fmptco1f1o - Mathbox for Thierry Arnoux

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Theorem fmptco1f1o 32725

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 8786	. . . . . . 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 6767	. . . . . . . 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 6679	. . . . . 6 ⊢ ((𝑓:𝐸⟶𝑅 ∧ 𝑇:𝐷⟶𝐸) → (𝑓 ∘ 𝑇):𝐷⟶𝑅)
17	11, 15, 16	syl2anc 590	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → (𝑓 ∘ 𝑇):𝐷⟶𝑅)
18		elmapg 8776	. . . . . 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 843	. . . 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 8786	. . . . . . 7 ⊢ (𝑔 ∈ (𝑅 ↑_m 𝐷) → 𝑔:𝐷⟶𝑅)
29	27, 28	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐵) → 𝑔:𝐷⟶𝑅)
30		f1ocnv 6779	. . . . . . . 8 ⊢ (𝑇:𝐷–1-1-onto→𝐸 → ^◡𝑇:𝐸–1-1-onto→𝐷)
31		f1of 6767	. . . . . . . 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 6679	. . . . . 6 ⊢ ((𝑔:𝐷⟶𝑅 ∧ ^◡𝑇:𝐸⟶𝐷) → (𝑔 ∘ ^◡𝑇):𝐸⟶𝑅)
35	29, 33, 34	syl2anc 590	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐵) → (𝑔 ∘ ^◡𝑇):𝐸⟶𝑅)
36		elmapg 8776	. . . . . 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 843	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐵) → (𝑔 ∘ ^◡𝑇) ∈ (𝑅 ↑_m 𝐸))
39	38, 8	eleqtrrdi 2850	. . 3 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐵) → (𝑔 ∘ ^◡𝑇) ∈ 𝐴)
40		coass 6217	. . . . . . 7 ⊢ ((𝑔 ∘ ^◡𝑇) ∘ 𝑇) = (𝑔 ∘ (^◡𝑇 ∘ 𝑇))
41	12	ad2antrr 732	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐴) ∧ 𝑔 ∈ 𝐵) → 𝑇:𝐷–1-1-onto→𝐸)
42		f1ococnv1 6796	. . . . . . . . . 10 ⊢ (𝑇:𝐷–1-1-onto→𝐸 → (^◡𝑇 ∘ 𝑇) = ( I ↾ 𝐷))
43	42	coeq2d 5804	. . . . . . . . 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 721	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐴) ∧ 𝑔 ∈ 𝐵) → 𝑔:𝐷⟶𝑅)
46		fcoi1 6701	. . . . . . . . 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 2774	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐴) ∧ 𝑔 ∈ 𝐵) → (𝑔 ∘ (^◡𝑇 ∘ 𝑇)) = 𝑔)
49	40, 48	eqtr2id 2787	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐴) ∧ 𝑔 ∈ 𝐵) → 𝑔 = ((𝑔 ∘ ^◡𝑇) ∘ 𝑇))
50	49	eqeq1d 2741	. . . . 5 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐴) ∧ 𝑔 ∈ 𝐵) → (𝑔 = (𝑓 ∘ 𝑇) ↔ ((𝑔 ∘ ^◡𝑇) ∘ 𝑇) = (𝑓 ∘ 𝑇)))
51		eqcom 2746	. . . . . 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 6774	. . . . . . 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 774	. . . . . . . 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 8802	. . . . . . 7 ⊢ (𝑓 ∈ (𝑅 ↑_m 𝐸) → 𝑓 Fn 𝐸)
58	56, 57	syl 17	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐴) ∧ 𝑔 ∈ 𝐵) → 𝑓 Fn 𝐸)
59		elmapfn 8802	. . . . . . . 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 721	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐴) ∧ 𝑔 ∈ 𝐵) → (𝑔 ∘ ^◡𝑇) Fn 𝐸)
62		cocan2 7236	. . . . . 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 1379	. . . . 5 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐴) ∧ 𝑔 ∈ 𝐵) → ((𝑓 ∘ 𝑇) = ((𝑔 ∘ ^◡𝑇) ∘ 𝑇) ↔ 𝑓 = (𝑔 ∘ ^◡𝑇)))
64	50, 52, 63	3bitrrd 307	. . . 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 32718	. 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 207 ∧ wa 396 = wceq 1547 ∈ wcel 2119 ↦ cmpt 5153 I cid 5512 ^◡ccnv 5617 ↾ cres 5620 ∘ ccom 5622 Fn wfn 6480 ⟶wf 6481 –onto→wfo 6483 –1-1-onto→wf1o 6484 (class class class)co 7356 ↑_m cmap 8763

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1917 ax-6 1974 ax-7 2015 ax-8 2121 ax-9 2129 ax-10 2152 ax-11 2168 ax-12 2189 ax-ext 2711 ax-sep 5218 ax-nul 5228 ax-pow 5294 ax-pr 5362 ax-un 7678

This theorem depends on definitions: df-bi 208 df-an 397 df-or 854 df-3an 1094 df-tru 1550 df-fal 1560 df-ex 1787 df-nf 1791 df-sb 2074 df-mo 2543 df-eu 2573 df-clab 2718 df-cleq 2731 df-clel 2814 df-nfc 2888 df-ne 2935 df-ral 3054 df-rex 3064 df-rab 3392 df-v 3433 df-sbc 3724 df-csb 3832 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4262 df-if 4455 df-pw 4531 df-sn 4556 df-pr 4558 df-op 4562 df-uni 4839 df-iun 4923 df-br 5073 df-opab 5135 df-mpt 5154 df-id 5513 df-xp 5624 df-rel 5625 df-cnv 5626 df-co 5627 df-dm 5628 df-rn 5629 df-res 5630 df-ima 5631 df-iota 6441 df-fun 6487 df-fn 6488 df-f 6489 df-f1 6490 df-fo 6491 df-f1o 6492 df-fv 6493 df-ov 7359 df-oprab 7360 df-mpo 7361 df-1st 7931 df-2nd 7932 df-map 8765

This theorem is referenced by: reprpmtf1o 34810

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