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Theorem rdgon 6004
Description: Evaluating the recursive definition generator produces an ordinal. There is a hypothesis that the characteristic function produces ordinals on ordinal arguments. (Contributed by Jim Kingdon, 26-Jul-2019.)
Hypotheses
Ref Expression
rdgon.1 (𝜑𝐹 Fn V)
rdgon.2 (𝜑𝐴 ∈ On)
rdgon.3 (𝜑 → ∀𝑥 ∈ On (𝐹𝑥) ∈ On)
Assertion
Ref Expression
rdgon ((𝜑𝐵 ∈ On) → (rec(𝐹, 𝐴)‘𝐵) ∈ On)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐹   𝜑,𝑥
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem rdgon
Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 fveq2 5206 . . . . 5 (𝑧 = 𝑥 → (rec(𝐹, 𝐴)‘𝑧) = (rec(𝐹, 𝐴)‘𝑥))
21eleq1d 2122 . . . 4 (𝑧 = 𝑥 → ((rec(𝐹, 𝐴)‘𝑧) ∈ On ↔ (rec(𝐹, 𝐴)‘𝑥) ∈ On))
32imbi2d 223 . . 3 (𝑧 = 𝑥 → ((𝜑 → (rec(𝐹, 𝐴)‘𝑧) ∈ On) ↔ (𝜑 → (rec(𝐹, 𝐴)‘𝑥) ∈ On)))
4 fveq2 5206 . . . . 5 (𝑧 = 𝐵 → (rec(𝐹, 𝐴)‘𝑧) = (rec(𝐹, 𝐴)‘𝐵))
54eleq1d 2122 . . . 4 (𝑧 = 𝐵 → ((rec(𝐹, 𝐴)‘𝑧) ∈ On ↔ (rec(𝐹, 𝐴)‘𝐵) ∈ On))
65imbi2d 223 . . 3 (𝑧 = 𝐵 → ((𝜑 → (rec(𝐹, 𝐴)‘𝑧) ∈ On) ↔ (𝜑 → (rec(𝐹, 𝐴)‘𝐵) ∈ On)))
7 r19.21v 2413 . . . 4 (∀𝑥𝑧 (𝜑 → (rec(𝐹, 𝐴)‘𝑥) ∈ On) ↔ (𝜑 → ∀𝑥𝑧 (rec(𝐹, 𝐴)‘𝑥) ∈ On))
8 rdgon.2 . . . . . . . . 9 (𝜑𝐴 ∈ On)
9 fvres 5226 . . . . . . . . . . . . . 14 (𝑥𝑧 → ((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥) = (rec(𝐹, 𝐴)‘𝑥))
109eleq1d 2122 . . . . . . . . . . . . 13 (𝑥𝑧 → (((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥) ∈ On ↔ (rec(𝐹, 𝐴)‘𝑥) ∈ On))
1110adantl 266 . . . . . . . . . . . 12 ((𝜑𝑥𝑧) → (((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥) ∈ On ↔ (rec(𝐹, 𝐴)‘𝑥) ∈ On))
12 rdgon.3 . . . . . . . . . . . . . . 15 (𝜑 → ∀𝑥 ∈ On (𝐹𝑥) ∈ On)
13 fveq2 5206 . . . . . . . . . . . . . . . . 17 (𝑥 = 𝑤 → (𝐹𝑥) = (𝐹𝑤))
1413eleq1d 2122 . . . . . . . . . . . . . . . 16 (𝑥 = 𝑤 → ((𝐹𝑥) ∈ On ↔ (𝐹𝑤) ∈ On))
1514cbvralv 2550 . . . . . . . . . . . . . . 15 (∀𝑥 ∈ On (𝐹𝑥) ∈ On ↔ ∀𝑤 ∈ On (𝐹𝑤) ∈ On)
1612, 15sylib 131 . . . . . . . . . . . . . 14 (𝜑 → ∀𝑤 ∈ On (𝐹𝑤) ∈ On)
17 fveq2 5206 . . . . . . . . . . . . . . . 16 (𝑤 = ((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥) → (𝐹𝑤) = (𝐹‘((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥)))
1817eleq1d 2122 . . . . . . . . . . . . . . 15 (𝑤 = ((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥) → ((𝐹𝑤) ∈ On ↔ (𝐹‘((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥)) ∈ On))
1918rspcv 2669 . . . . . . . . . . . . . 14 (((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥) ∈ On → (∀𝑤 ∈ On (𝐹𝑤) ∈ On → (𝐹‘((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥)) ∈ On))
2016, 19syl5com 29 . . . . . . . . . . . . 13 (𝜑 → (((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥) ∈ On → (𝐹‘((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥)) ∈ On))
2120adantr 265 . . . . . . . . . . . 12 ((𝜑𝑥𝑧) → (((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥) ∈ On → (𝐹‘((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥)) ∈ On))
2211, 21sylbird 163 . . . . . . . . . . 11 ((𝜑𝑥𝑧) → ((rec(𝐹, 𝐴)‘𝑥) ∈ On → (𝐹‘((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥)) ∈ On))
2322ralimdva 2404 . . . . . . . . . 10 (𝜑 → (∀𝑥𝑧 (rec(𝐹, 𝐴)‘𝑥) ∈ On → ∀𝑥𝑧 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥)) ∈ On))
24 vex 2577 . . . . . . . . . . 11 𝑧 ∈ V
25 iunon 5930 . . . . . . . . . . 11 ((𝑧 ∈ V ∧ ∀𝑥𝑧 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥)) ∈ On) → 𝑥𝑧 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥)) ∈ On)
2624, 25mpan 408 . . . . . . . . . 10 (∀𝑥𝑧 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥)) ∈ On → 𝑥𝑧 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥)) ∈ On)
2723, 26syl6 33 . . . . . . . . 9 (𝜑 → (∀𝑥𝑧 (rec(𝐹, 𝐴)‘𝑥) ∈ On → 𝑥𝑧 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥)) ∈ On))
28 onun2 4244 . . . . . . . . 9 ((𝐴 ∈ On ∧ 𝑥𝑧 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥)) ∈ On) → (𝐴 𝑥𝑧 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥))) ∈ On)
298, 27, 28syl6an 1339 . . . . . . . 8 (𝜑 → (∀𝑥𝑧 (rec(𝐹, 𝐴)‘𝑥) ∈ On → (𝐴 𝑥𝑧 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥))) ∈ On))
3029adantr 265 . . . . . . 7 ((𝜑𝑧 ∈ On) → (∀𝑥𝑧 (rec(𝐹, 𝐴)‘𝑥) ∈ On → (𝐴 𝑥𝑧 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥))) ∈ On))
31 rdgon.1 . . . . . . . . . 10 (𝜑𝐹 Fn V)
3231, 8jca 294 . . . . . . . . 9 (𝜑 → (𝐹 Fn V ∧ 𝐴 ∈ On))
33 rdgivallem 5999 . . . . . . . . . 10 ((𝐹 Fn V ∧ 𝐴 ∈ On ∧ 𝑧 ∈ On) → (rec(𝐹, 𝐴)‘𝑧) = (𝐴 𝑥𝑧 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥))))
34333expa 1115 . . . . . . . . 9 (((𝐹 Fn V ∧ 𝐴 ∈ On) ∧ 𝑧 ∈ On) → (rec(𝐹, 𝐴)‘𝑧) = (𝐴 𝑥𝑧 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥))))
3532, 34sylan 271 . . . . . . . 8 ((𝜑𝑧 ∈ On) → (rec(𝐹, 𝐴)‘𝑧) = (𝐴 𝑥𝑧 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥))))
3635eleq1d 2122 . . . . . . 7 ((𝜑𝑧 ∈ On) → ((rec(𝐹, 𝐴)‘𝑧) ∈ On ↔ (𝐴 𝑥𝑧 (𝐹‘((rec(𝐹, 𝐴) ↾ 𝑧)‘𝑥))) ∈ On))
3730, 36sylibrd 162 . . . . . 6 ((𝜑𝑧 ∈ On) → (∀𝑥𝑧 (rec(𝐹, 𝐴)‘𝑥) ∈ On → (rec(𝐹, 𝐴)‘𝑧) ∈ On))
3837expcom 113 . . . . 5 (𝑧 ∈ On → (𝜑 → (∀𝑥𝑧 (rec(𝐹, 𝐴)‘𝑥) ∈ On → (rec(𝐹, 𝐴)‘𝑧) ∈ On)))
3938a2d 26 . . . 4 (𝑧 ∈ On → ((𝜑 → ∀𝑥𝑧 (rec(𝐹, 𝐴)‘𝑥) ∈ On) → (𝜑 → (rec(𝐹, 𝐴)‘𝑧) ∈ On)))
407, 39syl5bi 145 . . 3 (𝑧 ∈ On → (∀𝑥𝑧 (𝜑 → (rec(𝐹, 𝐴)‘𝑥) ∈ On) → (𝜑 → (rec(𝐹, 𝐴)‘𝑧) ∈ On)))
413, 6, 40tfis3 4337 . 2 (𝐵 ∈ On → (𝜑 → (rec(𝐹, 𝐴)‘𝐵) ∈ On))
4241impcom 120 1 ((𝜑𝐵 ∈ On) → (rec(𝐹, 𝐴)‘𝐵) ∈ On)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 101  wb 102   = wceq 1259  wcel 1409  wral 2323  Vcvv 2574  cun 2943   ciun 3685  Oncon0 4128  cres 4375   Fn wfn 4925  cfv 4930  reccrdg 5987
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-13 1420  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-coll 3900  ax-sep 3903  ax-pow 3955  ax-pr 3972  ax-un 4198  ax-setind 4290
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-fal 1265  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ne 2221  df-ral 2328  df-rex 2329  df-reu 2330  df-rab 2332  df-v 2576  df-sbc 2788  df-csb 2881  df-dif 2948  df-un 2950  df-in 2952  df-ss 2959  df-nul 3253  df-pw 3389  df-sn 3409  df-pr 3410  df-op 3412  df-uni 3609  df-iun 3687  df-br 3793  df-opab 3847  df-mpt 3848  df-tr 3883  df-id 4058  df-iord 4131  df-on 4133  df-suc 4136  df-xp 4379  df-rel 4380  df-cnv 4381  df-co 4382  df-dm 4383  df-rn 4384  df-res 4385  df-ima 4386  df-iota 4895  df-fun 4932  df-fn 4933  df-f 4934  df-f1 4935  df-fo 4936  df-f1o 4937  df-fv 4938  df-recs 5951  df-irdg 5988
This theorem is referenced by:  oacl  6071  omcl  6072  oeicl  6073
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