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Theorem tfri1dALT 6020
Description: Alternate proof of tfri1d 6004 in terms of tfr1on 6019.

Although this does show that the tfr1on 6019 proof is general enough to also prove tfri1d 6004, the tfri1d 6004 proof is simpler in places because it does not need to deal with 𝑋 being any ordinal. For that reason, we have both proofs. (Proof modification is discouraged.) (New usage is discouraged.) (Contributed by Jim Kingdon, 20-Mar-2022.)

Hypotheses
Ref Expression
tfri1dALT.1 𝐹 = recs(𝐺)
tfri1dALT.2 (𝜑 → ∀𝑥(Fun 𝐺 ∧ (𝐺𝑥) ∈ V))
Assertion
Ref Expression
tfri1dALT (𝜑𝐹 Fn On)
Distinct variable group:   𝑥,𝐺
Allowed substitution hints:   𝜑(𝑥)   𝐹(𝑥)

Proof of Theorem tfri1dALT
Dummy variables 𝑧 𝑎 𝑏 𝑐 𝑓 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tfrfun 5989 . . . 4 Fun recs(𝐺)
2 tfri1dALT.1 . . . . 5 𝐹 = recs(𝐺)
32funeqi 4972 . . . 4 (Fun 𝐹 ↔ Fun recs(𝐺))
41, 3mpbir 144 . . 3 Fun 𝐹
54a1i 9 . 2 (𝜑 → Fun 𝐹)
6 eqid 2083 . . . . . 6 {𝑎 ∣ ∃𝑏 ∈ On (𝑎 Fn 𝑏 ∧ ∀𝑐𝑏 (𝑎𝑐) = (𝐺‘(𝑎𝑐)))} = {𝑎 ∣ ∃𝑏 ∈ On (𝑎 Fn 𝑏 ∧ ∀𝑐𝑏 (𝑎𝑐) = (𝐺‘(𝑎𝑐)))}
76tfrlem8 5987 . . . . 5 Ord dom recs(𝐺)
82dmeqi 4584 . . . . . 6 dom 𝐹 = dom recs(𝐺)
9 ordeq 4155 . . . . . 6 (dom 𝐹 = dom recs(𝐺) → (Ord dom 𝐹 ↔ Ord dom recs(𝐺)))
108, 9ax-mp 7 . . . . 5 (Ord dom 𝐹 ↔ Ord dom recs(𝐺))
117, 10mpbir 144 . . . 4 Ord dom 𝐹
12 ordsson 4264 . . . 4 (Ord dom 𝐹 → dom 𝐹 ⊆ On)
1311, 12mp1i 10 . . 3 (𝜑 → dom 𝐹 ⊆ On)
14 tfri1dALT.2 . . . . . . . . . 10 (𝜑 → ∀𝑥(Fun 𝐺 ∧ (𝐺𝑥) ∈ V))
15 simpl 107 . . . . . . . . . . 11 ((Fun 𝐺 ∧ (𝐺𝑥) ∈ V) → Fun 𝐺)
1615alimi 1385 . . . . . . . . . 10 (∀𝑥(Fun 𝐺 ∧ (𝐺𝑥) ∈ V) → ∀𝑥Fun 𝐺)
1714, 16syl 14 . . . . . . . . 9 (𝜑 → ∀𝑥Fun 𝐺)
181719.21bi 1491 . . . . . . . 8 (𝜑 → Fun 𝐺)
1918adantr 270 . . . . . . 7 ((𝜑𝑧 ∈ On) → Fun 𝐺)
20 ordon 4258 . . . . . . . 8 Ord On
2120a1i 9 . . . . . . 7 ((𝜑𝑧 ∈ On) → Ord On)
22 simpr 108 . . . . . . . . . . 11 ((Fun 𝐺 ∧ (𝐺𝑥) ∈ V) → (𝐺𝑥) ∈ V)
2322alimi 1385 . . . . . . . . . 10 (∀𝑥(Fun 𝐺 ∧ (𝐺𝑥) ∈ V) → ∀𝑥(𝐺𝑥) ∈ V)
24 fveq2 5229 . . . . . . . . . . . 12 (𝑥 = 𝑓 → (𝐺𝑥) = (𝐺𝑓))
2524eleq1d 2151 . . . . . . . . . . 11 (𝑥 = 𝑓 → ((𝐺𝑥) ∈ V ↔ (𝐺𝑓) ∈ V))
2625spv 1783 . . . . . . . . . 10 (∀𝑥(𝐺𝑥) ∈ V → (𝐺𝑓) ∈ V)
2714, 23, 263syl 17 . . . . . . . . 9 (𝜑 → (𝐺𝑓) ∈ V)
2827adantr 270 . . . . . . . 8 ((𝜑𝑧 ∈ On) → (𝐺𝑓) ∈ V)
29283ad2ant1 960 . . . . . . 7 (((𝜑𝑧 ∈ On) ∧ 𝑦 ∈ On ∧ 𝑓 Fn 𝑦) → (𝐺𝑓) ∈ V)
30 suceloni 4273 . . . . . . . . 9 (𝑦 ∈ On → suc 𝑦 ∈ On)
31 unon 4283 . . . . . . . . 9 On = On
3230, 31eleq2s 2177 . . . . . . . 8 (𝑦 On → suc 𝑦 ∈ On)
3332adantl 271 . . . . . . 7 (((𝜑𝑧 ∈ On) ∧ 𝑦 On) → suc 𝑦 ∈ On)
34 suceloni 4273 . . . . . . . 8 (𝑧 ∈ On → suc 𝑧 ∈ On)
3534adantl 271 . . . . . . 7 ((𝜑𝑧 ∈ On) → suc 𝑧 ∈ On)
362, 19, 21, 29, 33, 35tfr1on 6019 . . . . . 6 ((𝜑𝑧 ∈ On) → suc 𝑧 ⊆ dom 𝐹)
37 vex 2613 . . . . . . 7 𝑧 ∈ V
3837sucid 4200 . . . . . 6 𝑧 ∈ suc 𝑧
39 ssel2 3003 . . . . . 6 ((suc 𝑧 ⊆ dom 𝐹𝑧 ∈ suc 𝑧) → 𝑧 ∈ dom 𝐹)
4036, 38, 39sylancl 404 . . . . 5 ((𝜑𝑧 ∈ On) → 𝑧 ∈ dom 𝐹)
4140ex 113 . . . 4 (𝜑 → (𝑧 ∈ On → 𝑧 ∈ dom 𝐹))
4241ssrdv 3014 . . 3 (𝜑 → On ⊆ dom 𝐹)
4313, 42eqssd 3025 . 2 (𝜑 → dom 𝐹 = On)
44 df-fn 4955 . 2 (𝐹 Fn On ↔ (Fun 𝐹 ∧ dom 𝐹 = On))
455, 43, 44sylanbrc 408 1 (𝜑𝐹 Fn On)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102  wb 103  wal 1283   = wceq 1285  wcel 1434  {cab 2069  wral 2353  wrex 2354  Vcvv 2610  wss 2982   cuni 3621  Ord word 4145  Oncon0 4146  suc csuc 4148  dom cdm 4391  cres 4393  Fun wfun 4946   Fn wfn 4947  cfv 4952  recscrecs 5973
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-coll 3913  ax-sep 3916  ax-pow 3968  ax-pr 3992  ax-un 4216  ax-setind 4308
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ne 2250  df-ral 2358  df-rex 2359  df-reu 2360  df-rab 2362  df-v 2612  df-sbc 2825  df-csb 2918  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-iun 3700  df-br 3806  df-opab 3860  df-mpt 3861  df-tr 3896  df-id 4076  df-iord 4149  df-on 4151  df-suc 4154  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960  df-recs 5974
This theorem is referenced by: (None)
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