Theorem fcobij 30047

Description: Composing functions with a bijection yields a bijection between sets of functions. (Contributed by Thierry Arnoux, 25-Aug-2017.)

Hypotheses

Ref	Expression
fcobij.1	⊢ (𝜑 → 𝐺:𝑆–1-1-onto→𝑇)
fcobij.2	⊢ (𝜑 → 𝑅 ∈ 𝑈)
fcobij.3	⊢ (𝜑 → 𝑆 ∈ 𝑉)
fcobij.4	⊢ (𝜑 → 𝑇 ∈ 𝑊)

Assertion

Ref	Expression
fcobij	⊢ (𝜑 → (𝑓 ∈ (𝑆 ↑𝑚 𝑅) ↦ (𝐺 ∘ 𝑓)):(𝑆 ↑𝑚 𝑅)–1-1-onto→(𝑇 ↑𝑚 𝑅))

Distinct variable groups:   𝑓,𝐺   𝑅,𝑓   𝑆,𝑓   𝑇,𝑓   𝜑,𝑓

Allowed substitution hints:   𝑈(𝑓)   𝑉(𝑓)   𝑊(𝑓)

Proof of Theorem fcobij

Dummy variable ℎ is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eqid 2824	. 2 ⊢ (𝑓 ∈ (𝑆 ↑𝑚 𝑅) ↦ (𝐺 ∘ 𝑓)) = (𝑓 ∈ (𝑆 ↑𝑚 𝑅) ↦ (𝐺 ∘ 𝑓))
2		fcobij.1	. . . . . 6 ⊢ (𝜑 → 𝐺:𝑆–1-1-onto→𝑇)
3		f1of 6377	. . . . . 6 ⊢ (𝐺:𝑆–1-1-onto→𝑇 → 𝐺:𝑆⟶𝑇)
4	2, 3	syl 17	. . . . 5 ⊢ (𝜑 → 𝐺:𝑆⟶𝑇)
5	4	adantr 474	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑆 ↑𝑚 𝑅)) → 𝐺:𝑆⟶𝑇)
6		fcobij.3	. . . . . 6 ⊢ (𝜑 → 𝑆 ∈ 𝑉)
7		fcobij.2	. . . . . 6 ⊢ (𝜑 → 𝑅 ∈ 𝑈)
8	6, 7	elmapd 8135	. . . . 5 ⊢ (𝜑 → (𝑓 ∈ (𝑆 ↑𝑚 𝑅) ↔ 𝑓:𝑅⟶𝑆))
9	8	biimpa 470	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑆 ↑𝑚 𝑅)) → 𝑓:𝑅⟶𝑆)
10		fco 6294	. . . 4 ⊢ ((𝐺:𝑆⟶𝑇 ∧ 𝑓:𝑅⟶𝑆) → (𝐺 ∘ 𝑓):𝑅⟶𝑇)
11	5, 9, 10	syl2anc 581	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑆 ↑𝑚 𝑅)) → (𝐺 ∘ 𝑓):𝑅⟶𝑇)
12		fcobij.4	. . . . 5 ⊢ (𝜑 → 𝑇 ∈ 𝑊)
13	12, 7	elmapd 8135	. . . 4 ⊢ (𝜑 → ((𝐺 ∘ 𝑓) ∈ (𝑇 ↑𝑚 𝑅) ↔ (𝐺 ∘ 𝑓):𝑅⟶𝑇))
14	13	adantr 474	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑆 ↑𝑚 𝑅)) → ((𝐺 ∘ 𝑓) ∈ (𝑇 ↑𝑚 𝑅) ↔ (𝐺 ∘ 𝑓):𝑅⟶𝑇))
15	11, 14	mpbird 249	. 2 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑆 ↑𝑚 𝑅)) → (𝐺 ∘ 𝑓) ∈ (𝑇 ↑𝑚 𝑅))
16		f1ocnv 6389	. . . . . 6 ⊢ (𝐺:𝑆–1-1-onto→𝑇 → ◡𝐺:𝑇–1-1-onto→𝑆)
17		f1of 6377	. . . . . 6 ⊢ (◡𝐺:𝑇–1-1-onto→𝑆 → ◡𝐺:𝑇⟶𝑆)
18	2, 16, 17	3syl 18	. . . . 5 ⊢ (𝜑 → ◡𝐺:𝑇⟶𝑆)
19	18	adantr 474	. . . 4 ⊢ ((𝜑 ∧ ℎ ∈ (𝑇 ↑𝑚 𝑅)) → ◡𝐺:𝑇⟶𝑆)
20	12, 7	elmapd 8135	. . . . 5 ⊢ (𝜑 → (ℎ ∈ (𝑇 ↑𝑚 𝑅) ↔ ℎ:𝑅⟶𝑇))
21	20	biimpa 470	. . . 4 ⊢ ((𝜑 ∧ ℎ ∈ (𝑇 ↑𝑚 𝑅)) → ℎ:𝑅⟶𝑇)
22		fco 6294	. . . 4 ⊢ ((◡𝐺:𝑇⟶𝑆 ∧ ℎ:𝑅⟶𝑇) → (◡𝐺 ∘ ℎ):𝑅⟶𝑆)
23	19, 21, 22	syl2anc 581	. . 3 ⊢ ((𝜑 ∧ ℎ ∈ (𝑇 ↑𝑚 𝑅)) → (◡𝐺 ∘ ℎ):𝑅⟶𝑆)
24	6, 7	elmapd 8135	. . . 4 ⊢ (𝜑 → ((◡𝐺 ∘ ℎ) ∈ (𝑆 ↑𝑚 𝑅) ↔ (◡𝐺 ∘ ℎ):𝑅⟶𝑆))
25	24	adantr 474	. . 3 ⊢ ((𝜑 ∧ ℎ ∈ (𝑇 ↑𝑚 𝑅)) → ((◡𝐺 ∘ ℎ) ∈ (𝑆 ↑𝑚 𝑅) ↔ (◡𝐺 ∘ ℎ):𝑅⟶𝑆))
26	23, 25	mpbird 249	. 2 ⊢ ((𝜑 ∧ ℎ ∈ (𝑇 ↑𝑚 𝑅)) → (◡𝐺 ∘ ℎ) ∈ (𝑆 ↑𝑚 𝑅))
27		simpr 479	. . . . . 6 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝑆 ↑𝑚 𝑅) ∧ ℎ ∈ (𝑇 ↑𝑚 𝑅))) ∧ 𝑓 = (◡𝐺 ∘ ℎ)) → 𝑓 = (◡𝐺 ∘ ℎ))
28	27	coeq2d 5516	. . . . 5 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝑆 ↑𝑚 𝑅) ∧ ℎ ∈ (𝑇 ↑𝑚 𝑅))) ∧ 𝑓 = (◡𝐺 ∘ ℎ)) → (𝐺 ∘ 𝑓) = (𝐺 ∘ (◡𝐺 ∘ ℎ)))
29		coass 5894	. . . . 5 ⊢ ((𝐺 ∘ ◡𝐺) ∘ ℎ) = (𝐺 ∘ (◡𝐺 ∘ ℎ))
30	28, 29	syl6eqr 2878	. . . 4 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝑆 ↑𝑚 𝑅) ∧ ℎ ∈ (𝑇 ↑𝑚 𝑅))) ∧ 𝑓 = (◡𝐺 ∘ ℎ)) → (𝐺 ∘ 𝑓) = ((𝐺 ∘ ◡𝐺) ∘ ℎ))
31		simpll 785	. . . . . 6 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝑆 ↑𝑚 𝑅) ∧ ℎ ∈ (𝑇 ↑𝑚 𝑅))) ∧ 𝑓 = (◡𝐺 ∘ ℎ)) → 𝜑)
32		f1ococnv2 6403	. . . . . 6 ⊢ (𝐺:𝑆–1-1-onto→𝑇 → (𝐺 ∘ ◡𝐺) = ( I ↾ 𝑇))
33	31, 2, 32	3syl 18	. . . . 5 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝑆 ↑𝑚 𝑅) ∧ ℎ ∈ (𝑇 ↑𝑚 𝑅))) ∧ 𝑓 = (◡𝐺 ∘ ℎ)) → (𝐺 ∘ ◡𝐺) = ( I ↾ 𝑇))
34	33	coeq1d 5515	. . . 4 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝑆 ↑𝑚 𝑅) ∧ ℎ ∈ (𝑇 ↑𝑚 𝑅))) ∧ 𝑓 = (◡𝐺 ∘ ℎ)) → ((𝐺 ∘ ◡𝐺) ∘ ℎ) = (( I ↾ 𝑇) ∘ ℎ))
35		simplrr 798	. . . . . 6 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝑆 ↑𝑚 𝑅) ∧ ℎ ∈ (𝑇 ↑𝑚 𝑅))) ∧ 𝑓 = (◡𝐺 ∘ ℎ)) → ℎ ∈ (𝑇 ↑𝑚 𝑅))
36	31, 35, 21	syl2anc 581	. . . . 5 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝑆 ↑𝑚 𝑅) ∧ ℎ ∈ (𝑇 ↑𝑚 𝑅))) ∧ 𝑓 = (◡𝐺 ∘ ℎ)) → ℎ:𝑅⟶𝑇)
37		fcoi2 6315	. . . . 5 ⊢ (ℎ:𝑅⟶𝑇 → (( I ↾ 𝑇) ∘ ℎ) = ℎ)
38	36, 37	syl 17	. . . 4 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝑆 ↑𝑚 𝑅) ∧ ℎ ∈ (𝑇 ↑𝑚 𝑅))) ∧ 𝑓 = (◡𝐺 ∘ ℎ)) → (( I ↾ 𝑇) ∘ ℎ) = ℎ)
39	30, 34, 38	3eqtrrd 2865	. . 3 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝑆 ↑𝑚 𝑅) ∧ ℎ ∈ (𝑇 ↑𝑚 𝑅))) ∧ 𝑓 = (◡𝐺 ∘ ℎ)) → ℎ = (𝐺 ∘ 𝑓))
40		simpr 479	. . . . . 6 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝑆 ↑𝑚 𝑅) ∧ ℎ ∈ (𝑇 ↑𝑚 𝑅))) ∧ ℎ = (𝐺 ∘ 𝑓)) → ℎ = (𝐺 ∘ 𝑓))
41	40	coeq2d 5516	. . . . 5 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝑆 ↑𝑚 𝑅) ∧ ℎ ∈ (𝑇 ↑𝑚 𝑅))) ∧ ℎ = (𝐺 ∘ 𝑓)) → (◡𝐺 ∘ ℎ) = (◡𝐺 ∘ (𝐺 ∘ 𝑓)))
42		coass 5894	. . . . 5 ⊢ ((◡𝐺 ∘ 𝐺) ∘ 𝑓) = (◡𝐺 ∘ (𝐺 ∘ 𝑓))
43	41, 42	syl6eqr 2878	. . . 4 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝑆 ↑𝑚 𝑅) ∧ ℎ ∈ (𝑇 ↑𝑚 𝑅))) ∧ ℎ = (𝐺 ∘ 𝑓)) → (◡𝐺 ∘ ℎ) = ((◡𝐺 ∘ 𝐺) ∘ 𝑓))
44		simpll 785	. . . . . 6 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝑆 ↑𝑚 𝑅) ∧ ℎ ∈ (𝑇 ↑𝑚 𝑅))) ∧ ℎ = (𝐺 ∘ 𝑓)) → 𝜑)
45		f1ococnv1 6405	. . . . . 6 ⊢ (𝐺:𝑆–1-1-onto→𝑇 → (◡𝐺 ∘ 𝐺) = ( I ↾ 𝑆))
46	44, 2, 45	3syl 18	. . . . 5 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝑆 ↑𝑚 𝑅) ∧ ℎ ∈ (𝑇 ↑𝑚 𝑅))) ∧ ℎ = (𝐺 ∘ 𝑓)) → (◡𝐺 ∘ 𝐺) = ( I ↾ 𝑆))
47	46	coeq1d 5515	. . . 4 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝑆 ↑𝑚 𝑅) ∧ ℎ ∈ (𝑇 ↑𝑚 𝑅))) ∧ ℎ = (𝐺 ∘ 𝑓)) → ((◡𝐺 ∘ 𝐺) ∘ 𝑓) = (( I ↾ 𝑆) ∘ 𝑓))
48		simplrl 797	. . . . . 6 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝑆 ↑𝑚 𝑅) ∧ ℎ ∈ (𝑇 ↑𝑚 𝑅))) ∧ ℎ = (𝐺 ∘ 𝑓)) → 𝑓 ∈ (𝑆 ↑𝑚 𝑅))
49	44, 48, 9	syl2anc 581	. . . . 5 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝑆 ↑𝑚 𝑅) ∧ ℎ ∈ (𝑇 ↑𝑚 𝑅))) ∧ ℎ = (𝐺 ∘ 𝑓)) → 𝑓:𝑅⟶𝑆)
50		fcoi2 6315	. . . . 5 ⊢ (𝑓:𝑅⟶𝑆 → (( I ↾ 𝑆) ∘ 𝑓) = 𝑓)
51	49, 50	syl 17	. . . 4 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝑆 ↑𝑚 𝑅) ∧ ℎ ∈ (𝑇 ↑𝑚 𝑅))) ∧ ℎ = (𝐺 ∘ 𝑓)) → (( I ↾ 𝑆) ∘ 𝑓) = 𝑓)
52	43, 47, 51	3eqtrrd 2865	. . 3 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝑆 ↑𝑚 𝑅) ∧ ℎ ∈ (𝑇 ↑𝑚 𝑅))) ∧ ℎ = (𝐺 ∘ 𝑓)) → 𝑓 = (◡𝐺 ∘ ℎ))
53	39, 52	impbida 837	. 2 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝑆 ↑𝑚 𝑅) ∧ ℎ ∈ (𝑇 ↑𝑚 𝑅))) → (𝑓 = (◡𝐺 ∘ ℎ) ↔ ℎ = (𝐺 ∘ 𝑓)))
54	1, 15, 26, 53	f1o2d 7146	1 ⊢ (𝜑 → (𝑓 ∈ (𝑆 ↑𝑚 𝑅) ↦ (𝐺 ∘ 𝑓)):(𝑆 ↑𝑚 𝑅)–1-1-onto→(𝑇 ↑𝑚 𝑅))

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 386   = wceq 1658   ∈ wcel 2166   ↦ cmpt 4951   I cid 5248  ^◡ccnv 5340   ↾ cres 5343   ∘ ccom 5345  ⟶wf 6118  –1-1-onto→wf1o 6121  (class class class)co 6904   ↑_𝑚 cmap 8121

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1896  ax-4 1910  ax-5 2011  ax-6 2077  ax-7 2114  ax-8 2168  ax-9 2175  ax-10 2194  ax-11 2209  ax-12 2222  ax-13 2390  ax-ext 2802  ax-sep 5004  ax-nul 5012  ax-pow 5064  ax-pr 5126  ax-un 7208

This theorem depends on definitions:  df-bi 199  df-an 387  df-or 881  df-3an 1115  df-tru 1662  df-ex 1881  df-nf 1885  df-sb 2070  df-mo 2604  df-eu 2639  df-clab 2811  df-cleq 2817  df-clel 2820  df-nfc 2957  df-ral 3121  df-rex 3122  df-rab 3125  df-v 3415  df-sbc 3662  df-dif 3800  df-un 3802  df-in 3804  df-ss 3811  df-nul 4144  df-if 4306  df-pw 4379  df-sn 4397  df-pr 4399  df-op 4403  df-uni 4658  df-br 4873  df-opab 4935  df-mpt 4952  df-id 5249  df-xp 5347  df-rel 5348  df-cnv 5349  df-co 5350  df-dm 5351  df-rn 5352  df-res 5353  df-iota 6085  df-fun 6124  df-fn 6125  df-f 6126  df-f1 6127  df-fo 6128  df-f1o 6129  df-fv 6130  df-ov 6907  df-oprab 6908  df-mpt2 6909  df-map 8123

This theorem is referenced by: (None)