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Theorem tfr2b 8019
Description: Without assuming ax-rep 5157, 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 7487 . 2 (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On))
2 eqid 2801 . . . . 5 {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))} = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))}
32tfrlem15 8015 . . . 4 (𝐴 ∈ On → (𝐴 ∈ dom recs(𝐺) ↔ (recs(𝐺) ↾ 𝐴) ∈ V))
4 tfr.1 . . . . . 6 𝐹 = recs(𝐺)
54dmeqi 5741 . . . . 5 dom 𝐹 = dom recs(𝐺)
65eleq2i 2884 . . . 4 (𝐴 ∈ dom 𝐹𝐴 ∈ dom recs(𝐺))
74reseq1i 5818 . . . . 5 (𝐹𝐴) = (recs(𝐺) ↾ 𝐴)
87eleq1i 2883 . . . 4 ((𝐹𝐴) ∈ V ↔ (recs(𝐺) ↾ 𝐴) ∈ V)
93, 6, 83bitr4g 317 . . 3 (𝐴 ∈ On → (𝐴 ∈ dom 𝐹 ↔ (𝐹𝐴) ∈ V))
10 onprc 7483 . . . . . 6 ¬ On ∈ V
11 elex 3462 . . . . . 6 (On ∈ dom 𝐹 → On ∈ V)
1210, 11mto 200 . . . . 5 ¬ On ∈ dom 𝐹
13 eleq1 2880 . . . . 5 (𝐴 = On → (𝐴 ∈ dom 𝐹 ↔ On ∈ dom 𝐹))
1412, 13mtbiri 330 . . . 4 (𝐴 = On → ¬ 𝐴 ∈ dom 𝐹)
152tfrlem13 8013 . . . . . 6 ¬ recs(𝐺) ∈ V
164, 15eqneltri 2886 . . . . 5 ¬ 𝐹 ∈ V
17 reseq2 5817 . . . . . . 7 (𝐴 = On → (𝐹𝐴) = (𝐹 ↾ On))
184tfr1a 8017 . . . . . . . . . 10 (Fun 𝐹 ∧ Lim dom 𝐹)
1918simpli 487 . . . . . . . . 9 Fun 𝐹
20 funrel 6345 . . . . . . . . 9 (Fun 𝐹 → Rel 𝐹)
2119, 20ax-mp 5 . . . . . . . 8 Rel 𝐹
2218simpri 489 . . . . . . . . 9 Lim dom 𝐹
23 limord 6222 . . . . . . . . 9 (Lim dom 𝐹 → Ord dom 𝐹)
24 ordsson 7488 . . . . . . . . 9 (Ord dom 𝐹 → dom 𝐹 ⊆ On)
2522, 23, 24mp2b 10 . . . . . . . 8 dom 𝐹 ⊆ On
26 relssres 5863 . . . . . . . 8 ((Rel 𝐹 ∧ dom 𝐹 ⊆ On) → (𝐹 ↾ On) = 𝐹)
2721, 25, 26mp2an 691 . . . . . . 7 (𝐹 ↾ On) = 𝐹
2817, 27eqtrdi 2852 . . . . . 6 (𝐴 = On → (𝐹𝐴) = 𝐹)
2928eleq1d 2877 . . . . 5 (𝐴 = On → ((𝐹𝐴) ∈ V ↔ 𝐹 ∈ V))
3016, 29mtbiri 330 . . . 4 (𝐴 = On → ¬ (𝐹𝐴) ∈ V)
3114, 302falsed 380 . . 3 (𝐴 = On → (𝐴 ∈ dom 𝐹 ↔ (𝐹𝐴) ∈ V))
329, 31jaoi 854 . 2 ((𝐴 ∈ On ∨ 𝐴 = On) → (𝐴 ∈ dom 𝐹 ↔ (𝐹𝐴) ∈ V))
331, 32sylbi 220 1 (Ord 𝐴 → (𝐴 ∈ dom 𝐹 ↔ (𝐹𝐴) ∈ V))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399  wo 844   = wceq 1538  wcel 2112  {cab 2779  wral 3109  wrex 3110  Vcvv 3444  wss 3884  dom cdm 5523  cres 5525  Rel wrel 5528  Ord word 6162  Oncon0 6163  Lim wlim 6164  Fun wfun 6322   Fn wfn 6323  cfv 6328  recscrecs 7994
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298  ax-un 7445
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ne 2991  df-ral 3114  df-rex 3115  df-reu 3116  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-pss 3903  df-nul 4247  df-if 4429  df-pw 4502  df-sn 4529  df-pr 4531  df-tp 4533  df-op 4535  df-uni 4804  df-iun 4886  df-br 5034  df-opab 5096  df-mpt 5114  df-tr 5140  df-id 5428  df-eprel 5433  df-po 5442  df-so 5443  df-fr 5482  df-we 5484  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-pred 6120  df-ord 6166  df-on 6167  df-lim 6168  df-suc 6169  df-iota 6287  df-fun 6330  df-fn 6331  df-f 6332  df-f1 6333  df-fo 6334  df-f1o 6335  df-fv 6336  df-wrecs 7934  df-recs 7995
This theorem is referenced by:  ordtypelem3  8972  ordtypelem9  8978
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