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Theorem tfri1d 6198
Description: Principle of Transfinite Recursion, part 1 of 3. Theorem 7.41(1) of [TakeutiZaring] p. 47, with an additional condition.

The condition is that 𝐺 is defined "everywhere", which is stated here as (𝐺𝑥) ∈ V. Alternately, 𝑥 ∈ On∀𝑓(𝑓 Fn 𝑥𝑓 ∈ dom 𝐺) would suffice.

Given a function 𝐺 satisfying that condition, we define a class 𝐴 of all "acceptable" functions. The final function we're interested in is the union 𝐹 = recs(𝐺) of them. 𝐹 is then said to be defined by transfinite recursion. The purpose of the 3 parts of this theorem is to demonstrate properties of 𝐹. In this first part we show that 𝐹 is a function whose domain is all ordinal numbers. (Contributed by Jim Kingdon, 4-May-2019.) (Revised by Mario Carneiro, 24-May-2019.)

Hypotheses
Ref Expression
tfri1d.1 𝐹 = recs(𝐺)
tfri1d.2 (𝜑 → ∀𝑥(Fun 𝐺 ∧ (𝐺𝑥) ∈ V))
Assertion
Ref Expression
tfri1d (𝜑𝐹 Fn On)
Distinct variable group:   𝑥,𝐺
Allowed substitution hints:   𝜑(𝑥)   𝐹(𝑥)

Proof of Theorem tfri1d
Dummy variables 𝑓 𝑔 𝑢 𝑤 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2115 . . . . . 6 {𝑔 ∣ ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))} = {𝑔 ∣ ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))}
21tfrlem3 6174 . . . . 5 {𝑔 ∣ ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑢𝑧 (𝑔𝑢) = (𝐺‘(𝑔𝑢)))} = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝐺‘(𝑓𝑦)))}
3 tfri1d.2 . . . . 5 (𝜑 → ∀𝑥(Fun 𝐺 ∧ (𝐺𝑥) ∈ V))
42, 3tfrlemi14d 6196 . . . 4 (𝜑 → dom recs(𝐺) = On)
5 eqid 2115 . . . . 5 {𝑤 ∣ ∃𝑦 ∈ On (𝑤 Fn 𝑦 ∧ ∀𝑧𝑦 (𝑤𝑧) = (𝐺‘(𝑤𝑧)))} = {𝑤 ∣ ∃𝑦 ∈ On (𝑤 Fn 𝑦 ∧ ∀𝑧𝑦 (𝑤𝑧) = (𝐺‘(𝑤𝑧)))}
65tfrlem7 6180 . . . 4 Fun recs(𝐺)
74, 6jctil 308 . . 3 (𝜑 → (Fun recs(𝐺) ∧ dom recs(𝐺) = On))
8 df-fn 5094 . . 3 (recs(𝐺) Fn On ↔ (Fun recs(𝐺) ∧ dom recs(𝐺) = On))
97, 8sylibr 133 . 2 (𝜑 → recs(𝐺) Fn On)
10 tfri1d.1 . . 3 𝐹 = recs(𝐺)
1110fneq1i 5185 . 2 (𝐹 Fn On ↔ recs(𝐺) Fn On)
129, 11sylibr 133 1 (𝜑𝐹 Fn On)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 103  wal 1312   = wceq 1314  wcel 1463  {cab 2101  wral 2391  wrex 2392  Vcvv 2658  Oncon0 4253  dom cdm 4507  cres 4509  Fun wfun 5085   Fn wfn 5086  cfv 5091  recscrecs 6167
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420
This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-id 4183  df-iord 4256  df-on 4258  df-suc 4261  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-recs 6168
This theorem is referenced by:  tfri2d  6199  tfri1  6228  rdgifnon  6242  rdgifnon2  6243  frecfnom  6264
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