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Theorem tfri1 6360
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 (πΊβ€˜π‘₯) ∈ V. Alternately, βˆ€π‘₯ ∈ Onβˆ€π‘“(𝑓 Fn π‘₯ β†’ 𝑓 ∈ 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
tfri1.1 𝐹 = recs(𝐺)
tfri1.2 (Fun 𝐺 ∧ (πΊβ€˜π‘₯) ∈ V)
Assertion
Ref Expression
tfri1 𝐹 Fn On
Distinct variable group:   π‘₯,𝐺
Allowed substitution hint:   𝐹(π‘₯)

Proof of Theorem tfri1
StepHypRef Expression
1 tfri1.1 . . 3 𝐹 = recs(𝐺)
2 tfri1.2 . . . . 5 (Fun 𝐺 ∧ (πΊβ€˜π‘₯) ∈ V)
32ax-gen 1449 . . . 4 βˆ€π‘₯(Fun 𝐺 ∧ (πΊβ€˜π‘₯) ∈ V)
43a1i 9 . . 3 (⊀ β†’ βˆ€π‘₯(Fun 𝐺 ∧ (πΊβ€˜π‘₯) ∈ V))
51, 4tfri1d 6330 . 2 (⊀ β†’ 𝐹 Fn On)
65mptru 1362 1 𝐹 Fn On
Colors of variables: wff set class
Syntax hints:   ∧ wa 104  βˆ€wal 1351   = wceq 1353  βŠ€wtru 1354   ∈ wcel 2148  Vcvv 2737  Oncon0 4360  Fun wfun 5206   Fn wfn 5207  β€˜cfv 5212  recscrecs 6299
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-id 4290  df-iord 4363  df-on 4365  df-suc 4368  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-recs 6300
This theorem is referenced by:  tfri2  6361  tfri3  6362
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