Theorem fmptco1f1o 29434

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 477	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓 ∈ 𝐴)
8		fmptco1f1o.a	. . . . . . . 8 ⊢ 𝐴 = (𝑅 ↑𝑚 𝐸)
9	7, 8	syl6eleq 2711	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → 𝑓 ∈ (𝑅 ↑𝑚 𝐸))
10		elmapi 7879	. . . . . . 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 6137	. . . . . . . 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 6058	. . . . . 6 ⊢ ((𝑓:𝐸⟶𝑅 ∧ 𝑇:𝐷⟶𝐸) → (𝑓 ∘ 𝑇):𝐷⟶𝑅)
17	11, 15, 16	syl2anc 693	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → (𝑓 ∘ 𝑇):𝐷⟶𝑅)
18		elmapg 7870	. . . . . 6 ⊢ ((𝑅 ∈ 𝑋 ∧ 𝐷 ∈ 𝑉) → ((𝑓 ∘ 𝑇) ∈ (𝑅 ↑𝑚 𝐷) ↔ (𝑓 ∘ 𝑇):𝐷⟶𝑅))
19	18	biimpar 502	. . . . 5 ⊢ (((𝑅 ∈ 𝑋 ∧ 𝐷 ∈ 𝑉) ∧ (𝑓 ∘ 𝑇):𝐷⟶𝑅) → (𝑓 ∘ 𝑇) ∈ (𝑅 ↑𝑚 𝐷))
20	4, 6, 17, 19	syl21anc 1325	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝐴) → (𝑓 ∘ 𝑇) ∈ (𝑅 ↑𝑚 𝐷))
21		fmptco1f1o.b	. . . 4 ⊢ 𝐵 = (𝑅 ↑𝑚 𝐷)
22	20, 21	syl6eleqr 2712	. . 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 477	. . . . . . . 8 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐵) → 𝑔 ∈ 𝐵)
27	26, 21	syl6eleq 2711	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐵) → 𝑔 ∈ (𝑅 ↑𝑚 𝐷))
28		elmapi 7879	. . . . . . 7 ⊢ (𝑔 ∈ (𝑅 ↑𝑚 𝐷) → 𝑔:𝐷⟶𝑅)
29	27, 28	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐵) → 𝑔:𝐷⟶𝑅)
30		f1ocnv 6149	. . . . . . . 8 ⊢ (𝑇:𝐷–1-1-onto→𝐸 → ◡𝑇:𝐸–1-1-onto→𝐷)
31		f1of 6137	. . . . . . . 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 6058	. . . . . 6 ⊢ ((𝑔:𝐷⟶𝑅 ∧ ◡𝑇:𝐸⟶𝐷) → (𝑔 ∘ ◡𝑇):𝐸⟶𝑅)
35	29, 33, 34	syl2anc 693	. . . . 5 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐵) → (𝑔 ∘ ◡𝑇):𝐸⟶𝑅)
36		elmapg 7870	. . . . . 6 ⊢ ((𝑅 ∈ 𝑋 ∧ 𝐸 ∈ 𝑊) → ((𝑔 ∘ ◡𝑇) ∈ (𝑅 ↑𝑚 𝐸) ↔ (𝑔 ∘ ◡𝑇):𝐸⟶𝑅))
37	36	biimpar 502	. . . . 5 ⊢ (((𝑅 ∈ 𝑋 ∧ 𝐸 ∈ 𝑊) ∧ (𝑔 ∘ ◡𝑇):𝐸⟶𝑅) → (𝑔 ∘ ◡𝑇) ∈ (𝑅 ↑𝑚 𝐸))
38	23, 25, 35, 37	syl21anc 1325	. . . 4 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐵) → (𝑔 ∘ ◡𝑇) ∈ (𝑅 ↑𝑚 𝐸))
39	38, 8	syl6eleqr 2712	. . 3 ⊢ ((𝜑 ∧ 𝑔 ∈ 𝐵) → (𝑔 ∘ ◡𝑇) ∈ 𝐴)
40		coass 5654	. . . . . . 7 ⊢ ((𝑔 ∘ ◡𝑇) ∘ 𝑇) = (𝑔 ∘ (◡𝑇 ∘ 𝑇))
41	12	ad2antrr 762	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐴) ∧ 𝑔 ∈ 𝐵) → 𝑇:𝐷–1-1-onto→𝐸)
42		f1ococnv1 6165	. . . . . . . . . 10 ⊢ (𝑇:𝐷–1-1-onto→𝐸 → (◡𝑇 ∘ 𝑇) = ( I ↾ 𝐷))
43	42	coeq2d 5284	. . . . . . . . 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 751	. . . . . . . . 9 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐴) ∧ 𝑔 ∈ 𝐵) → 𝑔:𝐷⟶𝑅)
46		fcoi1 6078	. . . . . . . . 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 2656	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐴) ∧ 𝑔 ∈ 𝐵) → (𝑔 ∘ (◡𝑇 ∘ 𝑇)) = 𝑔)
49	40, 48	syl5req 2669	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐴) ∧ 𝑔 ∈ 𝐵) → 𝑔 = ((𝑔 ∘ ◡𝑇) ∘ 𝑇))
50	49	eqeq1d 2624	. . . . 5 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐴) ∧ 𝑔 ∈ 𝐵) → (𝑔 = (𝑓 ∘ 𝑇) ↔ ((𝑔 ∘ ◡𝑇) ∘ 𝑇) = (𝑓 ∘ 𝑇)))
51		eqcom 2629	. . . . . 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 6144	. . . . . . 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 792	. . . . . . . 8 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐴) ∧ 𝑔 ∈ 𝐵) → 𝑓 ∈ 𝐴)
56	55, 8	syl6eleq 2711	. . . . . . 7 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐴) ∧ 𝑔 ∈ 𝐵) → 𝑓 ∈ (𝑅 ↑𝑚 𝐸))
57		elmapfn 7880	. . . . . . 7 ⊢ (𝑓 ∈ (𝑅 ↑𝑚 𝐸) → 𝑓 Fn 𝐸)
58	56, 57	syl 17	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐴) ∧ 𝑔 ∈ 𝐵) → 𝑓 Fn 𝐸)
59		elmapfn 7880	. . . . . . . 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 751	. . . . . 6 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐴) ∧ 𝑔 ∈ 𝐵) → (𝑔 ∘ ◡𝑇) Fn 𝐸)
62		cocan2 6547	. . . . . 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 1326	. . . . 5 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐴) ∧ 𝑔 ∈ 𝐵) → ((𝑓 ∘ 𝑇) = ((𝑔 ∘ ◡𝑇) ∘ 𝑇) ↔ 𝑓 = (𝑔 ∘ ◡𝑇)))
64	50, 52, 63	3bitrrd 295	. . . 4 ⊢ (((𝜑 ∧ 𝑓 ∈ 𝐴) ∧ 𝑔 ∈ 𝐵) → (𝑓 = (𝑔 ∘ ◡𝑇) ↔ 𝑔 = (𝑓 ∘ 𝑇)))
65	64	anasss 679	. . 3 ⊢ ((𝜑 ∧ (𝑓 ∈ 𝐴 ∧ 𝑔 ∈ 𝐵)) → (𝑓 = (𝑔 ∘ ◡𝑇) ↔ 𝑔 = (𝑓 ∘ 𝑇)))
66	2, 22, 39, 65	f1o3d 29431	. 2 ⊢ (𝜑 → (𝐹:𝐴–1-1-onto→𝐵 ∧ ◡𝐹 = (𝑔 ∈ 𝐵 ↦ (𝑔 ∘ ◡𝑇))))
67	66	simpld 475	1 ⊢ (𝜑 → 𝐹:𝐴–1-1-onto→𝐵)

Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 384   = wceq 1483   ∈ wcel 1990   ↦ cmpt 4729   I cid 5023  ^◡ccnv 5113   ↾ cres 5116   ∘ ccom 5118   Fn wfn 5883  ⟶wf 5884  –onto→wfo 5886  –1-1-onto→wf1o 5887  (class class class)co 6650   ↑_𝑚 cmap 7857

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-map 7859

This theorem is referenced by:  reprpmtf1o  30704