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Theorem tfr2b 7489
Description: Without assuming ax-rep 4769, 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 6985 . 2 (Ord 𝐴 ↔ (𝐴 ∈ On ∨ 𝐴 = On))
2 eqid 2621 . . . . 5 {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))} = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))}
32tfrlem15 7485 . . . 4 (𝐴 ∈ On → (𝐴 ∈ dom recs(𝐺) ↔ (recs(𝐺) ↾ 𝐴) ∈ V))
4 tfr.1 . . . . . 6 𝐹 = recs(𝐺)
54dmeqi 5323 . . . . 5 dom 𝐹 = dom recs(𝐺)
65eleq2i 2692 . . . 4 (𝐴 ∈ dom 𝐹𝐴 ∈ dom recs(𝐺))
74reseq1i 5390 . . . . 5 (𝐹𝐴) = (recs(𝐺) ↾ 𝐴)
87eleq1i 2691 . . . 4 ((𝐹𝐴) ∈ V ↔ (recs(𝐺) ↾ 𝐴) ∈ V)
93, 6, 83bitr4g 303 . . 3 (𝐴 ∈ On → (𝐴 ∈ dom 𝐹 ↔ (𝐹𝐴) ∈ V))
10 onprc 6981 . . . . . 6 ¬ On ∈ V
11 elex 3210 . . . . . 6 (On ∈ dom 𝐹 → On ∈ V)
1210, 11mto 188 . . . . 5 ¬ On ∈ dom 𝐹
13 eleq1 2688 . . . . 5 (𝐴 = On → (𝐴 ∈ dom 𝐹 ↔ On ∈ dom 𝐹))
1412, 13mtbiri 317 . . . 4 (𝐴 = On → ¬ 𝐴 ∈ dom 𝐹)
152tfrlem13 7483 . . . . . 6 ¬ recs(𝐺) ∈ V
164eleq1i 2691 . . . . . 6 (𝐹 ∈ V ↔ recs(𝐺) ∈ V)
1715, 16mtbir 313 . . . . 5 ¬ 𝐹 ∈ V
18 reseq2 5389 . . . . . . 7 (𝐴 = On → (𝐹𝐴) = (𝐹 ↾ On))
194tfr1a 7487 . . . . . . . . . 10 (Fun 𝐹 ∧ Lim dom 𝐹)
2019simpli 474 . . . . . . . . 9 Fun 𝐹
21 funrel 5903 . . . . . . . . 9 (Fun 𝐹 → Rel 𝐹)
2220, 21ax-mp 5 . . . . . . . 8 Rel 𝐹
2319simpri 478 . . . . . . . . 9 Lim dom 𝐹
24 limord 5782 . . . . . . . . 9 (Lim dom 𝐹 → Ord dom 𝐹)
25 ordsson 6986 . . . . . . . . 9 (Ord dom 𝐹 → dom 𝐹 ⊆ On)
2623, 24, 25mp2b 10 . . . . . . . 8 dom 𝐹 ⊆ On
27 relssres 5435 . . . . . . . 8 ((Rel 𝐹 ∧ dom 𝐹 ⊆ On) → (𝐹 ↾ On) = 𝐹)
2822, 26, 27mp2an 708 . . . . . . 7 (𝐹 ↾ On) = 𝐹
2918, 28syl6eq 2671 . . . . . 6 (𝐴 = On → (𝐹𝐴) = 𝐹)
3029eleq1d 2685 . . . . 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 1482  wcel 1989  {cab 2607  wral 2911  wrex 2912  Vcvv 3198  wss 3572  dom cdm 5112  cres 5114  Rel wrel 5117  Ord word 5720  Oncon0 5721  Lim wlim 5722  Fun wfun 5880   Fn wfn 5881  cfv 5886  recscrecs 7464
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-8 1991  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pow 4841  ax-pr 4904  ax-un 6946
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-reu 2918  df-rab 2920  df-v 3200  df-sbc 3434  df-csb 3532  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-pss 3588  df-nul 3914  df-if 4085  df-pw 4158  df-sn 4176  df-pr 4178  df-tp 4180  df-op 4182  df-uni 4435  df-iun 4520  df-br 4652  df-opab 4711  df-mpt 4728  df-tr 4751  df-id 5022  df-eprel 5027  df-po 5033  df-so 5034  df-fr 5071  df-we 5073  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-pred 5678  df-ord 5724  df-on 5725  df-lim 5726  df-suc 5727  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-f1 5891  df-fo 5892  df-f1o 5893  df-fv 5894  df-wrecs 7404  df-recs 7465
This theorem is referenced by:  ordtypelem3  8422  ordtypelem9  8428
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