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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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