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Theorem tfr2b 8218
Description: Without assuming ax-rep 5214, 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 7626 . 2 (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On))
2 eqid 2740 . . . . 5 {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))} = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))}
32tfrlem15 8214 . . . 4 (𝐴 ∈ On → (𝐴 ∈ dom recs(𝐺) ↔ (recs(𝐺) ↾ 𝐴) ∈ V))
4 tfr.1 . . . . . 6 𝐹 = recs(𝐺)
54dmeqi 5812 . . . . 5 dom 𝐹 = dom recs(𝐺)
65eleq2i 2832 . . . 4 (𝐴 ∈ dom 𝐹𝐴 ∈ dom recs(𝐺))
74reseq1i 5886 . . . . 5 (𝐹𝐴) = (recs(𝐺) ↾ 𝐴)
87eleq1i 2831 . . . 4 ((𝐹𝐴) ∈ V ↔ (recs(𝐺) ↾ 𝐴) ∈ V)
93, 6, 83bitr4g 314 . . 3 (𝐴 ∈ On → (𝐴 ∈ dom 𝐹 ↔ (𝐹𝐴) ∈ V))
10 onprc 7622 . . . . . 6 ¬ On ∈ V
11 elex 3449 . . . . . 6 (On ∈ dom 𝐹 → On ∈ V)
1210, 11mto 196 . . . . 5 ¬ On ∈ dom 𝐹
13 eleq1 2828 . . . . 5 (𝐴 = On → (𝐴 ∈ dom 𝐹 ↔ On ∈ dom 𝐹))
1412, 13mtbiri 327 . . . 4 (𝐴 = On → ¬ 𝐴 ∈ dom 𝐹)
152tfrlem13 8212 . . . . . 6 ¬ recs(𝐺) ∈ V
164, 15eqneltri 2834 . . . . 5 ¬ 𝐹 ∈ V
17 reseq2 5885 . . . . . . 7 (𝐴 = On → (𝐹𝐴) = (𝐹 ↾ On))
184tfr1a 8216 . . . . . . . . . 10 (Fun 𝐹 ∧ Lim dom 𝐹)
1918simpli 484 . . . . . . . . 9 Fun 𝐹
20 funrel 6449 . . . . . . . . 9 (Fun 𝐹 → Rel 𝐹)
2119, 20ax-mp 5 . . . . . . . 8 Rel 𝐹
2218simpri 486 . . . . . . . . 9 Lim dom 𝐹
23 limord 6324 . . . . . . . . 9 (Lim dom 𝐹 → Ord dom 𝐹)
24 ordsson 7627 . . . . . . . . 9 (Ord dom 𝐹 → dom 𝐹 ⊆ On)
2522, 23, 24mp2b 10 . . . . . . . 8 dom 𝐹 ⊆ On
26 relssres 5931 . . . . . . . 8 ((Rel 𝐹 ∧ dom 𝐹 ⊆ On) → (𝐹 ↾ On) = 𝐹)
2721, 25, 26mp2an 689 . . . . . . 7 (𝐹 ↾ On) = 𝐹
2817, 27eqtrdi 2796 . . . . . 6 (𝐴 = On → (𝐹𝐴) = 𝐹)
2928eleq1d 2825 . . . . 5 (𝐴 = On → ((𝐹𝐴) ∈ V ↔ 𝐹 ∈ V))
3016, 29mtbiri 327 . . . 4 (𝐴 = On → ¬ (𝐹𝐴) ∈ V)
3114, 302falsed 377 . . 3 (𝐴 = On → (𝐴 ∈ dom 𝐹 ↔ (𝐹𝐴) ∈ V))
329, 31jaoi 854 . 2 ((𝐴 ∈ On ∨ 𝐴 = On) → (𝐴 ∈ dom 𝐹 ↔ (𝐹𝐴) ∈ V))
331, 32sylbi 216 1 (Ord 𝐴 → (𝐴 ∈ dom 𝐹 ↔ (𝐹𝐴) ∈ V))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 205  wa 396  wo 844   = wceq 1542  wcel 2110  {cab 2717  wral 3066  wrex 3067  Vcvv 3431  wss 3892  dom cdm 5590  cres 5592  Rel wrel 5595  Ord word 6264  Oncon0 6265  Lim wlim 6266  Fun wfun 6426   Fn wfn 6427  cfv 6432  recscrecs 8192
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2158  ax-12 2175  ax-ext 2711  ax-sep 5227  ax-nul 5234  ax-pr 5356  ax-un 7582
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-mo 2542  df-eu 2571  df-clab 2718  df-cleq 2732  df-clel 2818  df-nfc 2891  df-ne 2946  df-ral 3071  df-rex 3072  df-reu 3073  df-rab 3075  df-v 3433  df-sbc 3721  df-csb 3838  df-dif 3895  df-un 3897  df-in 3899  df-ss 3909  df-pss 3911  df-nul 4263  df-if 4466  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4846  df-iun 4932  df-br 5080  df-opab 5142  df-mpt 5163  df-tr 5197  df-id 5490  df-eprel 5496  df-po 5504  df-so 5505  df-fr 5545  df-we 5547  df-xp 5596  df-rel 5597  df-cnv 5598  df-co 5599  df-dm 5600  df-rn 5601  df-res 5602  df-ima 5603  df-pred 6201  df-ord 6268  df-on 6269  df-lim 6270  df-suc 6271  df-iota 6390  df-fun 6434  df-fn 6435  df-f 6436  df-f1 6437  df-fo 6438  df-f1o 6439  df-fv 6440  df-ov 7274  df-2nd 7825  df-frecs 8088  df-wrecs 8119  df-recs 8193
This theorem is referenced by:  ordtypelem3  9257  ordtypelem9  9263
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