MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tfr2b Structured version   Visualization version   GIF version

Theorem tfr2b 7351
Description: Without assuming ax-rep 4688, 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.)
Hypothesis
Ref Expression
tfr.1 𝐹 = recs(𝐺)
Assertion
Ref Expression
tfr2b (Ord 𝐴 → (𝐴 ∈ dom 𝐹 ↔ (𝐹𝐴) ∈ V))

Proof of Theorem tfr2b
Dummy variables 𝑥 𝑓 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ordeleqon 6852 . 2 (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On))
2 eqid 2604 . . . . 5 {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))} = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))}
32tfrlem15 7347 . . . 4 (𝐴 ∈ On → (𝐴 ∈ dom recs(𝐺) ↔ (recs(𝐺) ↾ 𝐴) ∈ V))
4 tfr.1 . . . . . 6 𝐹 = recs(𝐺)
54dmeqi 5229 . . . . 5 dom 𝐹 = dom recs(𝐺)
65eleq2i 2674 . . . 4 (𝐴 ∈ dom 𝐹𝐴 ∈ dom recs(𝐺))
74reseq1i 5295 . . . . 5 (𝐹𝐴) = (recs(𝐺) ↾ 𝐴)
87eleq1i 2673 . . . 4 ((𝐹𝐴) ∈ V ↔ (recs(𝐺) ↾ 𝐴) ∈ V)
93, 6, 83bitr4g 301 . . 3 (𝐴 ∈ On → (𝐴 ∈ dom 𝐹 ↔ (𝐹𝐴) ∈ V))
10 onprc 6848 . . . . . 6 ¬ On ∈ V
11 elex 3179 . . . . . 6 (On ∈ dom 𝐹 → On ∈ V)
1210, 11mto 186 . . . . 5 ¬ On ∈ dom 𝐹
13 eleq1 2670 . . . . 5 (𝐴 = On → (𝐴 ∈ dom 𝐹 ↔ On ∈ dom 𝐹))
1412, 13mtbiri 315 . . . 4 (𝐴 = On → ¬ 𝐴 ∈ dom 𝐹)
152tfrlem13 7345 . . . . . 6 ¬ recs(𝐺) ∈ V
164eleq1i 2673 . . . . . 6 (𝐹 ∈ V ↔ recs(𝐺) ∈ V)
1715, 16mtbir 311 . . . . 5 ¬ 𝐹 ∈ V
18 reseq2 5294 . . . . . . 7 (𝐴 = On → (𝐹𝐴) = (𝐹 ↾ On))
194tfr1a 7349 . . . . . . . . . 10 (Fun 𝐹 ∧ Lim dom 𝐹)
2019simpli 472 . . . . . . . . 9 Fun 𝐹
21 funrel 5802 . . . . . . . . 9 (Fun 𝐹 → Rel 𝐹)
2220, 21ax-mp 5 . . . . . . . 8 Rel 𝐹
2319simpri 476 . . . . . . . . 9 Lim dom 𝐹
24 limord 5682 . . . . . . . . 9 (Lim dom 𝐹 → Ord dom 𝐹)
25 ordsson 6853 . . . . . . . . 9 (Ord dom 𝐹 → dom 𝐹 ⊆ On)
2623, 24, 25mp2b 10 . . . . . . . 8 dom 𝐹 ⊆ On
27 relssres 5339 . . . . . . . 8 ((Rel 𝐹 ∧ dom 𝐹 ⊆ On) → (𝐹 ↾ On) = 𝐹)
2822, 26, 27mp2an 703 . . . . . . 7 (𝐹 ↾ On) = 𝐹
2918, 28syl6eq 2654 . . . . . 6 (𝐴 = On → (𝐹𝐴) = 𝐹)
3029eleq1d 2666 . . . . 5 (𝐴 = On → ((𝐹𝐴) ∈ V ↔ 𝐹 ∈ V))
3117, 30mtbiri 315 . . . 4 (𝐴 = On → ¬ (𝐹𝐴) ∈ V)
3214, 312falsed 364 . . 3 (𝐴 = On → (𝐴 ∈ dom 𝐹 ↔ (𝐹𝐴) ∈ V))
339, 32jaoi 392 . 2 ((𝐴 ∈ On ∨ 𝐴 = On) → (𝐴 ∈ dom 𝐹 ↔ (𝐹𝐴) ∈ V))
341, 33sylbi 205 1 (Ord 𝐴 → (𝐴 ∈ dom 𝐹 ↔ (𝐹𝐴) ∈ V))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 194  wo 381  wa 382   = wceq 1474  wcel 1975  {cab 2590  wral 2890  wrex 2891  Vcvv 3167  wss 3534  dom cdm 5023  cres 5025  Rel wrel 5028  Ord word 5620  Oncon0 5621  Lim wlim 5622  Fun wfun 5779   Fn wfn 5780  cfv 5785  recscrecs 7326
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1711  ax-4 1726  ax-5 1825  ax-6 1873  ax-7 1920  ax-8 1977  ax-9 1984  ax-10 2004  ax-11 2019  ax-12 2031  ax-13 2227  ax-ext 2584  ax-sep 4698  ax-nul 4707  ax-pow 4759  ax-pr 4823  ax-un 6819
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1866  df-eu 2456  df-mo 2457  df-clab 2591  df-cleq 2597  df-clel 2600  df-nfc 2734  df-ne 2776  df-ral 2895  df-rex 2896  df-reu 2897  df-rab 2899  df-v 3169  df-sbc 3397  df-csb 3494  df-dif 3537  df-un 3539  df-in 3541  df-ss 3548  df-pss 3550  df-nul 3869  df-if 4031  df-pw 4104  df-sn 4120  df-pr 4122  df-tp 4124  df-op 4126  df-uni 4362  df-iun 4446  df-br 4573  df-opab 4633  df-mpt 4634  df-tr 4670  df-eprel 4934  df-id 4938  df-po 4944  df-so 4945  df-fr 4982  df-we 4984  df-xp 5029  df-rel 5030  df-cnv 5031  df-co 5032  df-dm 5033  df-rn 5034  df-res 5035  df-ima 5036  df-pred 5578  df-ord 5624  df-on 5625  df-lim 5626  df-suc 5627  df-iota 5749  df-fun 5787  df-fn 5788  df-f 5789  df-f1 5790  df-fo 5791  df-f1o 5792  df-fv 5793  df-wrecs 7266  df-recs 7327
This theorem is referenced by:  ordtypelem3  8280  ordtypelem9  8286
  Copyright terms: Public domain W3C validator