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Theorem tfri1dALT 6351
Description: Alternate proof of tfri1d 6335 in terms of tfr1on 6350.

Although this does show that the tfr1on 6350 proof is general enough to also prove tfri1d 6335, the tfri1d 6335 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 6320 . . . 4 Fun recs(𝐺)
2 tfri1dALT.1 . . . . 5 𝐹 = recs(𝐺)
32funeqi 5237 . . . 4 (Fun 𝐹 ↔ Fun recs(𝐺))
41, 3mpbir 146 . . 3 Fun 𝐹
54a1i 9 . 2 (𝜑 → Fun 𝐹)
6 eqid 2177 . . . . . 6 {𝑎 ∣ ∃𝑏 ∈ On (𝑎 Fn 𝑏 ∧ ∀𝑐𝑏 (𝑎𝑐) = (𝐺‘(𝑎𝑐)))} = {𝑎 ∣ ∃𝑏 ∈ On (𝑎 Fn 𝑏 ∧ ∀𝑐𝑏 (𝑎𝑐) = (𝐺‘(𝑎𝑐)))}
76tfrlem8 6318 . . . . 5 Ord dom recs(𝐺)
82dmeqi 4828 . . . . . 6 dom 𝐹 = dom recs(𝐺)
9 ordeq 4372 . . . . . 6 (dom 𝐹 = dom recs(𝐺) → (Ord dom 𝐹 ↔ Ord dom recs(𝐺)))
108, 9ax-mp 5 . . . . 5 (Ord dom 𝐹 ↔ Ord dom recs(𝐺))
117, 10mpbir 146 . . . 4 Ord dom 𝐹
12 ordsson 4491 . . . 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 109 . . . . . . . . . . 11 ((Fun 𝐺 ∧ (𝐺𝑥) ∈ V) → Fun 𝐺)
1615alimi 1455 . . . . . . . . . 10 (∀𝑥(Fun 𝐺 ∧ (𝐺𝑥) ∈ V) → ∀𝑥Fun 𝐺)
1714, 16syl 14 . . . . . . . . 9 (𝜑 → ∀𝑥Fun 𝐺)
181719.21bi 1558 . . . . . . . 8 (𝜑 → Fun 𝐺)
1918adantr 276 . . . . . . 7 ((𝜑𝑧 ∈ On) → Fun 𝐺)
20 ordon 4485 . . . . . . . 8 Ord On
2120a1i 9 . . . . . . 7 ((𝜑𝑧 ∈ On) → Ord On)
22 simpr 110 . . . . . . . . . . 11 ((Fun 𝐺 ∧ (𝐺𝑥) ∈ V) → (𝐺𝑥) ∈ V)
2322alimi 1455 . . . . . . . . . 10 (∀𝑥(Fun 𝐺 ∧ (𝐺𝑥) ∈ V) → ∀𝑥(𝐺𝑥) ∈ V)
24 fveq2 5515 . . . . . . . . . . . 12 (𝑥 = 𝑓 → (𝐺𝑥) = (𝐺𝑓))
2524eleq1d 2246 . . . . . . . . . . 11 (𝑥 = 𝑓 → ((𝐺𝑥) ∈ V ↔ (𝐺𝑓) ∈ V))
2625spv 1860 . . . . . . . . . 10 (∀𝑥(𝐺𝑥) ∈ V → (𝐺𝑓) ∈ V)
2714, 23, 263syl 17 . . . . . . . . 9 (𝜑 → (𝐺𝑓) ∈ V)
2827adantr 276 . . . . . . . 8 ((𝜑𝑧 ∈ On) → (𝐺𝑓) ∈ V)
29283ad2ant1 1018 . . . . . . 7 (((𝜑𝑧 ∈ On) ∧ 𝑦 ∈ On ∧ 𝑓 Fn 𝑦) → (𝐺𝑓) ∈ V)
30 onsuc 4500 . . . . . . . . 9 (𝑦 ∈ On → suc 𝑦 ∈ On)
31 unon 4510 . . . . . . . . 9 On = On
3230, 31eleq2s 2272 . . . . . . . 8 (𝑦 On → suc 𝑦 ∈ On)
3332adantl 277 . . . . . . 7 (((𝜑𝑧 ∈ On) ∧ 𝑦 On) → suc 𝑦 ∈ On)
34 onsuc 4500 . . . . . . . 8 (𝑧 ∈ On → suc 𝑧 ∈ On)
3534adantl 277 . . . . . . 7 ((𝜑𝑧 ∈ On) → suc 𝑧 ∈ On)
362, 19, 21, 29, 33, 35tfr1on 6350 . . . . . 6 ((𝜑𝑧 ∈ On) → suc 𝑧 ⊆ dom 𝐹)
37 vex 2740 . . . . . . 7 𝑧 ∈ V
3837sucid 4417 . . . . . 6 𝑧 ∈ suc 𝑧
39 ssel2 3150 . . . . . 6 ((suc 𝑧 ⊆ dom 𝐹𝑧 ∈ suc 𝑧) → 𝑧 ∈ dom 𝐹)
4036, 38, 39sylancl 413 . . . . 5 ((𝜑𝑧 ∈ On) → 𝑧 ∈ dom 𝐹)
4140ex 115 . . . 4 (𝜑 → (𝑧 ∈ On → 𝑧 ∈ dom 𝐹))
4241ssrdv 3161 . . 3 (𝜑 → On ⊆ dom 𝐹)
4313, 42eqssd 3172 . 2 (𝜑 → dom 𝐹 = On)
44 df-fn 5219 . 2 (𝐹 Fn On ↔ (Fun 𝐹 ∧ dom 𝐹 = On))
455, 43, 44sylanbrc 417 1 (𝜑𝐹 Fn On)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wb 105  wal 1351   = wceq 1353  wcel 2148  {cab 2163  wral 2455  wrex 2456  Vcvv 2737  wss 3129   cuni 3809  Ord word 4362  Oncon0 4363  suc csuc 4365  dom cdm 4626  cres 4628  Fun wfun 5210   Fn wfn 5211  cfv 5216  recscrecs 6304
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 4118  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536
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 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-iun 3888  df-br 4004  df-opab 4065  df-mpt 4066  df-tr 4102  df-id 4293  df-iord 4366  df-on 4368  df-suc 4371  df-xp 4632  df-rel 4633  df-cnv 4634  df-co 4635  df-dm 4636  df-rn 4637  df-res 4638  df-ima 4639  df-iota 5178  df-fun 5218  df-fn 5219  df-f 5220  df-f1 5221  df-fo 5222  df-f1o 5223  df-fv 5224  df-recs 6305
This theorem is referenced by: (None)
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