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Theorem tfri1dALT 6098
Description: Alternate proof of tfri1d 6082 in terms of tfr1on 6097.

Although this does show that the tfr1on 6097 proof is general enough to also prove tfri1d 6082, the tfri1d 6082 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 6067 . . . 4 Fun recs(𝐺)
2 tfri1dALT.1 . . . . 5 𝐹 = recs(𝐺)
32funeqi 5022 . . . 4 (Fun 𝐹 ↔ Fun recs(𝐺))
41, 3mpbir 144 . . 3 Fun 𝐹
54a1i 9 . 2 (𝜑 → Fun 𝐹)
6 eqid 2088 . . . . . 6 {𝑎 ∣ ∃𝑏 ∈ On (𝑎 Fn 𝑏 ∧ ∀𝑐𝑏 (𝑎𝑐) = (𝐺‘(𝑎𝑐)))} = {𝑎 ∣ ∃𝑏 ∈ On (𝑎 Fn 𝑏 ∧ ∀𝑐𝑏 (𝑎𝑐) = (𝐺‘(𝑎𝑐)))}
76tfrlem8 6065 . . . . 5 Ord dom recs(𝐺)
82dmeqi 4625 . . . . . 6 dom 𝐹 = dom recs(𝐺)
9 ordeq 4190 . . . . . 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 4299 . . . 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 1389 . . . . . . . . . 10 (∀𝑥(Fun 𝐺 ∧ (𝐺𝑥) ∈ V) → ∀𝑥Fun 𝐺)
1714, 16syl 14 . . . . . . . . 9 (𝜑 → ∀𝑥Fun 𝐺)
181719.21bi 1495 . . . . . . . 8 (𝜑 → Fun 𝐺)
1918adantr 270 . . . . . . 7 ((𝜑𝑧 ∈ On) → Fun 𝐺)
20 ordon 4293 . . . . . . . 8 Ord On
2120a1i 9 . . . . . . 7 ((𝜑𝑧 ∈ On) → Ord On)
22 simpr 108 . . . . . . . . . . 11 ((Fun 𝐺 ∧ (𝐺𝑥) ∈ V) → (𝐺𝑥) ∈ V)
2322alimi 1389 . . . . . . . . . 10 (∀𝑥(Fun 𝐺 ∧ (𝐺𝑥) ∈ V) → ∀𝑥(𝐺𝑥) ∈ V)
24 fveq2 5289 . . . . . . . . . . . 12 (𝑥 = 𝑓 → (𝐺𝑥) = (𝐺𝑓))
2524eleq1d 2156 . . . . . . . . . . 11 (𝑥 = 𝑓 → ((𝐺𝑥) ∈ V ↔ (𝐺𝑓) ∈ V))
2625spv 1788 . . . . . . . . . 10 (∀𝑥(𝐺𝑥) ∈ V → (𝐺𝑓) ∈ V)
2714, 23, 263syl 17 . . . . . . . . 9 (𝜑 → (𝐺𝑓) ∈ V)
2827adantr 270 . . . . . . . 8 ((𝜑𝑧 ∈ On) → (𝐺𝑓) ∈ V)
29283ad2ant1 964 . . . . . . 7 (((𝜑𝑧 ∈ On) ∧ 𝑦 ∈ On ∧ 𝑓 Fn 𝑦) → (𝐺𝑓) ∈ V)
30 suceloni 4308 . . . . . . . . 9 (𝑦 ∈ On → suc 𝑦 ∈ On)
31 unon 4318 . . . . . . . . 9 On = On
3230, 31eleq2s 2182 . . . . . . . 8 (𝑦 On → suc 𝑦 ∈ On)
3332adantl 271 . . . . . . 7 (((𝜑𝑧 ∈ On) ∧ 𝑦 On) → suc 𝑦 ∈ On)
34 suceloni 4308 . . . . . . . 8 (𝑧 ∈ On → suc 𝑧 ∈ On)
3534adantl 271 . . . . . . 7 ((𝜑𝑧 ∈ On) → suc 𝑧 ∈ On)
362, 19, 21, 29, 33, 35tfr1on 6097 . . . . . 6 ((𝜑𝑧 ∈ On) → suc 𝑧 ⊆ dom 𝐹)
37 vex 2622 . . . . . . 7 𝑧 ∈ V
3837sucid 4235 . . . . . 6 𝑧 ∈ suc 𝑧
39 ssel2 3018 . . . . . 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 3029 . . 3 (𝜑 → On ⊆ dom 𝐹)
4313, 42eqssd 3040 . 2 (𝜑 → dom 𝐹 = On)
44 df-fn 5005 . 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 1287   = wceq 1289  wcel 1438  {cab 2074  wral 2359  wrex 2360  Vcvv 2619  wss 2997   cuni 3648  Ord word 4180  Oncon0 4181  suc csuc 4183  dom cdm 4428  cres 4430  Fun wfun 4996   Fn wfn 4997  cfv 5002  recscrecs 6051
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-13 1449  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-coll 3946  ax-sep 3949  ax-pow 4001  ax-pr 4027  ax-un 4251  ax-setind 4343
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-fal 1295  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ne 2256  df-ral 2364  df-rex 2365  df-reu 2366  df-rab 2368  df-v 2621  df-sbc 2839  df-csb 2932  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-uni 3649  df-iun 3727  df-br 3838  df-opab 3892  df-mpt 3893  df-tr 3929  df-id 4111  df-iord 4184  df-on 4186  df-suc 4189  df-xp 4434  df-rel 4435  df-cnv 4436  df-co 4437  df-dm 4438  df-rn 4439  df-res 4440  df-ima 4441  df-iota 4967  df-fun 5004  df-fn 5005  df-f 5006  df-f1 5007  df-fo 5008  df-f1o 5009  df-fv 5010  df-recs 6052
This theorem is referenced by: (None)
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