Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > tfr2b | Structured version Visualization version GIF version |
Description: Without assuming ax-rep 5164, 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 7544 | . 2 ⊢ (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On)) | |
2 | eqid 2736 | . . . . 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 8106 | . . . 4 ⊢ (𝐴 ∈ On → (𝐴 ∈ dom recs(𝐺) ↔ (recs(𝐺) ↾ 𝐴) ∈ V)) |
4 | tfr.1 | . . . . . 6 ⊢ 𝐹 = recs(𝐺) | |
5 | 4 | dmeqi 5758 | . . . . 5 ⊢ dom 𝐹 = dom recs(𝐺) |
6 | 5 | eleq2i 2822 | . . . 4 ⊢ (𝐴 ∈ dom 𝐹 ↔ 𝐴 ∈ dom recs(𝐺)) |
7 | 4 | reseq1i 5832 | . . . . 5 ⊢ (𝐹 ↾ 𝐴) = (recs(𝐺) ↾ 𝐴) |
8 | 7 | eleq1i 2821 | . . . 4 ⊢ ((𝐹 ↾ 𝐴) ∈ V ↔ (recs(𝐺) ↾ 𝐴) ∈ V) |
9 | 3, 6, 8 | 3bitr4g 317 | . . 3 ⊢ (𝐴 ∈ On → (𝐴 ∈ dom 𝐹 ↔ (𝐹 ↾ 𝐴) ∈ V)) |
10 | onprc 7540 | . . . . . 6 ⊢ ¬ On ∈ V | |
11 | elex 3416 | . . . . . 6 ⊢ (On ∈ dom 𝐹 → On ∈ V) | |
12 | 10, 11 | mto 200 | . . . . 5 ⊢ ¬ On ∈ dom 𝐹 |
13 | eleq1 2818 | . . . . 5 ⊢ (𝐴 = On → (𝐴 ∈ dom 𝐹 ↔ On ∈ dom 𝐹)) | |
14 | 12, 13 | mtbiri 330 | . . . 4 ⊢ (𝐴 = On → ¬ 𝐴 ∈ dom 𝐹) |
15 | 2 | tfrlem13 8104 | . . . . . 6 ⊢ ¬ recs(𝐺) ∈ V |
16 | 4, 15 | eqneltri 2824 | . . . . 5 ⊢ ¬ 𝐹 ∈ V |
17 | reseq2 5831 | . . . . . . 7 ⊢ (𝐴 = On → (𝐹 ↾ 𝐴) = (𝐹 ↾ On)) | |
18 | 4 | tfr1a 8108 | . . . . . . . . . 10 ⊢ (Fun 𝐹 ∧ Lim dom 𝐹) |
19 | 18 | simpli 487 | . . . . . . . . 9 ⊢ Fun 𝐹 |
20 | funrel 6375 | . . . . . . . . 9 ⊢ (Fun 𝐹 → Rel 𝐹) | |
21 | 19, 20 | ax-mp 5 | . . . . . . . 8 ⊢ Rel 𝐹 |
22 | 18 | simpri 489 | . . . . . . . . 9 ⊢ Lim dom 𝐹 |
23 | limord 6250 | . . . . . . . . 9 ⊢ (Lim dom 𝐹 → Ord dom 𝐹) | |
24 | ordsson 7545 | . . . . . . . . 9 ⊢ (Ord dom 𝐹 → dom 𝐹 ⊆ On) | |
25 | 22, 23, 24 | mp2b 10 | . . . . . . . 8 ⊢ dom 𝐹 ⊆ On |
26 | relssres 5877 | . . . . . . . 8 ⊢ ((Rel 𝐹 ∧ dom 𝐹 ⊆ On) → (𝐹 ↾ On) = 𝐹) | |
27 | 21, 25, 26 | mp2an 692 | . . . . . . 7 ⊢ (𝐹 ↾ On) = 𝐹 |
28 | 17, 27 | eqtrdi 2787 | . . . . . 6 ⊢ (𝐴 = On → (𝐹 ↾ 𝐴) = 𝐹) |
29 | 28 | eleq1d 2815 | . . . . 5 ⊢ (𝐴 = On → ((𝐹 ↾ 𝐴) ∈ V ↔ 𝐹 ∈ V)) |
30 | 16, 29 | mtbiri 330 | . . . 4 ⊢ (𝐴 = On → ¬ (𝐹 ↾ 𝐴) ∈ V) |
31 | 14, 30 | 2falsed 380 | . . 3 ⊢ (𝐴 = On → (𝐴 ∈ dom 𝐹 ↔ (𝐹 ↾ 𝐴) ∈ V)) |
32 | 9, 31 | jaoi 857 | . 2 ⊢ ((𝐴 ∈ On ∨ 𝐴 = On) → (𝐴 ∈ dom 𝐹 ↔ (𝐹 ↾ 𝐴) ∈ V)) |
33 | 1, 32 | sylbi 220 | 1 ⊢ (Ord 𝐴 → (𝐴 ∈ dom 𝐹 ↔ (𝐹 ↾ 𝐴) ∈ V)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∨ wo 847 = wceq 1543 ∈ wcel 2112 {cab 2714 ∀wral 3051 ∃wrex 3052 Vcvv 3398 ⊆ wss 3853 dom cdm 5536 ↾ cres 5538 Rel wrel 5541 Ord word 6190 Oncon0 6191 Lim wlim 6192 Fun wfun 6352 Fn wfn 6353 ‘cfv 6358 recscrecs 8085 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2018 ax-8 2114 ax-9 2122 ax-10 2143 ax-11 2160 ax-12 2177 ax-ext 2708 ax-sep 5177 ax-nul 5184 ax-pr 5307 ax-un 7501 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2073 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2728 df-clel 2809 df-nfc 2879 df-ne 2933 df-ral 3056 df-rex 3057 df-reu 3058 df-rab 3060 df-v 3400 df-sbc 3684 df-csb 3799 df-dif 3856 df-un 3858 df-in 3860 df-ss 3870 df-pss 3872 df-nul 4224 df-if 4426 df-pw 4501 df-sn 4528 df-pr 4530 df-tp 4532 df-op 4534 df-uni 4806 df-iun 4892 df-br 5040 df-opab 5102 df-mpt 5121 df-tr 5147 df-id 5440 df-eprel 5445 df-po 5453 df-so 5454 df-fr 5494 df-we 5496 df-xp 5542 df-rel 5543 df-cnv 5544 df-co 5545 df-dm 5546 df-rn 5547 df-res 5548 df-ima 5549 df-pred 6140 df-ord 6194 df-on 6195 df-lim 6196 df-suc 6197 df-iota 6316 df-fun 6360 df-fn 6361 df-f 6362 df-f1 6363 df-fo 6364 df-f1o 6365 df-fv 6366 df-wrecs 8025 df-recs 8086 |
This theorem is referenced by: ordtypelem3 9114 ordtypelem9 9120 |
Copyright terms: Public domain | W3C validator |