Theorem fcobijfs 30235

Description: Composing finitely supported functions with a bijection yields a bijection between sets of finitely supported functions. See also mapfien 8664. (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 4928	. . . . 5 ⊢ (ℎ = 𝑔 → (ℎ finSupp 𝑂 ↔ 𝑔 finSupp 𝑂))
3	2	cbvrabv 3405	. . . 4 ⊢ {ℎ ∈ (𝑆 ↑𝑚 𝑅) ∣ ℎ finSupp 𝑂} = {𝑔 ∈ (𝑆 ↑𝑚 𝑅) ∣ 𝑔 finSupp 𝑂}
4	1, 3	eqtr4i 2798	. . 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 6478	. . . 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	. . . 4 ⊢ (𝜑 → 𝑅 ∈ 𝑈)
11		elex 3426	. . . 4 ⊢ (𝑅 ∈ 𝑈 → 𝑅 ∈ V)
12	10, 11	syl 17	. . 3 ⊢ (𝜑 → 𝑅 ∈ V)
13		fcobij.3	. . . 4 ⊢ (𝜑 → 𝑆 ∈ 𝑉)
14		elex 3426	. . . 4 ⊢ (𝑆 ∈ 𝑉 → 𝑆 ∈ V)
15	13, 14	syl 17	. . 3 ⊢ (𝜑 → 𝑆 ∈ V)
16		fcobij.4	. . . 4 ⊢ (𝜑 → 𝑇 ∈ 𝑊)
17		elex 3426	. . . 4 ⊢ (𝑇 ∈ 𝑊 → 𝑇 ∈ V)
18	16, 17	syl 17	. . 3 ⊢ (𝜑 → 𝑇 ∈ V)
19		fcobijfs.5	. . 3 ⊢ (𝜑 → 𝑂 ∈ 𝑆)
20	4, 5, 6, 8, 9, 12, 15, 12, 18, 19	mapfien 8664	. 2 ⊢ (𝜑 → (𝑓 ∈ 𝑋 ↦ (𝐺 ∘ (𝑓 ∘ ( I ↾ 𝑅)))):𝑋–1-1-onto→𝑌)
21		ssrab2 3939	. . . . . . 7 ⊢ {𝑔 ∈ (𝑆 ↑𝑚 𝑅) ∣ 𝑔 finSupp 𝑂} ⊆ (𝑆 ↑𝑚 𝑅)
22	1, 21	eqsstri 3884	. . . . . 6 ⊢ 𝑋 ⊆ (𝑆 ↑𝑚 𝑅)
23	22	sseli 3847	. . . . 5 ⊢ (𝑓 ∈ 𝑋 → 𝑓 ∈ (𝑆 ↑𝑚 𝑅))
24		coass 5954	. . . . . 6 ⊢ ((𝐺 ∘ 𝑓) ∘ ( I ↾ 𝑅)) = (𝐺 ∘ (𝑓 ∘ ( I ↾ 𝑅)))
25		f1of 6441	. . . . . . . . 9 ⊢ (𝐺:𝑆–1-1-onto→𝑇 → 𝐺:𝑆⟶𝑇)
26	9, 25	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐺:𝑆⟶𝑇)
27		elmapi 8226	. . . . . . . 8 ⊢ (𝑓 ∈ (𝑆 ↑𝑚 𝑅) → 𝑓:𝑅⟶𝑆)
28		fco 6358	. . . . . . . 8 ⊢ ((𝐺:𝑆⟶𝑇 ∧ 𝑓:𝑅⟶𝑆) → (𝐺 ∘ 𝑓):𝑅⟶𝑇)
29	26, 27, 28	syl2an 587	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑆 ↑𝑚 𝑅)) → (𝐺 ∘ 𝑓):𝑅⟶𝑇)
30		fcoi1 6378	. . . . . . 7 ⊢ ((𝐺 ∘ 𝑓):𝑅⟶𝑇 → ((𝐺 ∘ 𝑓) ∘ ( I ↾ 𝑅)) = (𝐺 ∘ 𝑓))
31	29, 30	syl 17	. . . . . 6 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑆 ↑𝑚 𝑅)) → ((𝐺 ∘ 𝑓) ∘ ( I ↾ 𝑅)) = (𝐺 ∘ 𝑓))
32	24, 31	syl5eqr 2821	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑆 ↑𝑚 𝑅)) → (𝐺 ∘ (𝑓 ∘ ( I ↾ 𝑅))) = (𝐺 ∘ 𝑓))
33	23, 32	sylan2 584	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → (𝐺 ∘ (𝑓 ∘ ( I ↾ 𝑅))) = (𝐺 ∘ 𝑓))
34	33	mpteq2dva 5018	. . 3 ⊢ (𝜑 → (𝑓 ∈ 𝑋 ↦ (𝐺 ∘ (𝑓 ∘ ( I ↾ 𝑅)))) = (𝑓 ∈ 𝑋 ↦ (𝐺 ∘ 𝑓)))
35		f1oeq1 6430	. . 3 ⊢ ((𝑓 ∈ 𝑋 ↦ (𝐺 ∘ (𝑓 ∘ ( I ↾ 𝑅)))) = (𝑓 ∈ 𝑋 ↦ (𝐺 ∘ 𝑓)) → ((𝑓 ∈ 𝑋 ↦ (𝐺 ∘ (𝑓 ∘ ( I ↾ 𝑅)))):𝑋–1-1-onto→𝑌 ↔ (𝑓 ∈ 𝑋 ↦ (𝐺 ∘ 𝑓)):𝑋–1-1-onto→𝑌))
36	34, 35	syl 17	. 2 ⊢ (𝜑 → ((𝑓 ∈ 𝑋 ↦ (𝐺 ∘ (𝑓 ∘ ( I ↾ 𝑅)))):𝑋–1-1-onto→𝑌 ↔ (𝑓 ∈ 𝑋 ↦ (𝐺 ∘ 𝑓)):𝑋–1-1-onto→𝑌))
37	20, 36	mpbid 224	1 ⊢ (𝜑 → (𝑓 ∈ 𝑋 ↦ (𝐺 ∘ 𝑓)):𝑋–1-1-onto→𝑌)

Syntax hints:   → wi 4   ↔ wb 198   ∧ wa 387   = wceq 1508   ∈ wcel 2051  {crab 3085  Vcvv 3408   class class class wbr 4925   ↦ cmpt 5004   I cid 5307   ↾ cres 5405   ∘ ccom 5407  ⟶wf 6181  –1-1-onto→wf1o 6184  ‘cfv 6185  (class class class)co 6974   ↑_𝑚 cmap 8204   finSupp cfsupp 8626

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-13 2302  ax-ext 2743  ax-rep 5045  ax-sep 5056  ax-nul 5063  ax-pow 5115  ax-pr 5182  ax-un 7277

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-3or 1070  df-3an 1071  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-mo 2548  df-eu 2585  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-ne 2961  df-ral 3086  df-rex 3087  df-reu 3088  df-rab 3090  df-v 3410  df-sbc 3675  df-csb 3780  df-dif 3825  df-un 3827  df-in 3829  df-ss 3836  df-pss 3838  df-nul 4173  df-if 4345  df-pw 4418  df-sn 4436  df-pr 4438  df-tp 4440  df-op 4442  df-uni 4709  df-iun 4790  df-br 4926  df-opab 4988  df-mpt 5005  df-tr 5027  df-id 5308  df-eprel 5313  df-po 5322  df-so 5323  df-fr 5362  df-we 5364  df-xp 5409  df-rel 5410  df-cnv 5411  df-co 5412  df-dm 5413  df-rn 5414  df-res 5415  df-ima 5416  df-ord 6029  df-on 6030  df-lim 6031  df-suc 6032  df-iota 6149  df-fun 6187  df-fn 6188  df-f 6189  df-f1 6190  df-fo 6191  df-f1o 6192  df-fv 6193  df-ov 6977  df-oprab 6978  df-mpo 6979  df-om 7395  df-1st 7499  df-2nd 7500  df-supp 7632  df-1o 7903  df-er 8087  df-map 8206  df-en 8305  df-dom 8306  df-fin 8308  df-fsupp 8627

This theorem is referenced by:  eulerpartgbij  31307