fcobij - Mathbox for Thierry Arnoux

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Theorem fcobij 32872

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 2761	. 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 6802	. . . . . 6 ⊢ (𝐺:𝑆–1-1-onto→𝑇 → 𝐺:𝑆⟶𝑇)
4	2, 3	syl 17	. . . . 5 ⊢ (𝜑 → 𝐺:𝑆⟶𝑇)
5	4	adantr 484	. . . 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 8817	. . . . 5 ⊢ (𝜑 → (𝑓 ∈ (𝑆 ↑_m 𝑅) ↔ 𝑓:𝑅⟶𝑆))
9	8	biimpa 480	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑆 ↑_m 𝑅)) → 𝑓:𝑅⟶𝑆)
10		fco 6712	. . . 4 ⊢ ((𝐺:𝑆⟶𝑇 ∧ 𝑓:𝑅⟶𝑆) → (𝐺 ∘ 𝑓):𝑅⟶𝑇)
11	5, 9, 10	syl2anc 593	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑆 ↑_m 𝑅)) → (𝐺 ∘ 𝑓):𝑅⟶𝑇)
12		fcobij.4	. . . . 5 ⊢ (𝜑 → 𝑇 ∈ 𝑊)
13	12, 7	elmapd 8817	. . . 4 ⊢ (𝜑 → ((𝐺 ∘ 𝑓) ∈ (𝑇 ↑_m 𝑅) ↔ (𝐺 ∘ 𝑓):𝑅⟶𝑇))
14	13	adantr 484	. . 3 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑆 ↑_m 𝑅)) → ((𝐺 ∘ 𝑓) ∈ (𝑇 ↑_m 𝑅) ↔ (𝐺 ∘ 𝑓):𝑅⟶𝑇))
15	11, 14	mpbird 259	. 2 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑆 ↑_m 𝑅)) → (𝐺 ∘ 𝑓) ∈ (𝑇 ↑_m 𝑅))
16		f1ocnv 6815	. . . . . 6 ⊢ (𝐺:𝑆–1-1-onto→𝑇 → ^◡𝐺:𝑇–1-1-onto→𝑆)
17		f1of 6802	. . . . . 6 ⊢ (^◡𝐺:𝑇–1-1-onto→𝑆 → ^◡𝐺:𝑇⟶𝑆)
18	2, 16, 17	3syl 18	. . . . 5 ⊢ (𝜑 → ^◡𝐺:𝑇⟶𝑆)
19	18	adantr 484	. . . 4 ⊢ ((𝜑 ∧ ℎ ∈ (𝑇 ↑_m 𝑅)) → ^◡𝐺:𝑇⟶𝑆)
20	12, 7	elmapd 8817	. . . . 5 ⊢ (𝜑 → (ℎ ∈ (𝑇 ↑_m 𝑅) ↔ ℎ:𝑅⟶𝑇))
21	20	biimpa 480	. . . 4 ⊢ ((𝜑 ∧ ℎ ∈ (𝑇 ↑_m 𝑅)) → ℎ:𝑅⟶𝑇)
22		fco 6712	. . . 4 ⊢ ((^◡𝐺:𝑇⟶𝑆 ∧ ℎ:𝑅⟶𝑇) → (^◡𝐺 ∘ ℎ):𝑅⟶𝑆)
23	19, 21, 22	syl2anc 593	. . 3 ⊢ ((𝜑 ∧ ℎ ∈ (𝑇 ↑_m 𝑅)) → (^◡𝐺 ∘ ℎ):𝑅⟶𝑆)
24	6, 7	elmapd 8817	. . . 4 ⊢ (𝜑 → ((^◡𝐺 ∘ ℎ) ∈ (𝑆 ↑_m 𝑅) ↔ (^◡𝐺 ∘ ℎ):𝑅⟶𝑆))
25	24	adantr 484	. . 3 ⊢ ((𝜑 ∧ ℎ ∈ (𝑇 ↑_m 𝑅)) → ((^◡𝐺 ∘ ℎ) ∈ (𝑆 ↑_m 𝑅) ↔ (^◡𝐺 ∘ ℎ):𝑅⟶𝑆))
26	23, 25	mpbird 259	. 2 ⊢ ((𝜑 ∧ ℎ ∈ (𝑇 ↑_m 𝑅)) → (^◡𝐺 ∘ ℎ) ∈ (𝑆 ↑_m 𝑅))
27		simpr 488	. . . . . 6 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝑆 ↑_m 𝑅) ∧ ℎ ∈ (𝑇 ↑_m 𝑅))) ∧ 𝑓 = (^◡𝐺 ∘ ℎ)) → 𝑓 = (^◡𝐺 ∘ ℎ))
28	27	coeq2d 5832	. . . . 5 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝑆 ↑_m 𝑅) ∧ ℎ ∈ (𝑇 ↑_m 𝑅))) ∧ 𝑓 = (^◡𝐺 ∘ ℎ)) → (𝐺 ∘ 𝑓) = (𝐺 ∘ (^◡𝐺 ∘ ℎ)))
29		coass 6249	. . . . 5 ⊢ ((𝐺 ∘ ^◡𝐺) ∘ ℎ) = (𝐺 ∘ (^◡𝐺 ∘ ℎ))
30	28, 29	eqtr4di 2814	. . . 4 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝑆 ↑_m 𝑅) ∧ ℎ ∈ (𝑇 ↑_m 𝑅))) ∧ 𝑓 = (^◡𝐺 ∘ ℎ)) → (𝐺 ∘ 𝑓) = ((𝐺 ∘ ^◡𝐺) ∘ ℎ))
31		simpll 776	. . . . . 6 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝑆 ↑_m 𝑅) ∧ ℎ ∈ (𝑇 ↑_m 𝑅))) ∧ 𝑓 = (^◡𝐺 ∘ ℎ)) → 𝜑)
32		f1ococnv2 6830	. . . . . 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 5831	. . . 4 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝑆 ↑_m 𝑅) ∧ ℎ ∈ (𝑇 ↑_m 𝑅))) ∧ 𝑓 = (^◡𝐺 ∘ ℎ)) → ((𝐺 ∘ ^◡𝐺) ∘ ℎ) = (( I ↾ 𝑇) ∘ ℎ))
35		simplrr 787	. . . . . 6 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝑆 ↑_m 𝑅) ∧ ℎ ∈ (𝑇 ↑_m 𝑅))) ∧ 𝑓 = (^◡𝐺 ∘ ℎ)) → ℎ ∈ (𝑇 ↑_m 𝑅))
36	31, 35, 21	syl2anc 593	. . . . 5 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝑆 ↑_m 𝑅) ∧ ℎ ∈ (𝑇 ↑_m 𝑅))) ∧ 𝑓 = (^◡𝐺 ∘ ℎ)) → ℎ:𝑅⟶𝑇)
37		fcoi2 6735	. . . . 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 2801	. . 3 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝑆 ↑_m 𝑅) ∧ ℎ ∈ (𝑇 ↑_m 𝑅))) ∧ 𝑓 = (^◡𝐺 ∘ ℎ)) → ℎ = (𝐺 ∘ 𝑓))
40		simpr 488	. . . . . 6 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝑆 ↑_m 𝑅) ∧ ℎ ∈ (𝑇 ↑_m 𝑅))) ∧ ℎ = (𝐺 ∘ 𝑓)) → ℎ = (𝐺 ∘ 𝑓))
41	40	coeq2d 5832	. . . . 5 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝑆 ↑_m 𝑅) ∧ ℎ ∈ (𝑇 ↑_m 𝑅))) ∧ ℎ = (𝐺 ∘ 𝑓)) → (^◡𝐺 ∘ ℎ) = (^◡𝐺 ∘ (𝐺 ∘ 𝑓)))
42		coass 6249	. . . . 5 ⊢ ((^◡𝐺 ∘ 𝐺) ∘ 𝑓) = (^◡𝐺 ∘ (𝐺 ∘ 𝑓))
43	41, 42	eqtr4di 2814	. . . 4 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝑆 ↑_m 𝑅) ∧ ℎ ∈ (𝑇 ↑_m 𝑅))) ∧ ℎ = (𝐺 ∘ 𝑓)) → (^◡𝐺 ∘ ℎ) = ((^◡𝐺 ∘ 𝐺) ∘ 𝑓))
44		simpll 776	. . . . . 6 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝑆 ↑_m 𝑅) ∧ ℎ ∈ (𝑇 ↑_m 𝑅))) ∧ ℎ = (𝐺 ∘ 𝑓)) → 𝜑)
45		f1ococnv1 6832	. . . . . 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 5831	. . . 4 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝑆 ↑_m 𝑅) ∧ ℎ ∈ (𝑇 ↑_m 𝑅))) ∧ ℎ = (𝐺 ∘ 𝑓)) → ((^◡𝐺 ∘ 𝐺) ∘ 𝑓) = (( I ↾ 𝑆) ∘ 𝑓))
48		simplrl 786	. . . . . 6 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝑆 ↑_m 𝑅) ∧ ℎ ∈ (𝑇 ↑_m 𝑅))) ∧ ℎ = (𝐺 ∘ 𝑓)) → 𝑓 ∈ (𝑆 ↑_m 𝑅))
49	44, 48, 9	syl2anc 593	. . . . 5 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝑆 ↑_m 𝑅) ∧ ℎ ∈ (𝑇 ↑_m 𝑅))) ∧ ℎ = (𝐺 ∘ 𝑓)) → 𝑓:𝑅⟶𝑆)
50		fcoi2 6735	. . . . 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 2801	. . 3 ⊢ (((𝜑 ∧ (𝑓 ∈ (𝑆 ↑_m 𝑅) ∧ ℎ ∈ (𝑇 ↑_m 𝑅))) ∧ ℎ = (𝐺 ∘ 𝑓)) → 𝑓 = (^◡𝐺 ∘ ℎ))
53	39, 52	impbida 810	. 2 ⊢ ((𝜑 ∧ (𝑓 ∈ (𝑆 ↑_m 𝑅) ∧ ℎ ∈ (𝑇 ↑_m 𝑅))) → (𝑓 = (^◡𝐺 ∘ ℎ) ↔ ℎ = (𝐺 ∘ 𝑓)))
54	1, 15, 26, 53	f1o2d 7646	1 ⊢ (𝜑 → (𝑓 ∈ (𝑆 ↑_m 𝑅) ↦ (𝐺 ∘ 𝑓)):(𝑆 ↑_m 𝑅)–1-1-onto→(𝑇 ↑_m 𝑅))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 208 ∧ wa 399 = wceq 1559 ∈ wcel 2141 ↦ cmpt 5180 I cid 5539 ^◡ccnv 5644 ↾ cres 5647 ∘ ccom 5649 ⟶wf 6513 –1-1-onto→wf1o 6516 (class class class)co 7392 ↑_m cmap 8803

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1814 ax-4 1828 ax-5 1929 ax-6 1986 ax-7 2027 ax-8 2143 ax-9 2151 ax-10 2174 ax-11 2190 ax-12 2211 ax-ext 2733 ax-sep 5245 ax-pow 5321 ax-pr 5389 ax-un 7714

This theorem depends on definitions: df-bi 209 df-an 400 df-or 859 df-3an 1099 df-tru 1562 df-fal 1572 df-ex 1799 df-nf 1803 df-sb 2090 df-mo 2565 df-eu 2595 df-clab 2740 df-cleq 2753 df-clel 2836 df-nfc 2910 df-ral 3076 df-rex 3086 df-rab 3414 df-v 3455 df-sbc 3745 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-nul 4286 df-if 4480 df-pw 4556 df-sn 4582 df-pr 4584 df-op 4588 df-uni 4865 df-br 5100 df-opab 5162 df-mpt 5181 df-id 5540 df-xp 5651 df-rel 5652 df-cnv 5653 df-co 5654 df-dm 5655 df-rn 5656 df-res 5657 df-ima 5658 df-iota 6473 df-fun 6519 df-fn 6520 df-f 6521 df-f1 6522 df-fo 6523 df-f1o 6524 df-fv 6525 df-ov 7395 df-oprab 7396 df-mpo 7397 df-map 8805

This theorem is referenced by: (None)

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