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Theorem tfr2b 7644
Description: Without assuming ax-rep 4902, 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 7134 . 2 (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On))
2 eqid 2770 . . . . 5 {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))} = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))}
32tfrlem15 7640 . . . 4 (𝐴 ∈ On → (𝐴 ∈ dom recs(𝐺) ↔ (recs(𝐺) ↾ 𝐴) ∈ V))
4 tfr.1 . . . . . 6 𝐹 = recs(𝐺)
54dmeqi 5463 . . . . 5 dom 𝐹 = dom recs(𝐺)
65eleq2i 2841 . . . 4 (𝐴 ∈ dom 𝐹𝐴 ∈ dom recs(𝐺))
74reseq1i 5530 . . . . 5 (𝐹𝐴) = (recs(𝐺) ↾ 𝐴)
87eleq1i 2840 . . . 4 ((𝐹𝐴) ∈ V ↔ (recs(𝐺) ↾ 𝐴) ∈ V)
93, 6, 83bitr4g 303 . . 3 (𝐴 ∈ On → (𝐴 ∈ dom 𝐹 ↔ (𝐹𝐴) ∈ V))
10 onprc 7130 . . . . . 6 ¬ On ∈ V
11 elex 3361 . . . . . 6 (On ∈ dom 𝐹 → On ∈ V)
1210, 11mto 188 . . . . 5 ¬ On ∈ dom 𝐹
13 eleq1 2837 . . . . 5 (𝐴 = On → (𝐴 ∈ dom 𝐹 ↔ On ∈ dom 𝐹))
1412, 13mtbiri 316 . . . 4 (𝐴 = On → ¬ 𝐴 ∈ dom 𝐹)
152tfrlem13 7638 . . . . . 6 ¬ recs(𝐺) ∈ V
164eleq1i 2840 . . . . . 6 (𝐹 ∈ V ↔ recs(𝐺) ∈ V)
1715, 16mtbir 312 . . . . 5 ¬ 𝐹 ∈ V
18 reseq2 5529 . . . . . . 7 (𝐴 = On → (𝐹𝐴) = (𝐹 ↾ On))
194tfr1a 7642 . . . . . . . . . 10 (Fun 𝐹 ∧ Lim dom 𝐹)
2019simpli 470 . . . . . . . . 9 Fun 𝐹
21 funrel 6048 . . . . . . . . 9 (Fun 𝐹 → Rel 𝐹)
2220, 21ax-mp 5 . . . . . . . 8 Rel 𝐹
2319simpri 473 . . . . . . . . 9 Lim dom 𝐹
24 limord 5927 . . . . . . . . 9 (Lim dom 𝐹 → Ord dom 𝐹)
25 ordsson 7135 . . . . . . . . 9 (Ord dom 𝐹 → dom 𝐹 ⊆ On)
2623, 24, 25mp2b 10 . . . . . . . 8 dom 𝐹 ⊆ On
27 relssres 5578 . . . . . . . 8 ((Rel 𝐹 ∧ dom 𝐹 ⊆ On) → (𝐹 ↾ On) = 𝐹)
2822, 26, 27mp2an 664 . . . . . . 7 (𝐹 ↾ On) = 𝐹
2918, 28syl6eq 2820 . . . . . 6 (𝐴 = On → (𝐹𝐴) = 𝐹)
3029eleq1d 2834 . . . . 5 (𝐴 = On → ((𝐹𝐴) ∈ V ↔ 𝐹 ∈ V))
3117, 30mtbiri 316 . . . 4 (𝐴 = On → ¬ (𝐹𝐴) ∈ V)
3214, 312falsed 365 . . 3 (𝐴 = On → (𝐴 ∈ dom 𝐹 ↔ (𝐹𝐴) ∈ V))
339, 32jaoi 837 . 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  wa 382  wo 826   = wceq 1630  wcel 2144  {cab 2756  wral 3060  wrex 3061  Vcvv 3349  wss 3721  dom cdm 5249  cres 5251  Rel wrel 5254  Ord word 5865  Oncon0 5866  Lim wlim 5867  Fun wfun 6025   Fn wfn 6026  cfv 6031  recscrecs 7619
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-8 2146  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pow 4971  ax-pr 5034  ax-un 7095
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3or 1071  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-reu 3067  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-pss 3737  df-nul 4062  df-if 4224  df-pw 4297  df-sn 4315  df-pr 4317  df-tp 4319  df-op 4321  df-uni 4573  df-iun 4654  df-br 4785  df-opab 4845  df-mpt 4862  df-tr 4885  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-wrecs 7558  df-recs 7620
This theorem is referenced by:  ordtypelem3  8580  ordtypelem9  8586
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