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Mirrors > Home > MPE Home > Th. List > Mathboxes > psrbagres | Structured version Visualization version GIF version |
Description: Restrict a bag of variables in 𝐼 to a bag of variables in 𝐽 ⊆ 𝐼. (Contributed by SN, 10-Mar-2025.) |
Ref | Expression |
---|---|
psrbagres.d | ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} |
psrbagres.e | ⊢ 𝐸 = {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} |
psrbagres.i | ⊢ (𝜑 → 𝐼 ∈ 𝑉) |
psrbagres.j | ⊢ (𝜑 → 𝐽 ⊆ 𝐼) |
psrbagres.f | ⊢ (𝜑 → 𝐹 ∈ 𝐷) |
Ref | Expression |
---|---|
psrbagres | ⊢ (𝜑 → (𝐹 ↾ 𝐽) ∈ 𝐸) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | psrbagres.f | . . . 4 ⊢ (𝜑 → 𝐹 ∈ 𝐷) | |
2 | psrbagres.d | . . . . 5 ⊢ 𝐷 = {ℎ ∈ (ℕ0 ↑m 𝐼) ∣ (◡ℎ “ ℕ) ∈ Fin} | |
3 | 2 | psrbagf 21771 | . . . 4 ⊢ (𝐹 ∈ 𝐷 → 𝐹:𝐼⟶ℕ0) |
4 | 1, 3 | syl 17 | . . 3 ⊢ (𝜑 → 𝐹:𝐼⟶ℕ0) |
5 | psrbagres.j | . . 3 ⊢ (𝜑 → 𝐽 ⊆ 𝐼) | |
6 | 4, 5 | fssresd 6748 | . 2 ⊢ (𝜑 → (𝐹 ↾ 𝐽):𝐽⟶ℕ0) |
7 | 2 | psrbagfsupp 21773 | . . . . 5 ⊢ (𝐹 ∈ 𝐷 → 𝐹 finSupp 0) |
8 | 1, 7 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐹 finSupp 0) |
9 | 0zd 12566 | . . . 4 ⊢ (𝜑 → 0 ∈ ℤ) | |
10 | 8, 9 | fsuppres 9383 | . . 3 ⊢ (𝜑 → (𝐹 ↾ 𝐽) finSupp 0) |
11 | 1 | resexd 6018 | . . . 4 ⊢ (𝜑 → (𝐹 ↾ 𝐽) ∈ V) |
12 | fcdmnn0fsuppg 12527 | . . . 4 ⊢ (((𝐹 ↾ 𝐽) ∈ V ∧ (𝐹 ↾ 𝐽):𝐽⟶ℕ0) → ((𝐹 ↾ 𝐽) finSupp 0 ↔ (◡(𝐹 ↾ 𝐽) “ ℕ) ∈ Fin)) | |
13 | 11, 6, 12 | syl2anc 583 | . . 3 ⊢ (𝜑 → ((𝐹 ↾ 𝐽) finSupp 0 ↔ (◡(𝐹 ↾ 𝐽) “ ℕ) ∈ Fin)) |
14 | 10, 13 | mpbid 231 | . 2 ⊢ (𝜑 → (◡(𝐹 ↾ 𝐽) “ ℕ) ∈ Fin) |
15 | psrbagres.i | . . . 4 ⊢ (𝜑 → 𝐼 ∈ 𝑉) | |
16 | 15, 5 | ssexd 5314 | . . 3 ⊢ (𝜑 → 𝐽 ∈ V) |
17 | psrbagres.e | . . . 4 ⊢ 𝐸 = {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} | |
18 | 17 | psrbag 21770 | . . 3 ⊢ (𝐽 ∈ V → ((𝐹 ↾ 𝐽) ∈ 𝐸 ↔ ((𝐹 ↾ 𝐽):𝐽⟶ℕ0 ∧ (◡(𝐹 ↾ 𝐽) “ ℕ) ∈ Fin))) |
19 | 16, 18 | syl 17 | . 2 ⊢ (𝜑 → ((𝐹 ↾ 𝐽) ∈ 𝐸 ↔ ((𝐹 ↾ 𝐽):𝐽⟶ℕ0 ∧ (◡(𝐹 ↾ 𝐽) “ ℕ) ∈ Fin))) |
20 | 6, 14, 19 | mpbir2and 710 | 1 ⊢ (𝜑 → (𝐹 ↾ 𝐽) ∈ 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 = wceq 1533 ∈ wcel 2098 {crab 3424 Vcvv 3466 ⊆ wss 3940 class class class wbr 5138 ◡ccnv 5665 ↾ cres 5668 “ cima 5669 ⟶wf 6529 (class class class)co 7401 ↑m cmap 8815 Fincfn 8934 finSupp cfsupp 9356 0cc0 11105 ℕcn 12208 ℕ0cn0 12468 ℤcz 12554 |
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 2695 ax-sep 5289 ax-nul 5296 ax-pow 5353 ax-pr 5417 ax-un 7718 ax-cnex 11161 ax-resscn 11162 ax-1cn 11163 ax-icn 11164 ax-addcl 11165 ax-addrcl 11166 ax-mulcl 11167 ax-mulrcl 11168 ax-mulcom 11169 ax-addass 11170 ax-mulass 11171 ax-distr 11172 ax-i2m1 11173 ax-1ne0 11174 ax-1rid 11175 ax-rnegex 11176 ax-rrecex 11177 ax-cnre 11178 ax-pre-lttri 11179 ax-pre-lttrn 11180 ax-pre-ltadd 11181 |
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 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-nel 3039 df-ral 3054 df-rex 3063 df-reu 3369 df-rab 3425 df-v 3468 df-sbc 3770 df-csb 3886 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-pss 3959 df-nul 4315 df-if 4521 df-pw 4596 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-iun 4989 df-br 5139 df-opab 5201 df-mpt 5222 df-tr 5256 df-id 5564 df-eprel 5570 df-po 5578 df-so 5579 df-fr 5621 df-we 5623 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-pred 6290 df-ord 6357 df-on 6358 df-lim 6359 df-suc 6360 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-f1 6538 df-fo 6539 df-f1o 6540 df-fv 6541 df-ov 7404 df-oprab 7405 df-mpo 7406 df-om 7849 df-1st 7968 df-2nd 7969 df-supp 8141 df-frecs 8261 df-wrecs 8292 df-recs 8366 df-rdg 8405 df-1o 8461 df-er 8698 df-map 8817 df-en 8935 df-dom 8936 df-sdom 8937 df-fin 8938 df-fsupp 9357 df-pnf 11246 df-mnf 11247 df-xr 11248 df-ltxr 11249 df-le 11250 df-neg 11443 df-nn 12209 df-n0 12469 df-z 12555 |
This theorem is referenced by: selvvvval 41612 evlselvlem 41613 evlselv 41614 |
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