Theorem fcobij 33102

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 2766	. 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 6827	. . . . . 6 ⊢ (𝐺:𝑆–1-1-onto→𝑇 → 𝐺:𝑆⟶𝑇)
4	2, 3	syl 18	. . . . 5 ⊢ (𝜑 → 𝐺:𝑆⟶𝑇)
5	4	adantr 486	. . . 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 8846	. . . . 5 ⊢ (𝜑 → (𝑓 ∈ (𝑆 ↑m 𝑅) ↔ 𝑓:𝑅⟶𝑆))
9	8	biimpa 482	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑆 ↑m 𝑅)) → 𝑓:𝑅⟶𝑆)
10		fco 6737	. . . 4 ⊢ ((𝐺:𝑆⟶𝑇 ∧ 𝑓:𝑅⟶𝑆) → (𝐺 ∘ 𝑓):𝑅⟶𝑇)
11	5, 9, 10	syl2anc 596	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑆 ↑m 𝑅)) → (𝐺 ∘ 𝑓):𝑅⟶𝑇)
12		fcobij.4	. . . . 5 ⊢ (𝜑 → 𝑇 ∈ 𝑊)
13	12, 7	elmapd 8846	. . . 4 ⊢ (𝜑 → ((𝐺 ∘ 𝑓) ∈ (𝑇 ↑m 𝑅) ↔ (𝐺 ∘ 𝑓):𝑅⟶𝑇))
14	13	adantr 486	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑆 ↑m 𝑅)) → ((𝐺 ∘ 𝑓) ∈ (𝑇 ↑m 𝑅) ↔ (𝐺 ∘ 𝑓):𝑅⟶𝑇))
15	11, 14	mpbird 260	. 2 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑆 ↑m 𝑅)) → (𝐺 ∘ 𝑓) ∈ (𝑇 ↑m 𝑅))
16		f1ocnv 6840	. . . . . 6 ⊢ (𝐺:𝑆–1-1-onto→𝑇 → ◡𝐺:𝑇–1-1-onto→𝑆)
17		f1of 6827	. . . . . 6 ⊢ (◡𝐺:𝑇–1-1-onto→𝑆 → ◡𝐺:𝑇⟶𝑆)
18	2, 16, 17	3syl 19	. . . . 5 ⊢ (𝜑 → ◡𝐺:𝑇⟶𝑆)
19	18	adantr 486	. . . 4 ⊢ ((𝜑 ∧ ℎ ∈ (𝑇 ↑m 𝑅)) → ◡𝐺:𝑇⟶𝑆)
20	12, 7	elmapd 8846	. . . . 5 ⊢ (𝜑 → (ℎ ∈ (𝑇 ↑m 𝑅) ↔ ℎ:𝑅⟶𝑇))
21	20	biimpa 482	. . . 4 ⊢ ((𝜑 ∧ ℎ ∈ (𝑇 ↑m 𝑅)) → ℎ:𝑅⟶𝑇)
22		fco 6737	. . . 4 ⊢ ((◡𝐺:𝑇⟶𝑆 ∧ ℎ:𝑅⟶𝑇) → (◡𝐺 ∘ ℎ):𝑅⟶𝑆)
23	19, 21, 22	syl2anc 596	. . 3 ⊢ ((𝜑 ∧ ℎ ∈ (𝑇 ↑m 𝑅)) → (◡𝐺 ∘ ℎ):𝑅⟶𝑆)
24	6, 7	elmapd 8846	. . . 4 ⊢ (𝜑 → ((◡𝐺 ∘ ℎ) ∈ (𝑆 ↑m 𝑅) ↔ (◡𝐺 ∘ ℎ):𝑅⟶𝑆))
25	24	adantr 486	. . 3 ⊢ ((𝜑 ∧ ℎ ∈ (𝑇 ↑m 𝑅)) → ((◡𝐺 ∘ ℎ) ∈ (𝑆 ↑m 𝑅) ↔ (◡𝐺 ∘ ℎ):𝑅⟶𝑆))
26	23, 25	mpbird 260	. 2 ⊢ ((𝜑 ∧ ℎ ∈ (𝑇 ↑m 𝑅)) → (◡𝐺 ∘ ℎ) ∈ (𝑆 ↑m 𝑅))
27		simpr 490	. . . . . 6 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝑆 ↑m 𝑅) ∧ ℎ ∈ (𝑇 ↑m 𝑅))) ∧ 𝑓 = (◡𝐺 ∘ ℎ)) → 𝑓 = (◡𝐺 ∘ ℎ))
28	27	coeq2d 5853	. . . . 5 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝑆 ↑m 𝑅) ∧ ℎ ∈ (𝑇 ↑m 𝑅))) ∧ 𝑓 = (◡𝐺 ∘ ℎ)) → (𝐺 ∘ 𝑓) = (𝐺 ∘ (◡𝐺 ∘ ℎ)))
29		coass 6272	. . . . 5 ⊢ ((𝐺 ∘ ◡𝐺) ∘ ℎ) = (𝐺 ∘ (◡𝐺 ∘ ℎ))
30	28, 29	eqtr4di 2819	. . . 4 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝑆 ↑m 𝑅) ∧ ℎ ∈ (𝑇 ↑m 𝑅))) ∧ 𝑓 = (◡𝐺 ∘ ℎ)) → (𝐺 ∘ 𝑓) = ((𝐺 ∘ ◡𝐺) ∘ ℎ))
31		simpll 779	. . . . . 6 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝑆 ↑m 𝑅) ∧ ℎ ∈ (𝑇 ↑m 𝑅))) ∧ 𝑓 = (◡𝐺 ∘ ℎ)) → 𝜑)
32		f1ococnv2 6855	. . . . . 6 ⊢ (𝐺:𝑆–1-1-onto→𝑇 → (𝐺 ∘ ◡𝐺) = ( I ↾ 𝑇))
33	31, 2, 32	3syl 19	. . . . 5 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝑆 ↑m 𝑅) ∧ ℎ ∈ (𝑇 ↑m 𝑅))) ∧ 𝑓 = (◡𝐺 ∘ ℎ)) → (𝐺 ∘ ◡𝐺) = ( I ↾ 𝑇))
34	33	coeq1d 5852	. . . 4 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝑆 ↑m 𝑅) ∧ ℎ ∈ (𝑇 ↑m 𝑅))) ∧ 𝑓 = (◡𝐺 ∘ ℎ)) → ((𝐺 ∘ ◡𝐺) ∘ ℎ) = (( I ↾ 𝑇) ∘ ℎ))
35		simplrr 790	. . . . . 6 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝑆 ↑m 𝑅) ∧ ℎ ∈ (𝑇 ↑m 𝑅))) ∧ 𝑓 = (◡𝐺 ∘ ℎ)) → ℎ ∈ (𝑇 ↑m 𝑅))
36	31, 35, 21	syl2anc 596	. . . . 5 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝑆 ↑m 𝑅) ∧ ℎ ∈ (𝑇 ↑m 𝑅))) ∧ 𝑓 = (◡𝐺 ∘ ℎ)) → ℎ:𝑅⟶𝑇)
37		fcoi2 6760	. . . . 5 ⊢ (ℎ:𝑅⟶𝑇 → (( I ↾ 𝑇) ∘ ℎ) = ℎ)
38	36, 37	syl 18	. . . 4 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝑆 ↑m 𝑅) ∧ ℎ ∈ (𝑇 ↑m 𝑅))) ∧ 𝑓 = (◡𝐺 ∘ ℎ)) → (( I ↾ 𝑇) ∘ ℎ) = ℎ)
39	30, 34, 38	3eqtrrd 2806	. . 3 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝑆 ↑m 𝑅) ∧ ℎ ∈ (𝑇 ↑m 𝑅))) ∧ 𝑓 = (◡𝐺 ∘ ℎ)) → ℎ = (𝐺 ∘ 𝑓))
40		simpr 490	. . . . . 6 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝑆 ↑m 𝑅) ∧ ℎ ∈ (𝑇 ↑m 𝑅))) ∧ ℎ = (𝐺 ∘ 𝑓)) → ℎ = (𝐺 ∘ 𝑓))
41	40	coeq2d 5853	. . . . 5 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝑆 ↑m 𝑅) ∧ ℎ ∈ (𝑇 ↑m 𝑅))) ∧ ℎ = (𝐺 ∘ 𝑓)) → (◡𝐺 ∘ ℎ) = (◡𝐺 ∘ (𝐺 ∘ 𝑓)))
42		coass 6272	. . . . 5 ⊢ ((◡𝐺 ∘ 𝐺) ∘ 𝑓) = (◡𝐺 ∘ (𝐺 ∘ 𝑓))
43	41, 42	eqtr4di 2819	. . . 4 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝑆 ↑m 𝑅) ∧ ℎ ∈ (𝑇 ↑m 𝑅))) ∧ ℎ = (𝐺 ∘ 𝑓)) → (◡𝐺 ∘ ℎ) = ((◡𝐺 ∘ 𝐺) ∘ 𝑓))
44		simpll 779	. . . . . 6 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝑆 ↑m 𝑅) ∧ ℎ ∈ (𝑇 ↑m 𝑅))) ∧ ℎ = (𝐺 ∘ 𝑓)) → 𝜑)
45		f1ococnv1 6857	. . . . . 6 ⊢ (𝐺:𝑆–1-1-onto→𝑇 → (◡𝐺 ∘ 𝐺) = ( I ↾ 𝑆))
46	44, 2, 45	3syl 19	. . . . 5 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝑆 ↑m 𝑅) ∧ ℎ ∈ (𝑇 ↑m 𝑅))) ∧ ℎ = (𝐺 ∘ 𝑓)) → (◡𝐺 ∘ 𝐺) = ( I ↾ 𝑆))
47	46	coeq1d 5852	. . . 4 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝑆 ↑m 𝑅) ∧ ℎ ∈ (𝑇 ↑m 𝑅))) ∧ ℎ = (𝐺 ∘ 𝑓)) → ((◡𝐺 ∘ 𝐺) ∘ 𝑓) = (( I ↾ 𝑆) ∘ 𝑓))
48		simplrl 789	. . . . . 6 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝑆 ↑m 𝑅) ∧ ℎ ∈ (𝑇 ↑m 𝑅))) ∧ ℎ = (𝐺 ∘ 𝑓)) → 𝑓 ∈ (𝑆 ↑m 𝑅))
49	44, 48, 9	syl2anc 596	. . . . 5 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝑆 ↑m 𝑅) ∧ ℎ ∈ (𝑇 ↑m 𝑅))) ∧ ℎ = (𝐺 ∘ 𝑓)) → 𝑓:𝑅⟶𝑆)
50		fcoi2 6760	. . . . 5 ⊢ (𝑓:𝑅⟶𝑆 → (( I ↾ 𝑆) ∘ 𝑓) = 𝑓)
51	49, 50	syl 18	. . . 4 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝑆 ↑m 𝑅) ∧ ℎ ∈ (𝑇 ↑m 𝑅))) ∧ ℎ = (𝐺 ∘ 𝑓)) → (( I ↾ 𝑆) ∘ 𝑓) = 𝑓)
52	43, 47, 51	3eqtrrd 2806	. . 3 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝑆 ↑m 𝑅) ∧ ℎ ∈ (𝑇 ↑m 𝑅))) ∧ ℎ = (𝐺 ∘ 𝑓)) → 𝑓 = (◡𝐺 ∘ ℎ))
53	39, 52	impbida 813	. 2 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝑆 ↑m 𝑅) ∧ ℎ ∈ (𝑇 ↑m 𝑅))) → (𝑓 = (◡𝐺 ∘ ℎ) ↔ ℎ = (𝐺 ∘ 𝑓)))
54	1, 15, 26, 53	f1o2d 7677	1 ⊢ (𝜑 → (𝑓 ∈ (𝑆 ↑m 𝑅) ↦ (𝐺 ∘ 𝑓)):(𝑆 ↑m 𝑅)–1-1-onto→(𝑇 ↑m 𝑅))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146   ↦ cmpt 5197   I cid 5560  ^◡ccnv 5665   ↾ cres 5668   ∘ ccom 5670  ⟶wf 6539  –1-1-onto→wf1o 6542  (class class class)co 7423   ↑_m cmap 8833

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-pow 5341  ax-pr 5409  ax-un 7745

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-sbc 3748  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-pw 4569  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-mpt 5198  df-id 5561  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-iota 6499  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-ov 7426  df-oprab 7427  df-mpo 7428  df-map 8835

This theorem is used by: (None)