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Mirrors > Home > ILE Home > Th. List > frecsuclemdm | GIF version |
Description: Lemma for frecsuc 5991. (Contributed by Jim Kingdon, 15-Aug-2019.) |
Ref | Expression |
---|---|
frecsuclem1.h | ⊢ 𝐺 = (𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))}) |
Ref | Expression |
---|---|
frecsuclemdm | ⊢ ((∀𝑧(𝐹‘𝑧) ∈ V ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ ω) → dom (recs(𝐺) ↾ suc 𝐵) = suc 𝐵) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | nnon 4332 | . . . . 5 ⊢ (𝐵 ∈ ω → 𝐵 ∈ On) | |
2 | suceloni 4227 | . . . . 5 ⊢ (𝐵 ∈ On → suc 𝐵 ∈ On) | |
3 | onss 4219 | . . . . 5 ⊢ (suc 𝐵 ∈ On → suc 𝐵 ⊆ On) | |
4 | 1, 2, 3 | 3syl 17 | . . . 4 ⊢ (𝐵 ∈ ω → suc 𝐵 ⊆ On) |
5 | 4 | 3ad2ant3 927 | . . 3 ⊢ ((∀𝑧(𝐹‘𝑧) ∈ V ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ ω) → suc 𝐵 ⊆ On) |
6 | eqid 2040 | . . . . . . 7 ⊢ recs(𝐺) = recs(𝐺) | |
7 | frecsuclem1.h | . . . . . . . 8 ⊢ 𝐺 = (𝑔 ∈ V ↦ {𝑥 ∣ (∃𝑚 ∈ ω (dom 𝑔 = suc 𝑚 ∧ 𝑥 ∈ (𝐹‘(𝑔‘𝑚))) ∨ (dom 𝑔 = ∅ ∧ 𝑥 ∈ 𝐴))}) | |
8 | 7 | frectfr 5985 | . . . . . . 7 ⊢ ((∀𝑧(𝐹‘𝑧) ∈ V ∧ 𝐴 ∈ 𝑉) → ∀𝑦(Fun 𝐺 ∧ (𝐺‘𝑦) ∈ V)) |
9 | 6, 8 | tfri1d 5949 | . . . . . 6 ⊢ ((∀𝑧(𝐹‘𝑧) ∈ V ∧ 𝐴 ∈ 𝑉) → recs(𝐺) Fn On) |
10 | fndm 4998 | . . . . . 6 ⊢ (recs(𝐺) Fn On → dom recs(𝐺) = On) | |
11 | 9, 10 | syl 14 | . . . . 5 ⊢ ((∀𝑧(𝐹‘𝑧) ∈ V ∧ 𝐴 ∈ 𝑉) → dom recs(𝐺) = On) |
12 | 11 | sseq2d 2973 | . . . 4 ⊢ ((∀𝑧(𝐹‘𝑧) ∈ V ∧ 𝐴 ∈ 𝑉) → (suc 𝐵 ⊆ dom recs(𝐺) ↔ suc 𝐵 ⊆ On)) |
13 | 12 | 3adant3 924 | . . 3 ⊢ ((∀𝑧(𝐹‘𝑧) ∈ V ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ ω) → (suc 𝐵 ⊆ dom recs(𝐺) ↔ suc 𝐵 ⊆ On)) |
14 | 5, 13 | mpbird 156 | . 2 ⊢ ((∀𝑧(𝐹‘𝑧) ∈ V ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ ω) → suc 𝐵 ⊆ dom recs(𝐺)) |
15 | ssdmres 4633 | . 2 ⊢ (suc 𝐵 ⊆ dom recs(𝐺) ↔ dom (recs(𝐺) ↾ suc 𝐵) = suc 𝐵) | |
16 | 14, 15 | sylib 127 | 1 ⊢ ((∀𝑧(𝐹‘𝑧) ∈ V ∧ 𝐴 ∈ 𝑉 ∧ 𝐵 ∈ ω) → dom (recs(𝐺) ↾ suc 𝐵) = suc 𝐵) |
Colors of variables: wff set class |
Syntax hints: → wi 4 ∧ wa 97 ↔ wb 98 ∨ wo 629 ∧ w3a 885 ∀wal 1241 = wceq 1243 ∈ wcel 1393 {cab 2026 ∃wrex 2307 Vcvv 2557 ⊆ wss 2917 ∅c0 3224 ↦ cmpt 3818 Oncon0 4100 suc csuc 4102 ωcom 4313 dom cdm 4345 ↾ cres 4347 Fn wfn 4897 ‘cfv 4902 recscrecs 5919 |
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-in1 544 ax-in2 545 ax-io 630 ax-5 1336 ax-7 1337 ax-gen 1338 ax-ie1 1382 ax-ie2 1383 ax-8 1395 ax-10 1396 ax-11 1397 ax-i12 1398 ax-bndl 1399 ax-4 1400 ax-13 1404 ax-14 1405 ax-17 1419 ax-i9 1423 ax-ial 1427 ax-i5r 1428 ax-ext 2022 ax-coll 3872 ax-sep 3875 ax-nul 3883 ax-pow 3927 ax-pr 3944 ax-un 4170 ax-setind 4262 ax-iinf 4311 |
This theorem depends on definitions: df-bi 110 df-3an 887 df-tru 1246 df-fal 1249 df-nf 1350 df-sb 1646 df-eu 1903 df-mo 1904 df-clab 2027 df-cleq 2033 df-clel 2036 df-nfc 2167 df-ne 2206 df-ral 2311 df-rex 2312 df-reu 2313 df-rab 2315 df-v 2559 df-sbc 2765 df-csb 2853 df-dif 2920 df-un 2922 df-in 2924 df-ss 2931 df-nul 3225 df-pw 3361 df-sn 3381 df-pr 3382 df-op 3384 df-uni 3581 df-int 3616 df-iun 3659 df-br 3765 df-opab 3819 df-mpt 3820 df-tr 3855 df-id 4030 df-iord 4103 df-on 4105 df-suc 4108 df-iom 4314 df-xp 4351 df-rel 4352 df-cnv 4353 df-co 4354 df-dm 4355 df-rn 4356 df-res 4357 df-ima 4358 df-iota 4867 df-fun 4904 df-fn 4905 df-f 4906 df-f1 4907 df-fo 4908 df-f1o 4909 df-fv 4910 df-recs 5920 |
This theorem is referenced by: frecsuclem3 5990 |
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