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Mathbox for Thierry Arnoux |
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Mirrors > Home > MPE Home > Th. List > Mathboxes > fcobijfs | Structured version Visualization version GIF version |
Description: Composing finitely supported functions with a bijection yields a bijection between sets of finitely supported functions. See also mapfien 9446. (Contributed by Thierry Arnoux, 25-Aug-2017.) (Revised by Thierry Arnoux, 1-Sep-2019.) |
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} |
Ref | Expression |
---|---|
fcobijfs | ⊢ (𝜑 → (𝑓 ∈ 𝑋 ↦ (𝐺 ∘ 𝑓)):𝑋–1-1-onto→𝑌) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | fcobijfs.7 | . . . 4 ⊢ 𝑋 = {𝑔 ∈ (𝑆 ↑m 𝑅) ∣ 𝑔 finSupp 𝑂} | |
2 | breq1 5151 | . . . . 5 ⊢ (ℎ = 𝑔 → (ℎ finSupp 𝑂 ↔ 𝑔 finSupp 𝑂)) | |
3 | 2 | cbvrabv 3444 | . . . 4 ⊢ {ℎ ∈ (𝑆 ↑m 𝑅) ∣ ℎ finSupp 𝑂} = {𝑔 ∈ (𝑆 ↑m 𝑅) ∣ 𝑔 finSupp 𝑂} |
4 | 1, 3 | eqtr4i 2766 | . . 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 6887 | . . . 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 9446 | . 2 ⊢ (𝜑 → (𝑓 ∈ 𝑋 ↦ (𝐺 ∘ (𝑓 ∘ ( I ↾ 𝑅)))):𝑋–1-1-onto→𝑌) |
15 | 1 | ssrab3 4092 | . . . . . 6 ⊢ 𝑋 ⊆ (𝑆 ↑m 𝑅) |
16 | 15 | sseli 3991 | . . . . 5 ⊢ (𝑓 ∈ 𝑋 → 𝑓 ∈ (𝑆 ↑m 𝑅)) |
17 | coass 6287 | . . . . . 6 ⊢ ((𝐺 ∘ 𝑓) ∘ ( I ↾ 𝑅)) = (𝐺 ∘ (𝑓 ∘ ( I ↾ 𝑅))) | |
18 | f1of 6849 | . . . . . . . . 9 ⊢ (𝐺:𝑆–1-1-onto→𝑇 → 𝐺:𝑆⟶𝑇) | |
19 | 9, 18 | syl 17 | . . . . . . . 8 ⊢ (𝜑 → 𝐺:𝑆⟶𝑇) |
20 | elmapi 8888 | . . . . . . . 8 ⊢ (𝑓 ∈ (𝑆 ↑m 𝑅) → 𝑓:𝑅⟶𝑆) | |
21 | fco 6761 | . . . . . . . 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 6783 | . . . . . . 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 2789 | . . . . 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 5248 | . . 3 ⊢ (𝜑 → (𝑓 ∈ 𝑋 ↦ (𝐺 ∘ (𝑓 ∘ ( I ↾ 𝑅)))) = (𝑓 ∈ 𝑋 ↦ (𝐺 ∘ 𝑓))) |
28 | 27 | f1oeq1d 6844 | . 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) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1537 ∈ wcel 2106 {crab 3433 class class class wbr 5148 ↦ cmpt 5231 I cid 5582 ↾ cres 5691 ∘ ccom 5693 ⟶wf 6559 –1-1-onto→wf1o 6562 ‘cfv 6563 (class class class)co 7431 ↑m cmap 8865 finSupp cfsupp 9399 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-rep 5285 ax-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-reu 3379 df-rab 3434 df-v 3480 df-sbc 3792 df-csb 3909 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-pss 3983 df-nul 4340 df-if 4532 df-pw 4607 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-mpt 5232 df-tr 5266 df-id 5583 df-eprel 5589 df-po 5597 df-so 5598 df-fr 5641 df-we 5643 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-ord 6389 df-on 6390 df-lim 6391 df-suc 6392 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-f1 6568 df-fo 6569 df-f1o 6570 df-fv 6571 df-ov 7434 df-oprab 7435 df-mpo 7436 df-om 7888 df-1st 8013 df-2nd 8014 df-supp 8185 df-1o 8505 df-map 8867 df-en 8985 df-dom 8986 df-fin 8988 df-fsupp 9400 |
This theorem is referenced by: eulerpartgbij 34354 |
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