Theorem fcobij 31942

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 2732	. 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 6833	. . . . . 6 ⊢ (𝐺:𝑆–1-1-onto→𝑇 → 𝐺:𝑆⟶𝑇)
4	2, 3	syl 17	. . . . 5 ⊢ (𝜑 → 𝐺:𝑆⟶𝑇)
5	4	adantr 481	. . . 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 8833	. . . . 5 ⊢ (𝜑 → (𝑓 ∈ (𝑆 ↑m 𝑅) ↔ 𝑓:𝑅⟶𝑆))
9	8	biimpa 477	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑆 ↑m 𝑅)) → 𝑓:𝑅⟶𝑆)
10		fco 6741	. . . 4 ⊢ ((𝐺:𝑆⟶𝑇 ∧ 𝑓:𝑅⟶𝑆) → (𝐺 ∘ 𝑓):𝑅⟶𝑇)
11	5, 9, 10	syl2anc 584	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑆 ↑m 𝑅)) → (𝐺 ∘ 𝑓):𝑅⟶𝑇)
12		fcobij.4	. . . . 5 ⊢ (𝜑 → 𝑇 ∈ 𝑊)
13	12, 7	elmapd 8833	. . . 4 ⊢ (𝜑 → ((𝐺 ∘ 𝑓) ∈ (𝑇 ↑m 𝑅) ↔ (𝐺 ∘ 𝑓):𝑅⟶𝑇))
14	13	adantr 481	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑆 ↑m 𝑅)) → ((𝐺 ∘ 𝑓) ∈ (𝑇 ↑m 𝑅) ↔ (𝐺 ∘ 𝑓):𝑅⟶𝑇))
15	11, 14	mpbird 256	. 2 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑆 ↑m 𝑅)) → (𝐺 ∘ 𝑓) ∈ (𝑇 ↑m 𝑅))
16		f1ocnv 6845	. . . . . 6 ⊢ (𝐺:𝑆–1-1-onto→𝑇 → ◡𝐺:𝑇–1-1-onto→𝑆)
17		f1of 6833	. . . . . 6 ⊢ (◡𝐺:𝑇–1-1-onto→𝑆 → ◡𝐺:𝑇⟶𝑆)
18	2, 16, 17	3syl 18	. . . . 5 ⊢ (𝜑 → ◡𝐺:𝑇⟶𝑆)
19	18	adantr 481	. . . 4 ⊢ ((𝜑 ∧ ℎ ∈ (𝑇 ↑m 𝑅)) → ◡𝐺:𝑇⟶𝑆)
20	12, 7	elmapd 8833	. . . . 5 ⊢ (𝜑 → (ℎ ∈ (𝑇 ↑m 𝑅) ↔ ℎ:𝑅⟶𝑇))
21	20	biimpa 477	. . . 4 ⊢ ((𝜑 ∧ ℎ ∈ (𝑇 ↑m 𝑅)) → ℎ:𝑅⟶𝑇)
22		fco 6741	. . . 4 ⊢ ((◡𝐺:𝑇⟶𝑆 ∧ ℎ:𝑅⟶𝑇) → (◡𝐺 ∘ ℎ):𝑅⟶𝑆)
23	19, 21, 22	syl2anc 584	. . 3 ⊢ ((𝜑 ∧ ℎ ∈ (𝑇 ↑m 𝑅)) → (◡𝐺 ∘ ℎ):𝑅⟶𝑆)
24	6, 7	elmapd 8833	. . . 4 ⊢ (𝜑 → ((◡𝐺 ∘ ℎ) ∈ (𝑆 ↑m 𝑅) ↔ (◡𝐺 ∘ ℎ):𝑅⟶𝑆))
25	24	adantr 481	. . 3 ⊢ ((𝜑 ∧ ℎ ∈ (𝑇 ↑m 𝑅)) → ((◡𝐺 ∘ ℎ) ∈ (𝑆 ↑m 𝑅) ↔ (◡𝐺 ∘ ℎ):𝑅⟶𝑆))
26	23, 25	mpbird 256	. 2 ⊢ ((𝜑 ∧ ℎ ∈ (𝑇 ↑m 𝑅)) → (◡𝐺 ∘ ℎ) ∈ (𝑆 ↑m 𝑅))
27		simpr 485	. . . . . 6 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝑆 ↑m 𝑅) ∧ ℎ ∈ (𝑇 ↑m 𝑅))) ∧ 𝑓 = (◡𝐺 ∘ ℎ)) → 𝑓 = (◡𝐺 ∘ ℎ))
28	27	coeq2d 5862	. . . . 5 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝑆 ↑m 𝑅) ∧ ℎ ∈ (𝑇 ↑m 𝑅))) ∧ 𝑓 = (◡𝐺 ∘ ℎ)) → (𝐺 ∘ 𝑓) = (𝐺 ∘ (◡𝐺 ∘ ℎ)))
29		coass 6264	. . . . 5 ⊢ ((𝐺 ∘ ◡𝐺) ∘ ℎ) = (𝐺 ∘ (◡𝐺 ∘ ℎ))
30	28, 29	eqtr4di 2790	. . . 4 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝑆 ↑m 𝑅) ∧ ℎ ∈ (𝑇 ↑m 𝑅))) ∧ 𝑓 = (◡𝐺 ∘ ℎ)) → (𝐺 ∘ 𝑓) = ((𝐺 ∘ ◡𝐺) ∘ ℎ))
31		simpll 765	. . . . . 6 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝑆 ↑m 𝑅) ∧ ℎ ∈ (𝑇 ↑m 𝑅))) ∧ 𝑓 = (◡𝐺 ∘ ℎ)) → 𝜑)
32		f1ococnv2 6860	. . . . . 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 5861	. . . 4 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝑆 ↑m 𝑅) ∧ ℎ ∈ (𝑇 ↑m 𝑅))) ∧ 𝑓 = (◡𝐺 ∘ ℎ)) → ((𝐺 ∘ ◡𝐺) ∘ ℎ) = (( I ↾ 𝑇) ∘ ℎ))
35		simplrr 776	. . . . . 6 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝑆 ↑m 𝑅) ∧ ℎ ∈ (𝑇 ↑m 𝑅))) ∧ 𝑓 = (◡𝐺 ∘ ℎ)) → ℎ ∈ (𝑇 ↑m 𝑅))
36	31, 35, 21	syl2anc 584	. . . . 5 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝑆 ↑m 𝑅) ∧ ℎ ∈ (𝑇 ↑m 𝑅))) ∧ 𝑓 = (◡𝐺 ∘ ℎ)) → ℎ:𝑅⟶𝑇)
37		fcoi2 6766	. . . . 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 2777	. . 3 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝑆 ↑m 𝑅) ∧ ℎ ∈ (𝑇 ↑m 𝑅))) ∧ 𝑓 = (◡𝐺 ∘ ℎ)) → ℎ = (𝐺 ∘ 𝑓))
40		simpr 485	. . . . . 6 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝑆 ↑m 𝑅) ∧ ℎ ∈ (𝑇 ↑m 𝑅))) ∧ ℎ = (𝐺 ∘ 𝑓)) → ℎ = (𝐺 ∘ 𝑓))
41	40	coeq2d 5862	. . . . 5 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝑆 ↑m 𝑅) ∧ ℎ ∈ (𝑇 ↑m 𝑅))) ∧ ℎ = (𝐺 ∘ 𝑓)) → (◡𝐺 ∘ ℎ) = (◡𝐺 ∘ (𝐺 ∘ 𝑓)))
42		coass 6264	. . . . 5 ⊢ ((◡𝐺 ∘ 𝐺) ∘ 𝑓) = (◡𝐺 ∘ (𝐺 ∘ 𝑓))
43	41, 42	eqtr4di 2790	. . . 4 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝑆 ↑m 𝑅) ∧ ℎ ∈ (𝑇 ↑m 𝑅))) ∧ ℎ = (𝐺 ∘ 𝑓)) → (◡𝐺 ∘ ℎ) = ((◡𝐺 ∘ 𝐺) ∘ 𝑓))
44		simpll 765	. . . . . 6 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝑆 ↑m 𝑅) ∧ ℎ ∈ (𝑇 ↑m 𝑅))) ∧ ℎ = (𝐺 ∘ 𝑓)) → 𝜑)
45		f1ococnv1 6862	. . . . . 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 5861	. . . 4 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝑆 ↑m 𝑅) ∧ ℎ ∈ (𝑇 ↑m 𝑅))) ∧ ℎ = (𝐺 ∘ 𝑓)) → ((◡𝐺 ∘ 𝐺) ∘ 𝑓) = (( I ↾ 𝑆) ∘ 𝑓))
48		simplrl 775	. . . . . 6 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝑆 ↑m 𝑅) ∧ ℎ ∈ (𝑇 ↑m 𝑅))) ∧ ℎ = (𝐺 ∘ 𝑓)) → 𝑓 ∈ (𝑆 ↑m 𝑅))
49	44, 48, 9	syl2anc 584	. . . . 5 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝑆 ↑m 𝑅) ∧ ℎ ∈ (𝑇 ↑m 𝑅))) ∧ ℎ = (𝐺 ∘ 𝑓)) → 𝑓:𝑅⟶𝑆)
50		fcoi2 6766	. . . . 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 2777	. . 3 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝑆 ↑m 𝑅) ∧ ℎ ∈ (𝑇 ↑m 𝑅))) ∧ ℎ = (𝐺 ∘ 𝑓)) → 𝑓 = (◡𝐺 ∘ ℎ))
53	39, 52	impbida 799	. 2 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝑆 ↑m 𝑅) ∧ ℎ ∈ (𝑇 ↑m 𝑅))) → (𝑓 = (◡𝐺 ∘ ℎ) ↔ ℎ = (𝐺 ∘ 𝑓)))
54	1, 15, 26, 53	f1o2d 7659	1 ⊢ (𝜑 → (𝑓 ∈ (𝑆 ↑m 𝑅) ↦ (𝐺 ∘ 𝑓)):(𝑆 ↑m 𝑅)–1-1-onto→(𝑇 ↑m 𝑅))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1541   ∈ wcel 2106   ↦ cmpt 5231   I cid 5573  ^◡ccnv 5675   ↾ cres 5678   ∘ ccom 5680  ⟶wf 6539  –1-1-onto→wf1o 6542  (class class class)co 7408   ↑_m cmap 8819

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  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 2703  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7724

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-nfc 2885  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-sbc 3778  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-br 5149  df-opab 5211  df-mpt 5232  df-id 5574  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-iota 6495  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7411  df-oprab 7412  df-mpo 7413  df-map 8821

This theorem is referenced by: (None)