Theorem fcobijfs 32737

Description: Composing finitely supported functions with a bijection yields a bijection between sets of finitely supported functions. See also mapfien 9477. (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 5169	. . . . 5 ⊢ (ℎ = 𝑔 → (ℎ finSupp 𝑂 ↔ 𝑔 finSupp 𝑂))
3	2	cbvrabv 3454	. . . 4 ⊢ {ℎ ∈ (𝑆 ↑m 𝑅) ∣ ℎ finSupp 𝑂} = {𝑔 ∈ (𝑆 ↑m 𝑅) ∣ 𝑔 finSupp 𝑂}
4	1, 3	eqtr4i 2771	. . 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 6900	. . . 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 9477	. 2 ⊢ (𝜑 → (𝑓 ∈ 𝑋 ↦ (𝐺 ∘ (𝑓 ∘ ( I ↾ 𝑅)))):𝑋–1-1-onto→𝑌)
15	1	ssrab3 4105	. . . . . 6 ⊢ 𝑋 ⊆ (𝑆 ↑m 𝑅)
16	15	sseli 4004	. . . . 5 ⊢ (𝑓 ∈ 𝑋 → 𝑓 ∈ (𝑆 ↑m 𝑅))
17		coass 6296	. . . . . 6 ⊢ ((𝐺 ∘ 𝑓) ∘ ( I ↾ 𝑅)) = (𝐺 ∘ (𝑓 ∘ ( I ↾ 𝑅)))
18		f1of 6862	. . . . . . . . 9 ⊢ (𝐺:𝑆–1-1-onto→𝑇 → 𝐺:𝑆⟶𝑇)
19	9, 18	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐺:𝑆⟶𝑇)
20		elmapi 8907	. . . . . . . 8 ⊢ (𝑓 ∈ (𝑆 ↑m 𝑅) → 𝑓:𝑅⟶𝑆)
21		fco 6771	. . . . . . . 8 ⊢ ((𝐺:𝑆⟶𝑇 ∧ 𝑓:𝑅⟶𝑆) → (𝐺 ∘ 𝑓):𝑅⟶𝑇)
22	19, 20, 21	syl2an 595	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑆 ↑m 𝑅)) → (𝐺 ∘ 𝑓):𝑅⟶𝑇)
23		fcoi1 6795	. . . . . . 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 2794	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑆 ↑m 𝑅)) → (𝐺 ∘ (𝑓 ∘ ( I ↾ 𝑅))) = (𝐺 ∘ 𝑓))
26	16, 25	sylan2 592	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → (𝐺 ∘ (𝑓 ∘ ( I ↾ 𝑅))) = (𝐺 ∘ 𝑓))
27	26	mpteq2dva 5266	. . 3 ⊢ (𝜑 → (𝑓 ∈ 𝑋 ↦ (𝐺 ∘ (𝑓 ∘ ( I ↾ 𝑅)))) = (𝑓 ∈ 𝑋 ↦ (𝐺 ∘ 𝑓)))
28	27	f1oeq1d 6857	. 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 1537   ∈ wcel 2108  {crab 3443   class class class wbr 5166   ↦ cmpt 5249   I cid 5592   ↾ cres 5702   ∘ ccom 5704  ⟶wf 6569  –1-1-onto→wf1o 6572  ‘cfv 6573  (class class class)co 7448   ↑_m cmap 8884   finSupp cfsupp 9431

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-rep 5303  ax-sep 5317  ax-nul 5324  ax-pow 5383  ax-pr 5447  ax-un 7770

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3or 1088  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-reu 3389  df-rab 3444  df-v 3490  df-sbc 3805  df-csb 3922  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-pss 3996  df-nul 4353  df-if 4549  df-pw 4624  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-mpt 5250  df-tr 5284  df-id 5593  df-eprel 5599  df-po 5607  df-so 5608  df-fr 5652  df-we 5654  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-ord 6398  df-on 6399  df-lim 6400  df-suc 6401  df-iota 6525  df-fun 6575  df-fn 6576  df-f 6577  df-f1 6578  df-fo 6579  df-f1o 6580  df-fv 6581  df-ov 7451  df-oprab 7452  df-mpo 7453  df-om 7904  df-1st 8030  df-2nd 8031  df-supp 8202  df-1o 8522  df-map 8886  df-en 9004  df-dom 9005  df-fin 9007  df-fsupp 9432

This theorem is referenced by:  eulerpartgbij  34337