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Theorem tfri1dALT 6214
 Description: Alternate proof of tfri1d 6198 in terms of tfr1on 6213. Although this does show that the tfr1on 6213 proof is general enough to also prove tfri1d 6198, the tfri1d 6198 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 6183 . . . 4 Fun recs(𝐺)
2 tfri1dALT.1 . . . . 5 𝐹 = recs(𝐺)
32funeqi 5112 . . . 4 (Fun 𝐹 ↔ Fun recs(𝐺))
41, 3mpbir 145 . . 3 Fun 𝐹
54a1i 9 . 2 (𝜑 → Fun 𝐹)
6 eqid 2115 . . . . . 6 {𝑎 ∣ ∃𝑏 ∈ On (𝑎 Fn 𝑏 ∧ ∀𝑐𝑏 (𝑎𝑐) = (𝐺‘(𝑎𝑐)))} = {𝑎 ∣ ∃𝑏 ∈ On (𝑎 Fn 𝑏 ∧ ∀𝑐𝑏 (𝑎𝑐) = (𝐺‘(𝑎𝑐)))}
76tfrlem8 6181 . . . . 5 Ord dom recs(𝐺)
82dmeqi 4708 . . . . . 6 dom 𝐹 = dom recs(𝐺)
9 ordeq 4262 . . . . . 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 145 . . . 4 Ord dom 𝐹
12 ordsson 4376 . . . 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 108 . . . . . . . . . . 11 ((Fun 𝐺 ∧ (𝐺𝑥) ∈ V) → Fun 𝐺)
1615alimi 1414 . . . . . . . . . 10 (∀𝑥(Fun 𝐺 ∧ (𝐺𝑥) ∈ V) → ∀𝑥Fun 𝐺)
1714, 16syl 14 . . . . . . . . 9 (𝜑 → ∀𝑥Fun 𝐺)
181719.21bi 1520 . . . . . . . 8 (𝜑 → Fun 𝐺)
1918adantr 272 . . . . . . 7 ((𝜑𝑧 ∈ On) → Fun 𝐺)
20 ordon 4370 . . . . . . . 8 Ord On
2120a1i 9 . . . . . . 7 ((𝜑𝑧 ∈ On) → Ord On)
22 simpr 109 . . . . . . . . . . 11 ((Fun 𝐺 ∧ (𝐺𝑥) ∈ V) → (𝐺𝑥) ∈ V)
2322alimi 1414 . . . . . . . . . 10 (∀𝑥(Fun 𝐺 ∧ (𝐺𝑥) ∈ V) → ∀𝑥(𝐺𝑥) ∈ V)
24 fveq2 5387 . . . . . . . . . . . 12 (𝑥 = 𝑓 → (𝐺𝑥) = (𝐺𝑓))
2524eleq1d 2184 . . . . . . . . . . 11 (𝑥 = 𝑓 → ((𝐺𝑥) ∈ V ↔ (𝐺𝑓) ∈ V))
2625spv 1814 . . . . . . . . . 10 (∀𝑥(𝐺𝑥) ∈ V → (𝐺𝑓) ∈ V)
2714, 23, 263syl 17 . . . . . . . . 9 (𝜑 → (𝐺𝑓) ∈ V)
2827adantr 272 . . . . . . . 8 ((𝜑𝑧 ∈ On) → (𝐺𝑓) ∈ V)
29283ad2ant1 985 . . . . . . 7 (((𝜑𝑧 ∈ On) ∧ 𝑦 ∈ On ∧ 𝑓 Fn 𝑦) → (𝐺𝑓) ∈ V)
30 suceloni 4385 . . . . . . . . 9 (𝑦 ∈ On → suc 𝑦 ∈ On)
31 unon 4395 . . . . . . . . 9 On = On
3230, 31eleq2s 2210 . . . . . . . 8 (𝑦 On → suc 𝑦 ∈ On)
3332adantl 273 . . . . . . 7 (((𝜑𝑧 ∈ On) ∧ 𝑦 On) → suc 𝑦 ∈ On)
34 suceloni 4385 . . . . . . . 8 (𝑧 ∈ On → suc 𝑧 ∈ On)
3534adantl 273 . . . . . . 7 ((𝜑𝑧 ∈ On) → suc 𝑧 ∈ On)
362, 19, 21, 29, 33, 35tfr1on 6213 . . . . . 6 ((𝜑𝑧 ∈ On) → suc 𝑧 ⊆ dom 𝐹)
37 vex 2661 . . . . . . 7 𝑧 ∈ V
3837sucid 4307 . . . . . 6 𝑧 ∈ suc 𝑧
39 ssel2 3060 . . . . . 6 ((suc 𝑧 ⊆ dom 𝐹𝑧 ∈ suc 𝑧) → 𝑧 ∈ dom 𝐹)
4036, 38, 39sylancl 407 . . . . 5 ((𝜑𝑧 ∈ On) → 𝑧 ∈ dom 𝐹)
4140ex 114 . . . 4 (𝜑 → (𝑧 ∈ On → 𝑧 ∈ dom 𝐹))
4241ssrdv 3071 . . 3 (𝜑 → On ⊆ dom 𝐹)
4313, 42eqssd 3082 . 2 (𝜑 → dom 𝐹 = On)
44 df-fn 5094 . 2 (𝐹 Fn On ↔ (Fun 𝐹 ∧ dom 𝐹 = On))
455, 43, 44sylanbrc 411 1 (𝜑𝐹 Fn On)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  ∀wal 1312   = wceq 1314   ∈ wcel 1463  {cab 2101  ∀wral 2391  ∃wrex 2392  Vcvv 2658   ⊆ wss 3039  ∪ cuni 3704  Ord word 4252  Oncon0 4253  suc csuc 4255  dom cdm 4507   ↾ cres 4509  Fun wfun 5085   Fn wfn 5086  ‘cfv 5091  recscrecs 6167 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420 This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-id 4183  df-iord 4256  df-on 4258  df-suc 4261  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-recs 6168 This theorem is referenced by: (None)
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