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Theorem tfri1dALT 6352
Description: Alternate proof of tfri1d 6336 in terms of tfr1on 6351.

Although this does show that the tfr1on 6351 proof is general enough to also prove tfri1d 6336, the tfri1d 6336 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 6321 . . . 4 Fun recs(𝐺)
2 tfri1dALT.1 . . . . 5 𝐹 = recs(𝐺)
32funeqi 5238 . . . 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 6319 . . . . 5 Ord dom recs(𝐺)
82dmeqi 4829 . . . . . 6 dom 𝐹 = dom recs(𝐺)
9 ordeq 4373 . . . . . 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 4492 . . . 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 4486 . . . . . . . 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 5516 . . . . . . . . . . . 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 4501 . . . . . . . . 9 (𝑦 ∈ On → suc 𝑦 ∈ On)
31 unon 4511 . . . . . . . . 9 On = On
3230, 31eleq2s 2272 . . . . . . . 8 (𝑦 On → suc 𝑦 ∈ On)
3332adantl 277 . . . . . . 7 (((𝜑𝑧 ∈ On) ∧ 𝑦 On) → suc 𝑦 ∈ On)
34 onsuc 4501 . . . . . . . 8 (𝑧 ∈ On → suc 𝑧 ∈ On)
3534adantl 277 . . . . . . 7 ((𝜑𝑧 ∈ On) → suc 𝑧 ∈ On)
362, 19, 21, 29, 33, 35tfr1on 6351 . . . . . 6 ((𝜑𝑧 ∈ On) → suc 𝑧 ⊆ dom 𝐹)
37 vex 2741 . . . . . . 7 𝑧 ∈ V
3837sucid 4418 . . . . . 6 𝑧 ∈ suc 𝑧
39 ssel2 3151 . . . . . 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 3162 . . 3 (𝜑 → On ⊆ dom 𝐹)
4313, 42eqssd 3173 . 2 (𝜑 → dom 𝐹 = On)
44 df-fn 5220 . 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 2738  wss 3130   cuni 3810  Ord word 4363  Oncon0 4364  suc csuc 4366  dom cdm 4627  cres 4629  Fun wfun 5211   Fn wfn 5212  cfv 5217  recscrecs 6305
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 4119  ax-sep 4122  ax-pow 4175  ax-pr 4210  ax-un 4434  ax-setind 4537
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 2740  df-sbc 2964  df-csb 3059  df-dif 3132  df-un 3134  df-in 3136  df-ss 3143  df-nul 3424  df-pw 3578  df-sn 3599  df-pr 3600  df-op 3602  df-uni 3811  df-iun 3889  df-br 4005  df-opab 4066  df-mpt 4067  df-tr 4103  df-id 4294  df-iord 4367  df-on 4369  df-suc 4372  df-xp 4633  df-rel 4634  df-cnv 4635  df-co 4636  df-dm 4637  df-rn 4638  df-res 4639  df-ima 4640  df-iota 5179  df-fun 5219  df-fn 5220  df-f 5221  df-f1 5222  df-fo 5223  df-f1o 5224  df-fv 5225  df-recs 6306
This theorem is referenced by: (None)
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