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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 21956 | . . . 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 6776 | . 2 ⊢ (𝜑 → (𝐹 ↾ 𝐽):𝐽⟶ℕ0) |
7 | 2 | psrbagfsupp 21957 | . . . . 5 ⊢ (𝐹 ∈ 𝐷 → 𝐹 finSupp 0) |
8 | 1, 7 | syl 17 | . . . 4 ⊢ (𝜑 → 𝐹 finSupp 0) |
9 | 0zd 12623 | . . . 4 ⊢ (𝜑 → 0 ∈ ℤ) | |
10 | 8, 9 | fsuppres 9431 | . . 3 ⊢ (𝜑 → (𝐹 ↾ 𝐽) finSupp 0) |
11 | 1 | resexd 6048 | . . . 4 ⊢ (𝜑 → (𝐹 ↾ 𝐽) ∈ V) |
12 | fcdmnn0fsuppg 12584 | . . . 4 ⊢ (((𝐹 ↾ 𝐽) ∈ V ∧ (𝐹 ↾ 𝐽):𝐽⟶ℕ0) → ((𝐹 ↾ 𝐽) finSupp 0 ↔ (◡(𝐹 ↾ 𝐽) “ ℕ) ∈ Fin)) | |
13 | 11, 6, 12 | syl2anc 584 | . . 3 ⊢ (𝜑 → ((𝐹 ↾ 𝐽) finSupp 0 ↔ (◡(𝐹 ↾ 𝐽) “ ℕ) ∈ Fin)) |
14 | 10, 13 | mpbid 232 | . 2 ⊢ (𝜑 → (◡(𝐹 ↾ 𝐽) “ ℕ) ∈ Fin) |
15 | psrbagres.i | . . . 4 ⊢ (𝜑 → 𝐼 ∈ 𝑉) | |
16 | 15, 5 | ssexd 5330 | . . 3 ⊢ (𝜑 → 𝐽 ∈ V) |
17 | psrbagres.e | . . . 4 ⊢ 𝐸 = {𝑔 ∈ (ℕ0 ↑m 𝐽) ∣ (◡𝑔 “ ℕ) ∈ Fin} | |
18 | 17 | psrbag 21955 | . . 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 713 | 1 ⊢ (𝜑 → (𝐹 ↾ 𝐽) ∈ 𝐸) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 {crab 3433 Vcvv 3478 ⊆ wss 3963 class class class wbr 5148 ◡ccnv 5688 ↾ cres 5691 “ cima 5692 ⟶wf 6559 (class class class)co 7431 ↑m cmap 8865 Fincfn 8984 finSupp cfsupp 9399 0cc0 11153 ℕcn 12264 ℕ0cn0 12524 ℤcz 12611 |
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-sep 5302 ax-nul 5312 ax-pow 5371 ax-pr 5438 ax-un 7754 ax-cnex 11209 ax-resscn 11210 ax-1cn 11211 ax-icn 11212 ax-addcl 11213 ax-addrcl 11214 ax-mulcl 11215 ax-mulrcl 11216 ax-mulcom 11217 ax-addass 11218 ax-mulass 11219 ax-distr 11220 ax-i2m1 11221 ax-1ne0 11222 ax-1rid 11223 ax-rnegex 11224 ax-rrecex 11225 ax-cnre 11226 ax-pre-lttri 11227 ax-pre-lttrn 11228 ax-pre-ltadd 11229 |
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-nel 3045 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-pred 6323 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-frecs 8305 df-wrecs 8336 df-recs 8410 df-rdg 8449 df-1o 8505 df-er 8744 df-map 8867 df-en 8985 df-dom 8986 df-sdom 8987 df-fin 8988 df-fsupp 9400 df-pnf 11295 df-mnf 11296 df-xr 11297 df-ltxr 11298 df-le 11299 df-neg 11493 df-nn 12265 df-n0 12525 df-z 12612 |
This theorem is referenced by: selvvvval 42572 evlselvlem 42573 evlselv 42574 |
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