝐸⟶𝐷) → (𝑔 ∘ ^◡𝑇):𝐸⟶𝑅)
35	29, 33, 34	syl2anc 584	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐵) → (𝑔 ∘ ^◡𝑇):𝐸⟶𝑅)
36		elmapg 8628	. . . . . 6 ⊢ ((𝑅 ∈ 𝑋 ∧ 𝐸 ∈ 𝑊) → ((𝑔 ∘ ^◡𝑇) ∈ (𝑅 ↑_m 𝐸) ↔ (𝑔 ∘ ^◡𝑇):𝐸⟶𝑅))
37	36	biimpar 478	. . . . 5 ⊢ (((𝑅 ∈ 𝑋 ∧ 𝐸 ∈ 𝑊) ∧ (𝑔 ∘ ^◡𝑇):𝐸⟶𝑅) → (𝑔 ∘ ^◡𝑇) ∈ (𝑅 ↑_m 𝐸))
38	23, 25, 35, 37	syl21anc 835	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐵) → (𝑔 ∘ ^◡𝑇) ∈ (𝑅 ↑_m 𝐸))
39	38, 8	eleqtrrdi 2850	. . 3 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐵) → (𝑔 ∘ ^◡𝑇) ∈ 𝐴)
40		coass 6169	. . . . . . 7 ⊢ ((𝑔 ∘ ^◡𝑇) ∘ 𝑇) = (𝑔 ∘ (^◡𝑇 ∘ 𝑇))
41	12	ad2antrr 723	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐴) ∧ 𝑔 ∈ 𝐵) → 𝑇:𝐷–1-1-onto→𝐸)
42		f1ococnv1 6745	. . . . . . . . . 10 ⊢ (𝑇:𝐷–1-1-onto→𝐸 → (^◡𝑇 ∘ 𝑇) = ( I ↾ 𝐷))
43	42	coeq2d 5771	. . . . . . . . 9 ⊢ (𝑇:𝐷–1-1-onto→𝐸 → (𝑔 ∘ (^◡𝑇 ∘ 𝑇)) = (𝑔 ∘ ( I ↾ 𝐷)))
44	41, 43	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐴) ∧ 𝑔 ∈ 𝐵) → (𝑔 ∘ (^◡𝑇 ∘ 𝑇)) = (𝑔 ∘ ( I ↾ 𝐷)))
45	29	adantlr 712	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐴) ∧ 𝑔 ∈ 𝐵) → 𝑔:𝐷⟶𝑅)
46		fcoi1 6648	. . . . . . . . 9 ⊢ (𝑔:𝐷⟶𝑅 → (𝑔 ∘ ( I ↾ 𝐷)) = 𝑔)
47	45, 46	syl 17	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐴) ∧ 𝑔 ∈ 𝐵) → (𝑔 ∘ ( I ↾ 𝐷)) = 𝑔)
48	44, 47	eqtrd 2778	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐴) ∧ 𝑔 ∈ 𝐵) → (𝑔 ∘ (^◡𝑇 ∘ 𝑇)) = 𝑔)
49	40, 48	eqtr2id 2791	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐴) ∧ 𝑔 ∈ 𝐵) → 𝑔 = ((𝑔 ∘ ^◡𝑇) ∘ 𝑇))
50	49	eqeq1d 2740	. . . . 5 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐴) ∧ 𝑔 ∈ 𝐵) → (𝑔 = (𝑓 ∘ 𝑇) ↔ ((𝑔 ∘ ^◡𝑇) ∘ 𝑇) = (𝑓 ∘ 𝑇)))
51		eqcom 2745	. . . . . 6 ⊢ (((𝑔 ∘ ^◡𝑇) ∘ 𝑇) = (𝑓 ∘ 𝑇) ↔ (𝑓 ∘ 𝑇) = ((𝑔 ∘ ^◡𝑇) ∘ 𝑇))
52	51	a1i 11	. . . . 5 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐴) ∧ 𝑔 ∈ 𝐵) → (((𝑔 ∘ ^◡𝑇) ∘ 𝑇) = (𝑓 ∘ 𝑇) ↔ (𝑓 ∘ 𝑇) = ((𝑔 ∘ ^◡𝑇) ∘ 𝑇)))
53		f1ofo 6723	. . . . . . 7 ⊢ (𝑇:𝐷–1-1-onto→𝐸 → 𝑇:𝐷–onto→𝐸)
54	41, 53	syl 17	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐴) ∧ 𝑔 ∈ 𝐵) → 𝑇:𝐷–onto→𝐸)
55		simplr 766	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐴) ∧ 𝑔 ∈ 𝐵) → 𝑓 ∈ 𝐴)
56	55, 8	eleqtrdi 2849	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐴) ∧ 𝑔 ∈ 𝐵) → 𝑓 ∈ (𝑅 ↑_m 𝐸))
57		elmapfn 8653	. . . . . . 7 ⊢ (𝑓 ∈ (𝑅 ↑_m 𝐸) → 𝑓 Fn 𝐸)
58	56, 57	syl 17	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐴) ∧ 𝑔 ∈ 𝐵) → 𝑓 Fn 𝐸)
59		elmapfn 8653	. . . . . . . 8 ⊢ ((𝑔 ∘ ^◡𝑇) ∈ (𝑅 ↑_m 𝐸) → (𝑔 ∘ ^◡𝑇) Fn 𝐸)
60	38, 59	syl 17	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐵) → (𝑔 ∘ ^◡𝑇) Fn 𝐸)
61	60	adantlr 712	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐴) ∧ 𝑔 ∈ 𝐵) → (𝑔 ∘ ^◡𝑇) Fn 𝐸)
62		cocan2 7164	. . . . . 6 ⊢ ((𝑇:𝐷–onto→𝐸 ∧ 𝑓 Fn 𝐸 ∧ (𝑔 ∘ ^◡𝑇) Fn 𝐸) → ((𝑓 ∘ 𝑇) = ((𝑔 ∘ ^◡𝑇) ∘ 𝑇) ↔ 𝑓 = (𝑔 ∘ ^◡𝑇)))
63	54, 58, 61, 62	syl3anc 1370	. . . . 5 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐴) ∧ 𝑔 ∈ 𝐵) → ((𝑓 ∘ 𝑇) = ((𝑔 ∘ ^◡𝑇) ∘ 𝑇) ↔ 𝑓 = (𝑔 ∘ ^◡𝑇)))
64	50, 52, 63	3bitrrd 306	. . . 4 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐴) ∧ 𝑔 ∈ 𝐵) → (𝑓 = (𝑔 ∘ ^◡𝑇) ↔ 𝑔 = (𝑓 ∘ 𝑇)))
65	64	anasss 467	. . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐵)) → (𝑓 = (𝑔 ∘ ^◡𝑇) ↔ 𝑔 = (𝑓 ∘ 𝑇)))
66	2, 22, 39, 65	f1o3d 30962	. 2 ⊢ (𝜑 → (𝐹:𝐴–1-1-onto→𝐵 ∧ ^◡𝐹 = (𝑔 ∈ 𝐵 ↦ (𝑔 ∘ ^◡𝑇))))
67	66	simpld 495	1 ⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1539 ∈ wcel 2106 ↦ cmpt 5157 I cid 5488 ^◡ccnv 5588 ↾ cres 5591 ∘ ccom 5593 Fn wfn 6428 ⟶wf 6429 –onto→wfo 6431 –1-1-onto→wf1o 6432 (class class class)co 7275 ↑_m cmap 8615

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-sep 5223 ax-nul 5230 ax-pow 5288 ax-pr 5352 ax-un 7588

This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 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-ral 3069 df-rex 3070 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-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-id 5489 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-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-ov 7278 df-oprab 7279 df-mpo 7280 df-1st 7831 df-2nd 7832 df-map 8617

This theorem is referenced by: reprpmtf1o 32606

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