![]() |
Intuitionistic Logic Explorer |
< Previous
Next >
Nearby theorems |
|
Mirrors > Home > ILE Home > Th. List > frectfr | GIF version |
Description: Lemma to connect
transfinite recursion theorems with finite recursion.
That is, given the conditions 𝐹 Fn V and A ∈ 𝑉 on
frec(𝐹, A), we want to be able to apply tfri1d 5890 or tfri2d 5891,
and this lemma lets us satisfy hypotheses of those theorems.
(Contributed by Jim Kingdon, 15-Aug-2019.) |
Ref | Expression |
---|---|
frectfr.1 | ⊢ 𝐺 = (g ∈ V ↦ {x ∣ (∃𝑚 ∈ 𝜔 (dom g = suc 𝑚 ∧ x ∈ (𝐹‘(g‘𝑚))) ∨ (dom g = ∅ ∧ x ∈ A))}) |
Ref | Expression |
---|---|
frectfr | ⊢ ((∀z(𝐹‘z) ∈ V ∧ A ∈ 𝑉) → ∀y(Fun 𝐺 ∧ (𝐺‘y) ∈ V)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | vex 2554 | . . . . . . . 8 ⊢ g ∈ V | |
2 | 1 | a1i 9 | . . . . . . 7 ⊢ ((∀z(𝐹‘z) ∈ V ∧ A ∈ 𝑉) → g ∈ V) |
3 | simpl 102 | . . . . . . 7 ⊢ ((∀z(𝐹‘z) ∈ V ∧ A ∈ 𝑉) → ∀z(𝐹‘z) ∈ V) | |
4 | simpr 103 | . . . . . . 7 ⊢ ((∀z(𝐹‘z) ∈ V ∧ A ∈ 𝑉) → A ∈ 𝑉) | |
5 | 2, 3, 4 | frecabex 5923 | . . . . . 6 ⊢ ((∀z(𝐹‘z) ∈ V ∧ A ∈ 𝑉) → {x ∣ (∃𝑚 ∈ 𝜔 (dom g = suc 𝑚 ∧ x ∈ (𝐹‘(g‘𝑚))) ∨ (dom g = ∅ ∧ x ∈ A))} ∈ V) |
6 | 5 | ralrimivw 2387 | . . . . 5 ⊢ ((∀z(𝐹‘z) ∈ V ∧ A ∈ 𝑉) → ∀g ∈ V {x ∣ (∃𝑚 ∈ 𝜔 (dom g = suc 𝑚 ∧ x ∈ (𝐹‘(g‘𝑚))) ∨ (dom g = ∅ ∧ x ∈ A))} ∈ V) |
7 | frectfr.1 | . . . . . 6 ⊢ 𝐺 = (g ∈ V ↦ {x ∣ (∃𝑚 ∈ 𝜔 (dom g = suc 𝑚 ∧ x ∈ (𝐹‘(g‘𝑚))) ∨ (dom g = ∅ ∧ x ∈ A))}) | |
8 | 7 | fnmpt 4968 | . . . . 5 ⊢ (∀g ∈ V {x ∣ (∃𝑚 ∈ 𝜔 (dom g = suc 𝑚 ∧ x ∈ (𝐹‘(g‘𝑚))) ∨ (dom g = ∅ ∧ x ∈ A))} ∈ V → 𝐺 Fn V) |
9 | 6, 8 | syl 14 | . . . 4 ⊢ ((∀z(𝐹‘z) ∈ V ∧ A ∈ 𝑉) → 𝐺 Fn V) |
10 | vex 2554 | . . . 4 ⊢ y ∈ V | |
11 | funfvex 5135 | . . . . 5 ⊢ ((Fun 𝐺 ∧ y ∈ dom 𝐺) → (𝐺‘y) ∈ V) | |
12 | 11 | funfni 4942 | . . . 4 ⊢ ((𝐺 Fn V ∧ y ∈ V) → (𝐺‘y) ∈ V) |
13 | 9, 10, 12 | sylancl 392 | . . 3 ⊢ ((∀z(𝐹‘z) ∈ V ∧ A ∈ 𝑉) → (𝐺‘y) ∈ V) |
14 | 7 | funmpt2 4882 | . . 3 ⊢ Fun 𝐺 |
15 | 13, 14 | jctil 295 | . 2 ⊢ ((∀z(𝐹‘z) ∈ V ∧ A ∈ 𝑉) → (Fun 𝐺 ∧ (𝐺‘y) ∈ V)) |
16 | 15 | alrimiv 1751 | 1 ⊢ ((∀z(𝐹‘z) ∈ V ∧ A ∈ 𝑉) → ∀y(Fun 𝐺 ∧ (𝐺‘y) ∈ V)) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ∨ wo 628 ∀wal 1240 = wceq 1242 ∈ wcel 1390 {cab 2023 ∀wral 2300 ∃wrex 2301 Vcvv 2551 ∅c0 3218 ↦ cmpt 3809 suc csuc 4068 𝜔com 4256 dom cdm 4288 Fun wfun 4839 Fn wfn 4840 ‘cfv 4845 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-13 1401 ax-14 1402 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 ax-coll 3863 ax-sep 3866 ax-pow 3918 ax-pr 3935 ax-un 4136 ax-iinf 4254 |
This theorem depends on definitions: df-bi 110 df-3an 886 df-tru 1245 df-nf 1347 df-sb 1643 df-eu 1900 df-mo 1901 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-ral 2305 df-rex 2306 df-reu 2307 df-rab 2309 df-v 2553 df-sbc 2759 df-csb 2847 df-un 2916 df-in 2918 df-ss 2925 df-pw 3353 df-sn 3373 df-pr 3374 df-op 3376 df-uni 3572 df-int 3607 df-iun 3650 df-br 3756 df-opab 3810 df-mpt 3811 df-id 4021 df-iom 4257 df-xp 4294 df-rel 4295 df-cnv 4296 df-co 4297 df-dm 4298 df-rn 4299 df-res 4300 df-ima 4301 df-iota 4810 df-fun 4847 df-fn 4848 df-f 4849 df-f1 4850 df-fo 4851 df-f1o 4852 df-fv 4853 |
This theorem is referenced by: frecfnom 5925 frecsuclem1 5926 frecsuclemdm 5927 frecsuclem3 5929 |
Copyright terms: Public domain | W3C validator |