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Mirrors > Home > MPE Home > Th. List > tfr2b | Structured version Visualization version GIF version |
Description: Without assuming ax-rep 5190, 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 7503 | . 2 ⊢ (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On)) | |
2 | eqid 2821 | . . . . 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 8028 | . . . 4 ⊢ (𝐴 ∈ On → (𝐴 ∈ dom recs(𝐺) ↔ (recs(𝐺) ↾ 𝐴) ∈ V)) |
4 | tfr.1 | . . . . . 6 ⊢ 𝐹 = recs(𝐺) | |
5 | 4 | dmeqi 5773 | . . . . 5 ⊢ dom 𝐹 = dom recs(𝐺) |
6 | 5 | eleq2i 2904 | . . . 4 ⊢ (𝐴 ∈ dom 𝐹 ↔ 𝐴 ∈ dom recs(𝐺)) |
7 | 4 | reseq1i 5849 | . . . . 5 ⊢ (𝐹 ↾ 𝐴) = (recs(𝐺) ↾ 𝐴) |
8 | 7 | eleq1i 2903 | . . . 4 ⊢ ((𝐹 ↾ 𝐴) ∈ V ↔ (recs(𝐺) ↾ 𝐴) ∈ V) |
9 | 3, 6, 8 | 3bitr4g 316 | . . 3 ⊢ (𝐴 ∈ On → (𝐴 ∈ dom 𝐹 ↔ (𝐹 ↾ 𝐴) ∈ V)) |
10 | onprc 7499 | . . . . . 6 ⊢ ¬ On ∈ V | |
11 | elex 3512 | . . . . . 6 ⊢ (On ∈ dom 𝐹 → On ∈ V) | |
12 | 10, 11 | mto 199 | . . . . 5 ⊢ ¬ On ∈ dom 𝐹 |
13 | eleq1 2900 | . . . . 5 ⊢ (𝐴 = On → (𝐴 ∈ dom 𝐹 ↔ On ∈ dom 𝐹)) | |
14 | 12, 13 | mtbiri 329 | . . . 4 ⊢ (𝐴 = On → ¬ 𝐴 ∈ dom 𝐹) |
15 | 2 | tfrlem13 8026 | . . . . . 6 ⊢ ¬ recs(𝐺) ∈ V |
16 | 4, 15 | eqneltri 2906 | . . . . 5 ⊢ ¬ 𝐹 ∈ V |
17 | reseq2 5848 | . . . . . . 7 ⊢ (𝐴 = On → (𝐹 ↾ 𝐴) = (𝐹 ↾ On)) | |
18 | 4 | tfr1a 8030 | . . . . . . . . . 10 ⊢ (Fun 𝐹 ∧ Lim dom 𝐹) |
19 | 18 | simpli 486 | . . . . . . . . 9 ⊢ Fun 𝐹 |
20 | funrel 6372 | . . . . . . . . 9 ⊢ (Fun 𝐹 → Rel 𝐹) | |
21 | 19, 20 | ax-mp 5 | . . . . . . . 8 ⊢ Rel 𝐹 |
22 | 18 | simpri 488 | . . . . . . . . 9 ⊢ Lim dom 𝐹 |
23 | limord 6250 | . . . . . . . . 9 ⊢ (Lim dom 𝐹 → Ord dom 𝐹) | |
24 | ordsson 7504 | . . . . . . . . 9 ⊢ (Ord dom 𝐹 → dom 𝐹 ⊆ On) | |
25 | 22, 23, 24 | mp2b 10 | . . . . . . . 8 ⊢ dom 𝐹 ⊆ On |
26 | relssres 5893 | . . . . . . . 8 ⊢ ((Rel 𝐹 ∧ dom 𝐹 ⊆ On) → (𝐹 ↾ On) = 𝐹) | |
27 | 21, 25, 26 | mp2an 690 | . . . . . . 7 ⊢ (𝐹 ↾ On) = 𝐹 |
28 | 17, 27 | syl6eq 2872 | . . . . . 6 ⊢ (𝐴 = On → (𝐹 ↾ 𝐴) = 𝐹) |
29 | 28 | eleq1d 2897 | . . . . 5 ⊢ (𝐴 = On → ((𝐹 ↾ 𝐴) ∈ V ↔ 𝐹 ∈ V)) |
30 | 16, 29 | mtbiri 329 | . . . 4 ⊢ (𝐴 = On → ¬ (𝐹 ↾ 𝐴) ∈ V) |
31 | 14, 30 | 2falsed 379 | . . 3 ⊢ (𝐴 = On → (𝐴 ∈ dom 𝐹 ↔ (𝐹 ↾ 𝐴) ∈ V)) |
32 | 9, 31 | jaoi 853 | . 2 ⊢ ((𝐴 ∈ On ∨ 𝐴 = On) → (𝐴 ∈ dom 𝐹 ↔ (𝐹 ↾ 𝐴) ∈ V)) |
33 | 1, 32 | sylbi 219 | 1 ⊢ (Ord 𝐴 → (𝐴 ∈ dom 𝐹 ↔ (𝐹 ↾ 𝐴) ∈ V)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 208 ∧ wa 398 ∨ wo 843 = wceq 1537 ∈ wcel 2114 {cab 2799 ∀wral 3138 ∃wrex 3139 Vcvv 3494 ⊆ wss 3936 dom cdm 5555 ↾ cres 5557 Rel wrel 5560 Ord word 6190 Oncon0 6191 Lim wlim 6192 Fun wfun 6349 Fn wfn 6350 ‘cfv 6355 recscrecs 8007 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pow 5266 ax-pr 5330 ax-un 7461 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rab 3147 df-v 3496 df-sbc 3773 df-csb 3884 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-pss 3954 df-nul 4292 df-if 4468 df-pw 4541 df-sn 4568 df-pr 4570 df-tp 4572 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-mpt 5147 df-tr 5173 df-id 5460 df-eprel 5465 df-po 5474 df-so 5475 df-fr 5514 df-we 5516 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6314 df-fun 6357 df-fn 6358 df-f 6359 df-f1 6360 df-fo 6361 df-f1o 6362 df-fv 6363 df-wrecs 7947 df-recs 8008 |
This theorem is referenced by: ordtypelem3 8984 ordtypelem9 8990 |
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