Theorem fcobijfs 32710

Description: Composing finitely supported functions with a bijection yields a bijection between sets of finitely supported functions. See also mapfien 9301. (Contributed by Thierry Arnoux, 25-Aug-2017.) (Revised by Thierry Arnoux, 1-Sep-2019.)

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)
fcobijfs.5	⊢ (𝜑 → 𝑂 ∈ 𝑆)
fcobijfs.6	⊢ 𝑄 = (𝐺‘𝑂)
fcobijfs.7	⊢ 𝑋 = {𝑔 ∈ (𝑆 ↑m 𝑅) ∣ 𝑔 finSupp 𝑂}
fcobijfs.8	⊢ 𝑌 = {ℎ ∈ (𝑇 ↑m 𝑅) ∣ ℎ finSupp 𝑄}

Assertion

Ref	Expression
fcobijfs	⊢ (𝜑 → (𝑓 ∈ 𝑋 ↦ (𝐺 ∘ 𝑓)):𝑋–1-1-onto→𝑌)

Distinct variable groups:   𝑓,ℎ,𝐺   𝑅,𝑓,ℎ   𝑆,𝑓,ℎ   𝑇,𝑓,ℎ   𝜑,𝑓,ℎ   𝑓,𝑂,ℎ   𝑄,𝑓,ℎ   𝑔,ℎ,𝑂   𝑅,𝑔   𝑆,𝑔   𝑓,𝑋   𝑓,𝑌

Allowed substitution hints:   𝜑(𝑔)   𝑄(𝑔)   𝑇(𝑔)   𝑈(𝑓,𝑔,ℎ)   𝐺(𝑔)   𝑉(𝑓,𝑔,ℎ)   𝑊(𝑓,𝑔,ℎ)   𝑋(𝑔,ℎ)   𝑌(𝑔,ℎ)

Proof of Theorem fcobijfs

Step	Hyp	Ref	Expression
1		fcobijfs.7	. . . 4 ⊢ 𝑋 = {𝑔 ∈ (𝑆 ↑m 𝑅) ∣ 𝑔 finSupp 𝑂}
2		breq1 5098	. . . . 5 ⊢ (ℎ = 𝑔 → (ℎ finSupp 𝑂 ↔ 𝑔 finSupp 𝑂))
3	2	cbvrabv 3406	. . . 4 ⊢ {ℎ ∈ (𝑆 ↑m 𝑅) ∣ ℎ finSupp 𝑂} = {𝑔 ∈ (𝑆 ↑m 𝑅) ∣ 𝑔 finSupp 𝑂}
4	1, 3	eqtr4i 2759	. . 3 ⊢ 𝑋 = {ℎ ∈ (𝑆 ↑m 𝑅) ∣ ℎ finSupp 𝑂}
5		fcobijfs.8	. . 3 ⊢ 𝑌 = {ℎ ∈ (𝑇 ↑m 𝑅) ∣ ℎ finSupp 𝑄}
6		fcobijfs.6	. . 3 ⊢ 𝑄 = (𝐺‘𝑂)
7		f1oi 6808	. . . 4 ⊢ ( I ↾ 𝑅):𝑅–1-1-onto→𝑅
8	7	a1i 11	. . 3 ⊢ (𝜑 → ( I ↾ 𝑅):𝑅–1-1-onto→𝑅)
9		fcobij.1	. . 3 ⊢ (𝜑 → 𝐺:𝑆–1-1-onto→𝑇)
10		fcobij.2	. . 3 ⊢ (𝜑 → 𝑅 ∈ 𝑈)
11		fcobij.3	. . 3 ⊢ (𝜑 → 𝑆 ∈ 𝑉)
12		fcobij.4	. . 3 ⊢ (𝜑 → 𝑇 ∈ 𝑊)
13		fcobijfs.5	. . 3 ⊢ (𝜑 → 𝑂 ∈ 𝑆)
14	4, 5, 6, 8, 9, 10, 11, 10, 12, 13	mapfien 9301	. 2 ⊢ (𝜑 → (𝑓 ∈ 𝑋 ↦ (𝐺 ∘ (𝑓 ∘ ( I ↾ 𝑅)))):𝑋–1-1-onto→𝑌)
15	1	ssrab3 4031	. . . . . 6 ⊢ 𝑋 ⊆ (𝑆 ↑m 𝑅)
16	15	sseli 3926	. . . . 5 ⊢ (𝑓 ∈ 𝑋 → 𝑓 ∈ (𝑆 ↑m 𝑅))
17		coass 6220	. . . . . 6 ⊢ ((𝐺 ∘ 𝑓) ∘ ( I ↾ 𝑅)) = (𝐺 ∘ (𝑓 ∘ ( I ↾ 𝑅)))
18		f1of 6770	. . . . . . . . 9 ⊢ (𝐺:𝑆–1-1-onto→𝑇 → 𝐺:𝑆⟶𝑇)
19	9, 18	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐺:𝑆⟶𝑇)
20		elmapi 8781	. . . . . . . 8 ⊢ (𝑓 ∈ (𝑆 ↑m 𝑅) → 𝑓:𝑅⟶𝑆)
21		fco 6682	. . . . . . . 8 ⊢ ((𝐺:𝑆⟶𝑇 ∧ 𝑓:𝑅⟶𝑆) → (𝐺 ∘ 𝑓):𝑅⟶𝑇)
22	19, 20, 21	syl2an 596	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑆 ↑m 𝑅)) → (𝐺 ∘ 𝑓):𝑅⟶𝑇)
23		fcoi1 6704	. . . . . . 7 ⊢ ((𝐺 ∘ 𝑓):𝑅⟶𝑇 → ((𝐺 ∘ 𝑓) ∘ ( I ↾ 𝑅)) = (𝐺 ∘ 𝑓))
24	22, 23	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑆 ↑m 𝑅)) → ((𝐺 ∘ 𝑓) ∘ ( I ↾ 𝑅)) = (𝐺 ∘ 𝑓))
25	17, 24	eqtr3id 2782	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑆 ↑m 𝑅)) → (𝐺 ∘ (𝑓 ∘ ( I ↾ 𝑅))) = (𝐺 ∘ 𝑓))
26	16, 25	sylan2 593	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → (𝐺 ∘ (𝑓 ∘ ( I ↾ 𝑅))) = (𝐺 ∘ 𝑓))
27	26	mpteq2dva 5188	. . 3 ⊢ (𝜑 → (𝑓 ∈ 𝑋 ↦ (𝐺 ∘ (𝑓 ∘ ( I ↾ 𝑅)))) = (𝑓 ∈ 𝑋 ↦ (𝐺 ∘ 𝑓)))
28	27	f1oeq1d 6765	. 2 ⊢ (𝜑 → ((𝑓 ∈ 𝑋 ↦ (𝐺 ∘ (𝑓 ∘ ( I ↾ 𝑅)))):𝑋–1-1-onto→𝑌 ↔ (𝑓 ∈ 𝑋 ↦ (𝐺 ∘ 𝑓)):𝑋–1-1-onto→𝑌))
29	14, 28	mpbid 232	1 ⊢ (𝜑 → (𝑓 ∈ 𝑋 ↦ (𝐺 ∘ 𝑓)):𝑋–1-1-onto→𝑌)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2113  {crab 3396   class class class wbr 5095   ↦ cmpt 5176   I cid 5515   ↾ cres 5623   ∘ ccom 5625  ⟶wf 6484  –1-1-onto→wf1o 6487  ‘cfv 6488  (class class class)co 7354   ↑_m cmap 8758   finSupp cfsupp 9254

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-rep 5221  ax-sep 5238  ax-nul 5248  ax-pow 5307  ax-pr 5374  ax-un 7676

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-ral 3049  df-rex 3058  df-reu 3348  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-pss 3918  df-nul 4283  df-if 4477  df-pw 4553  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4861  df-iun 4945  df-br 5096  df-opab 5158  df-mpt 5177  df-tr 5203  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-ord 6316  df-on 6317  df-lim 6318  df-suc 6319  df-iota 6444  df-fun 6490  df-fn 6491  df-f 6492  df-f1 6493  df-fo 6494  df-f1o 6495  df-fv 6496  df-ov 7357  df-oprab 7358  df-mpo 7359  df-om 7805  df-1st 7929  df-2nd 7930  df-supp 8099  df-1o 8393  df-map 8760  df-en 8878  df-dom 8879  df-fin 8881  df-fsupp 9255

This theorem is referenced by:  eulerpartgbij  34408