Theorem fcobijfs 32452

Description: Composing finitely supported functions with a bijection yields a bijection between sets of finitely supported functions. See also mapfien 9402. (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 5144	. . . . 5 ⊢ (ℎ = 𝑔 → (ℎ finSupp 𝑂 ↔ 𝑔 finSupp 𝑂))
3	2	cbvrabv 3436	. . . 4 ⊢ {ℎ ∈ (𝑆 ↑m 𝑅) ∣ ℎ finSupp 𝑂} = {𝑔 ∈ (𝑆 ↑m 𝑅) ∣ 𝑔 finSupp 𝑂}
4	1, 3	eqtr4i 2757	. . 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 6864	. . . 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 9402	. 2 ⊢ (𝜑 → (𝑓 ∈ 𝑋 ↦ (𝐺 ∘ (𝑓 ∘ ( I ↾ 𝑅)))):𝑋–1-1-onto→𝑌)
15	1	ssrab3 4075	. . . . . 6 ⊢ 𝑋 ⊆ (𝑆 ↑m 𝑅)
16	15	sseli 3973	. . . . 5 ⊢ (𝑓 ∈ 𝑋 → 𝑓 ∈ (𝑆 ↑m 𝑅))
17		coass 6257	. . . . . 6 ⊢ ((𝐺 ∘ 𝑓) ∘ ( I ↾ 𝑅)) = (𝐺 ∘ (𝑓 ∘ ( I ↾ 𝑅)))
18		f1of 6826	. . . . . . . . 9 ⊢ (𝐺:𝑆–1-1-onto→𝑇 → 𝐺:𝑆⟶𝑇)
19	9, 18	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐺:𝑆⟶𝑇)
20		elmapi 8842	. . . . . . . 8 ⊢ (𝑓 ∈ (𝑆 ↑m 𝑅) → 𝑓:𝑅⟶𝑆)
21		fco 6734	. . . . . . . 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 6758	. . . . . . 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 2780	. . . . 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 5241	. . 3 ⊢ (𝜑 → (𝑓 ∈ 𝑋 ↦ (𝐺 ∘ (𝑓 ∘ ( I ↾ 𝑅)))) = (𝑓 ∈ 𝑋 ↦ (𝐺 ∘ 𝑓)))
28	27	f1oeq1d 6821	. 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 231	1 ⊢ (𝜑 → (𝑓 ∈ 𝑋 ↦ (𝐺 ∘ 𝑓)):𝑋–1-1-onto→𝑌)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1533   ∈ wcel 2098  {crab 3426   class class class wbr 5141   ↦ cmpt 5224   I cid 5566   ↾ cres 5671   ∘ ccom 5673  ⟶wf 6532  –1-1-onto→wf1o 6535  ‘cfv 6536  (class class class)co 7404   ↑_m cmap 8819   finSupp cfsupp 9360

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pow 5356  ax-pr 5420  ax-un 7721

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-ord 6360  df-on 6361  df-lim 6362  df-suc 6363  df-iota 6488  df-fun 6538  df-fn 6539  df-f 6540  df-f1 6541  df-fo 6542  df-f1o 6543  df-fv 6544  df-ov 7407  df-oprab 7408  df-mpo 7409  df-om 7852  df-1st 7971  df-2nd 7972  df-supp 8144  df-1o 8464  df-map 8821  df-en 8939  df-fin 8942  df-fsupp 9361

This theorem is referenced by:  eulerpartgbij  33900