Theorem fcobijfs 32781

Description: Composing finitely supported functions with a bijection yields a bijection between sets of finitely supported functions. See also mapfien 9315. (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 5102	. . . . 5 ⊢ (ℎ = 𝑔 → (ℎ finSupp 𝑂 ↔ 𝑔 finSupp 𝑂))
3	2	cbvrabv 3410	. . . 4 ⊢ {ℎ ∈ (𝑆 ↑m 𝑅) ∣ ℎ finSupp 𝑂} = {𝑔 ∈ (𝑆 ↑m 𝑅) ∣ 𝑔 finSupp 𝑂}
4	1, 3	eqtr4i 2763	. . 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 6813	. . . 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 9315	. 2 ⊢ (𝜑 → (𝑓 ∈ 𝑋 ↦ (𝐺 ∘ (𝑓 ∘ ( I ↾ 𝑅)))):𝑋–1-1-onto→𝑌)
15	1	ssrab3 4035	. . . . . 6 ⊢ 𝑋 ⊆ (𝑆 ↑m 𝑅)
16	15	sseli 3930	. . . . 5 ⊢ (𝑓 ∈ 𝑋 → 𝑓 ∈ (𝑆 ↑m 𝑅))
17		coass 6225	. . . . . 6 ⊢ ((𝐺 ∘ 𝑓) ∘ ( I ↾ 𝑅)) = (𝐺 ∘ (𝑓 ∘ ( I ↾ 𝑅)))
18		f1of 6775	. . . . . . . . 9 ⊢ (𝐺:𝑆–1-1-onto→𝑇 → 𝐺:𝑆⟶𝑇)
19	9, 18	syl 17	. . . . . . . 8 ⊢ (𝜑 → 𝐺:𝑆⟶𝑇)
20		elmapi 8790	. . . . . . . 8 ⊢ (𝑓 ∈ (𝑆 ↑m 𝑅) → 𝑓:𝑅⟶𝑆)
21		fco 6687	. . . . . . . 8 ⊢ ((𝐺:𝑆⟶𝑇 ∧ 𝑓:𝑅⟶𝑆) → (𝐺 ∘ 𝑓):𝑅⟶𝑇)
22	19, 20, 21	syl2an 597	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑆 ↑m 𝑅)) → (𝐺 ∘ 𝑓):𝑅⟶𝑇)
23		fcoi1 6709	. . . . . . 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 2786	. . . . 5 ⊢ ((𝜑 ∧ 𝑓 ∈ (𝑆 ↑m 𝑅)) → (𝐺 ∘ (𝑓 ∘ ( I ↾ 𝑅))) = (𝐺 ∘ 𝑓))
26	16, 25	sylan2 594	. . . 4 ⊢ ((𝜑 ∧ 𝑓 ∈ 𝑋) → (𝐺 ∘ (𝑓 ∘ ( I ↾ 𝑅))) = (𝐺 ∘ 𝑓))
27	26	mpteq2dva 5192	. . 3 ⊢ (𝜑 → (𝑓 ∈ 𝑋 ↦ (𝐺 ∘ (𝑓 ∘ ( I ↾ 𝑅)))) = (𝑓 ∈ 𝑋 ↦ (𝐺 ∘ 𝑓)))
28	27	f1oeq1d 6770	. 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 1542   ∈ wcel 2114  {crab 3400   class class class wbr 5099   ↦ cmpt 5180   I cid 5519   ↾ cres 5627   ∘ ccom 5629  ⟶wf 6489  –1-1-onto→wf1o 6492  ‘cfv 6493  (class class class)co 7360   ↑_m cmap 8767   finSupp cfsupp 9268

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5225  ax-sep 5242  ax-nul 5252  ax-pow 5311  ax-pr 5378  ax-un 7682

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3062  df-reu 3352  df-rab 3401  df-v 3443  df-sbc 3742  df-csb 3851  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-pss 3922  df-nul 4287  df-if 4481  df-pw 4557  df-sn 4582  df-pr 4584  df-op 4588  df-uni 4865  df-iun 4949  df-br 5100  df-opab 5162  df-mpt 5181  df-tr 5207  df-id 5520  df-eprel 5525  df-po 5533  df-so 5534  df-fr 5578  df-we 5580  df-xp 5631  df-rel 5632  df-cnv 5633  df-co 5634  df-dm 5635  df-rn 5636  df-res 5637  df-ima 5638  df-ord 6321  df-on 6322  df-lim 6323  df-suc 6324  df-iota 6449  df-fun 6495  df-fn 6496  df-f 6497  df-f1 6498  df-fo 6499  df-f1o 6500  df-fv 6501  df-ov 7363  df-oprab 7364  df-mpo 7365  df-om 7811  df-1st 7935  df-2nd 7936  df-supp 8105  df-1o 8399  df-map 8769  df-en 8888  df-dom 8889  df-fin 8891  df-fsupp 9269

This theorem is referenced by:  eulerpartgbij  34510