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Theorem tfr2b 7477
Description: Without assuming ax-rep 4762, 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 6973 . 2 (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On))
2 eqid 2620 . . . . 5 {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))} = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))}
32tfrlem15 7473 . . . 4 (𝐴 ∈ On → (𝐴 ∈ dom recs(𝐺) ↔ (recs(𝐺) ↾ 𝐴) ∈ V))
4 tfr.1 . . . . . 6 𝐹 = recs(𝐺)
54dmeqi 5314 . . . . 5 dom 𝐹 = dom recs(𝐺)
65eleq2i 2691 . . . 4 (𝐴 ∈ dom 𝐹𝐴 ∈ dom recs(𝐺))
74reseq1i 5381 . . . . 5 (𝐹𝐴) = (recs(𝐺) ↾ 𝐴)
87eleq1i 2690 . . . 4 ((𝐹𝐴) ∈ V ↔ (recs(𝐺) ↾ 𝐴) ∈ V)
93, 6, 83bitr4g 303 . . 3 (𝐴 ∈ On → (𝐴 ∈ dom 𝐹 ↔ (𝐹𝐴) ∈ V))
10 onprc 6969 . . . . . 6 ¬ On ∈ V
11 elex 3207 . . . . . 6 (On ∈ dom 𝐹 → On ∈ V)
1210, 11mto 188 . . . . 5 ¬ On ∈ dom 𝐹
13 eleq1 2687 . . . . 5 (𝐴 = On → (𝐴 ∈ dom 𝐹 ↔ On ∈ dom 𝐹))
1412, 13mtbiri 317 . . . 4 (𝐴 = On → ¬ 𝐴 ∈ dom 𝐹)
152tfrlem13 7471 . . . . . 6 ¬ recs(𝐺) ∈ V
164eleq1i 2690 . . . . . 6 (𝐹 ∈ V ↔ recs(𝐺) ∈ V)
1715, 16mtbir 313 . . . . 5 ¬ 𝐹 ∈ V
18 reseq2 5380 . . . . . . 7 (𝐴 = On → (𝐹𝐴) = (𝐹 ↾ On))
194tfr1a 7475 . . . . . . . . . 10 (Fun 𝐹 ∧ Lim dom 𝐹)
2019simpli 474 . . . . . . . . 9 Fun 𝐹
21 funrel 5893 . . . . . . . . 9 (Fun 𝐹 → Rel 𝐹)
2220, 21ax-mp 5 . . . . . . . 8 Rel 𝐹
2319simpri 478 . . . . . . . . 9 Lim dom 𝐹
24 limord 5772 . . . . . . . . 9 (Lim dom 𝐹 → Ord dom 𝐹)
25 ordsson 6974 . . . . . . . . 9 (Ord dom 𝐹 → dom 𝐹 ⊆ On)
2623, 24, 25mp2b 10 . . . . . . . 8 dom 𝐹 ⊆ On
27 relssres 5425 . . . . . . . 8 ((Rel 𝐹 ∧ dom 𝐹 ⊆ On) → (𝐹 ↾ On) = 𝐹)
2822, 26, 27mp2an 707 . . . . . . 7 (𝐹 ↾ On) = 𝐹
2918, 28syl6eq 2670 . . . . . 6 (𝐴 = On → (𝐹𝐴) = 𝐹)
3029eleq1d 2684 . . . . 5 (𝐴 = On → ((𝐹𝐴) ∈ V ↔ 𝐹 ∈ V))
3117, 30mtbiri 317 . . . 4 (𝐴 = On → ¬ (𝐹𝐴) ∈ V)
3214, 312falsed 366 . . 3 (𝐴 = On → (𝐴 ∈ dom 𝐹 ↔ (𝐹𝐴) ∈ V))
339, 32jaoi 394 . 2 ((𝐴 ∈ On ∨ 𝐴 = On) → (𝐴 ∈ dom 𝐹 ↔ (𝐹𝐴) ∈ V))
341, 33sylbi 207 1 (Ord 𝐴 → (𝐴 ∈ dom 𝐹 ↔ (𝐹𝐴) ∈ V))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wo 383  wa 384   = wceq 1481  wcel 1988  {cab 2606  wral 2909  wrex 2910  Vcvv 3195  wss 3567  dom cdm 5104  cres 5106  Rel wrel 5109  Ord word 5710  Oncon0 5711  Lim wlim 5712  Fun wfun 5870   Fn wfn 5871  cfv 5876  recscrecs 7452
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1720  ax-4 1735  ax-5 1837  ax-6 1886  ax-7 1933  ax-8 1990  ax-9 1997  ax-10 2017  ax-11 2032  ax-12 2045  ax-13 2244  ax-ext 2600  ax-sep 4772  ax-nul 4780  ax-pow 4834  ax-pr 4897  ax-un 6934
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1037  df-3an 1038  df-tru 1484  df-ex 1703  df-nf 1708  df-sb 1879  df-eu 2472  df-mo 2473  df-clab 2607  df-cleq 2613  df-clel 2616  df-nfc 2751  df-ne 2792  df-ral 2914  df-rex 2915  df-reu 2916  df-rab 2918  df-v 3197  df-sbc 3430  df-csb 3527  df-dif 3570  df-un 3572  df-in 3574  df-ss 3581  df-pss 3583  df-nul 3908  df-if 4078  df-pw 4151  df-sn 4169  df-pr 4171  df-tp 4173  df-op 4175  df-uni 4428  df-iun 4513  df-br 4645  df-opab 4704  df-mpt 4721  df-tr 4744  df-id 5014  df-eprel 5019  df-po 5025  df-so 5026  df-fr 5063  df-we 5065  df-xp 5110  df-rel 5111  df-cnv 5112  df-co 5113  df-dm 5114  df-rn 5115  df-res 5116  df-ima 5117  df-pred 5668  df-ord 5714  df-on 5715  df-lim 5716  df-suc 5717  df-iota 5839  df-fun 5878  df-fn 5879  df-f 5880  df-f1 5881  df-fo 5882  df-f1o 5883  df-fv 5884  df-wrecs 7392  df-recs 7453
This theorem is referenced by:  ordtypelem3  8410  ordtypelem9  8416
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