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Theorem tfr2b 8110
Description: Without assuming ax-rep 5164, 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 7544 . 2 (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On))
2 eqid 2736 . . . . 5 {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))} = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))}
32tfrlem15 8106 . . . 4 (𝐴 ∈ On → (𝐴 ∈ dom recs(𝐺) ↔ (recs(𝐺) ↾ 𝐴) ∈ V))
4 tfr.1 . . . . . 6 𝐹 = recs(𝐺)
54dmeqi 5758 . . . . 5 dom 𝐹 = dom recs(𝐺)
65eleq2i 2822 . . . 4 (𝐴 ∈ dom 𝐹𝐴 ∈ dom recs(𝐺))
74reseq1i 5832 . . . . 5 (𝐹𝐴) = (recs(𝐺) ↾ 𝐴)
87eleq1i 2821 . . . 4 ((𝐹𝐴) ∈ V ↔ (recs(𝐺) ↾ 𝐴) ∈ V)
93, 6, 83bitr4g 317 . . 3 (𝐴 ∈ On → (𝐴 ∈ dom 𝐹 ↔ (𝐹𝐴) ∈ V))
10 onprc 7540 . . . . . 6 ¬ On ∈ V
11 elex 3416 . . . . . 6 (On ∈ dom 𝐹 → On ∈ V)
1210, 11mto 200 . . . . 5 ¬ On ∈ dom 𝐹
13 eleq1 2818 . . . . 5 (𝐴 = On → (𝐴 ∈ dom 𝐹 ↔ On ∈ dom 𝐹))
1412, 13mtbiri 330 . . . 4 (𝐴 = On → ¬ 𝐴 ∈ dom 𝐹)
152tfrlem13 8104 . . . . . 6 ¬ recs(𝐺) ∈ V
164, 15eqneltri 2824 . . . . 5 ¬ 𝐹 ∈ V
17 reseq2 5831 . . . . . . 7 (𝐴 = On → (𝐹𝐴) = (𝐹 ↾ On))
184tfr1a 8108 . . . . . . . . . 10 (Fun 𝐹 ∧ Lim dom 𝐹)
1918simpli 487 . . . . . . . . 9 Fun 𝐹
20 funrel 6375 . . . . . . . . 9 (Fun 𝐹 → Rel 𝐹)
2119, 20ax-mp 5 . . . . . . . 8 Rel 𝐹
2218simpri 489 . . . . . . . . 9 Lim dom 𝐹
23 limord 6250 . . . . . . . . 9 (Lim dom 𝐹 → Ord dom 𝐹)
24 ordsson 7545 . . . . . . . . 9 (Ord dom 𝐹 → dom 𝐹 ⊆ On)
2522, 23, 24mp2b 10 . . . . . . . 8 dom 𝐹 ⊆ On
26 relssres 5877 . . . . . . . 8 ((Rel 𝐹 ∧ dom 𝐹 ⊆ On) → (𝐹 ↾ On) = 𝐹)
2721, 25, 26mp2an 692 . . . . . . 7 (𝐹 ↾ On) = 𝐹
2817, 27eqtrdi 2787 . . . . . 6 (𝐴 = On → (𝐹𝐴) = 𝐹)
2928eleq1d 2815 . . . . 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 857 . 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 847   = wceq 1543  wcel 2112  {cab 2714  wral 3051  wrex 3052  Vcvv 3398  wss 3853  dom cdm 5536  cres 5538  Rel wrel 5541  Ord word 6190  Oncon0 6191  Lim wlim 6192  Fun wfun 6352   Fn wfn 6353  cfv 6358  recscrecs 8085
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-sep 5177  ax-nul 5184  ax-pr 5307  ax-un 7501
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ne 2933  df-ral 3056  df-rex 3057  df-reu 3058  df-rab 3060  df-v 3400  df-sbc 3684  df-csb 3799  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-pss 3872  df-nul 4224  df-if 4426  df-pw 4501  df-sn 4528  df-pr 4530  df-tp 4532  df-op 4534  df-uni 4806  df-iun 4892  df-br 5040  df-opab 5102  df-mpt 5121  df-tr 5147  df-id 5440  df-eprel 5445  df-po 5453  df-so 5454  df-fr 5494  df-we 5496  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-pred 6140  df-ord 6194  df-on 6195  df-lim 6196  df-suc 6197  df-iota 6316  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-wrecs 8025  df-recs 8086
This theorem is referenced by:  ordtypelem3  9114  ordtypelem9  9120
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