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Mirrors > Home > MPE Home > Th. List > tfr2b | Structured version Visualization version GIF version |
Description: Without assuming ax-rep 4902, we can show that all proper initial subsets of recs are sets, while nothing larger is a set. (Contributed by Mario Carneiro, 24-Jun-2015.) |
Ref | Expression |
---|---|
tfr.1 | ⊢ 𝐹 = recs(𝐺) |
Ref | Expression |
---|---|
tfr2b | ⊢ (Ord 𝐴 → (𝐴 ∈ dom 𝐹 ↔ (𝐹 ↾ 𝐴) ∈ V)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | ordeleqon 7134 | . 2 ⊢ (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On)) | |
2 | eqid 2770 | . . . . 5 ⊢ {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))} = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐺‘(𝑓 ↾ 𝑦)))} | |
3 | 2 | tfrlem15 7640 | . . . 4 ⊢ (𝐴 ∈ On → (𝐴 ∈ dom recs(𝐺) ↔ (recs(𝐺) ↾ 𝐴) ∈ V)) |
4 | tfr.1 | . . . . . 6 ⊢ 𝐹 = recs(𝐺) | |
5 | 4 | dmeqi 5463 | . . . . 5 ⊢ dom 𝐹 = dom recs(𝐺) |
6 | 5 | eleq2i 2841 | . . . 4 ⊢ (𝐴 ∈ dom 𝐹 ↔ 𝐴 ∈ dom recs(𝐺)) |
7 | 4 | reseq1i 5530 | . . . . 5 ⊢ (𝐹 ↾ 𝐴) = (recs(𝐺) ↾ 𝐴) |
8 | 7 | eleq1i 2840 | . . . 4 ⊢ ((𝐹 ↾ 𝐴) ∈ V ↔ (recs(𝐺) ↾ 𝐴) ∈ V) |
9 | 3, 6, 8 | 3bitr4g 303 | . . 3 ⊢ (𝐴 ∈ On → (𝐴 ∈ dom 𝐹 ↔ (𝐹 ↾ 𝐴) ∈ V)) |
10 | onprc 7130 | . . . . . 6 ⊢ ¬ On ∈ V | |
11 | elex 3361 | . . . . . 6 ⊢ (On ∈ dom 𝐹 → On ∈ V) | |
12 | 10, 11 | mto 188 | . . . . 5 ⊢ ¬ On ∈ dom 𝐹 |
13 | eleq1 2837 | . . . . 5 ⊢ (𝐴 = On → (𝐴 ∈ dom 𝐹 ↔ On ∈ dom 𝐹)) | |
14 | 12, 13 | mtbiri 316 | . . . 4 ⊢ (𝐴 = On → ¬ 𝐴 ∈ dom 𝐹) |
15 | 2 | tfrlem13 7638 | . . . . . 6 ⊢ ¬ recs(𝐺) ∈ V |
16 | 4 | eleq1i 2840 | . . . . . 6 ⊢ (𝐹 ∈ V ↔ recs(𝐺) ∈ V) |
17 | 15, 16 | mtbir 312 | . . . . 5 ⊢ ¬ 𝐹 ∈ V |
18 | reseq2 5529 | . . . . . . 7 ⊢ (𝐴 = On → (𝐹 ↾ 𝐴) = (𝐹 ↾ On)) | |
19 | 4 | tfr1a 7642 | . . . . . . . . . 10 ⊢ (Fun 𝐹 ∧ Lim dom 𝐹) |
20 | 19 | simpli 470 | . . . . . . . . 9 ⊢ Fun 𝐹 |
21 | funrel 6048 | . . . . . . . . 9 ⊢ (Fun 𝐹 → Rel 𝐹) | |
22 | 20, 21 | ax-mp 5 | . . . . . . . 8 ⊢ Rel 𝐹 |
23 | 19 | simpri 473 | . . . . . . . . 9 ⊢ Lim dom 𝐹 |
24 | limord 5927 | . . . . . . . . 9 ⊢ (Lim dom 𝐹 → Ord dom 𝐹) | |
25 | ordsson 7135 | . . . . . . . . 9 ⊢ (Ord dom 𝐹 → dom 𝐹 ⊆ On) | |
26 | 23, 24, 25 | mp2b 10 | . . . . . . . 8 ⊢ dom 𝐹 ⊆ On |
27 | relssres 5578 | . . . . . . . 8 ⊢ ((Rel 𝐹 ∧ dom 𝐹 ⊆ On) → (𝐹 ↾ On) = 𝐹) | |
28 | 22, 26, 27 | mp2an 664 | . . . . . . 7 ⊢ (𝐹 ↾ On) = 𝐹 |
29 | 18, 28 | syl6eq 2820 | . . . . . 6 ⊢ (𝐴 = On → (𝐹 ↾ 𝐴) = 𝐹) |
30 | 29 | eleq1d 2834 | . . . . 5 ⊢ (𝐴 = On → ((𝐹 ↾ 𝐴) ∈ V ↔ 𝐹 ∈ V)) |
31 | 17, 30 | mtbiri 316 | . . . 4 ⊢ (𝐴 = On → ¬ (𝐹 ↾ 𝐴) ∈ V) |
32 | 14, 31 | 2falsed 365 | . . 3 ⊢ (𝐴 = On → (𝐴 ∈ dom 𝐹 ↔ (𝐹 ↾ 𝐴) ∈ V)) |
33 | 9, 32 | jaoi 837 | . 2 ⊢ ((𝐴 ∈ On ∨ 𝐴 = On) → (𝐴 ∈ dom 𝐹 ↔ (𝐹 ↾ 𝐴) ∈ V)) |
34 | 1, 33 | sylbi 207 | 1 ⊢ (Ord 𝐴 → (𝐴 ∈ dom 𝐹 ↔ (𝐹 ↾ 𝐴) ∈ V)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 196 ∧ wa 382 ∨ wo 826 = wceq 1630 ∈ wcel 2144 {cab 2756 ∀wral 3060 ∃wrex 3061 Vcvv 3349 ⊆ wss 3721 dom cdm 5249 ↾ cres 5251 Rel wrel 5254 Ord word 5865 Oncon0 5866 Lim wlim 5867 Fun wfun 6025 Fn wfn 6026 ‘cfv 6031 recscrecs 7619 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1869 ax-4 1884 ax-5 1990 ax-6 2056 ax-7 2092 ax-8 2146 ax-9 2153 ax-10 2173 ax-11 2189 ax-12 2202 ax-13 2407 ax-ext 2750 ax-sep 4912 ax-nul 4920 ax-pow 4971 ax-pr 5034 ax-un 7095 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 827 df-3or 1071 df-3an 1072 df-tru 1633 df-ex 1852 df-nf 1857 df-sb 2049 df-eu 2621 df-mo 2622 df-clab 2757 df-cleq 2763 df-clel 2766 df-nfc 2901 df-ne 2943 df-ral 3065 df-rex 3066 df-reu 3067 df-rab 3069 df-v 3351 df-sbc 3586 df-csb 3681 df-dif 3724 df-un 3726 df-in 3728 df-ss 3735 df-pss 3737 df-nul 4062 df-if 4224 df-pw 4297 df-sn 4315 df-pr 4317 df-tp 4319 df-op 4321 df-uni 4573 df-iun 4654 df-br 4785 df-opab 4845 df-mpt 4862 df-tr 4885 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-wrecs 7558 df-recs 7620 |
This theorem is referenced by: ordtypelem3 8580 ordtypelem9 8586 |
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