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Theorem ttrclselem1 9669
Description: Lemma for ttrclse 9671. Show that all finite ordinal function values of 𝐹 are subsets of 𝐴. (Contributed by Scott Fenton, 31-Oct-2024.)
Hypothesis
Ref Expression
ttrclselem.1 𝐹 = rec((𝑏 ∈ V ↦ 𝑤𝑏 Pred(𝑅, 𝐴, 𝑤)), Pred(𝑅, 𝐴, 𝑋))
Assertion
Ref Expression
ttrclselem1 (𝑁 ∈ ω → (𝐹𝑁) ⊆ 𝐴)
Distinct variable groups:   𝐴,𝑏,𝑤   𝑅,𝑏,𝑤   𝑋,𝑏
Allowed substitution hints:   𝐹(𝑤,𝑏)   𝑁(𝑤,𝑏)   𝑋(𝑤)

Proof of Theorem ttrclselem1
Dummy variables 𝑛 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 nn0suc 7836 . 2 (𝑁 ∈ ω → (𝑁 = ∅ ∨ ∃𝑛 ∈ ω 𝑁 = suc 𝑛))
2 ttrclselem.1 . . . . . 6 𝐹 = rec((𝑏 ∈ V ↦ 𝑤𝑏 Pred(𝑅, 𝐴, 𝑤)), Pred(𝑅, 𝐴, 𝑋))
32fveq1i 6847 . . . . 5 (𝐹𝑁) = (rec((𝑏 ∈ V ↦ 𝑤𝑏 Pred(𝑅, 𝐴, 𝑤)), Pred(𝑅, 𝐴, 𝑋))‘𝑁)
4 fveq2 6846 . . . . 5 (𝑁 = ∅ → (rec((𝑏 ∈ V ↦ 𝑤𝑏 Pred(𝑅, 𝐴, 𝑤)), Pred(𝑅, 𝐴, 𝑋))‘𝑁) = (rec((𝑏 ∈ V ↦ 𝑤𝑏 Pred(𝑅, 𝐴, 𝑤)), Pred(𝑅, 𝐴, 𝑋))‘∅))
53, 4eqtrid 2785 . . . 4 (𝑁 = ∅ → (𝐹𝑁) = (rec((𝑏 ∈ V ↦ 𝑤𝑏 Pred(𝑅, 𝐴, 𝑤)), Pred(𝑅, 𝐴, 𝑋))‘∅))
6 rdg0g 8377 . . . . . 6 (Pred(𝑅, 𝐴, 𝑋) ∈ V → (rec((𝑏 ∈ V ↦ 𝑤𝑏 Pred(𝑅, 𝐴, 𝑤)), Pred(𝑅, 𝐴, 𝑋))‘∅) = Pred(𝑅, 𝐴, 𝑋))
7 predss 6265 . . . . . 6 Pred(𝑅, 𝐴, 𝑋) ⊆ 𝐴
86, 7eqsstrdi 4002 . . . . 5 (Pred(𝑅, 𝐴, 𝑋) ∈ V → (rec((𝑏 ∈ V ↦ 𝑤𝑏 Pred(𝑅, 𝐴, 𝑤)), Pred(𝑅, 𝐴, 𝑋))‘∅) ⊆ 𝐴)
9 rdg0n 8384 . . . . . 6 (¬ Pred(𝑅, 𝐴, 𝑋) ∈ V → (rec((𝑏 ∈ V ↦ 𝑤𝑏 Pred(𝑅, 𝐴, 𝑤)), Pred(𝑅, 𝐴, 𝑋))‘∅) = ∅)
10 0ss 4360 . . . . . 6 ∅ ⊆ 𝐴
119, 10eqsstrdi 4002 . . . . 5 (¬ Pred(𝑅, 𝐴, 𝑋) ∈ V → (rec((𝑏 ∈ V ↦ 𝑤𝑏 Pred(𝑅, 𝐴, 𝑤)), Pred(𝑅, 𝐴, 𝑋))‘∅) ⊆ 𝐴)
128, 11pm2.61i 182 . . . 4 (rec((𝑏 ∈ V ↦ 𝑤𝑏 Pred(𝑅, 𝐴, 𝑤)), Pred(𝑅, 𝐴, 𝑋))‘∅) ⊆ 𝐴
135, 12eqsstrdi 4002 . . 3 (𝑁 = ∅ → (𝐹𝑁) ⊆ 𝐴)
14 nnon 7812 . . . . . . 7 (𝑛 ∈ ω → 𝑛 ∈ On)
15 nfcv 2904 . . . . . . . . 9 𝑏Pred(𝑅, 𝐴, 𝑋)
16 nfcv 2904 . . . . . . . . 9 𝑏𝑛
17 nfmpt1 5217 . . . . . . . . . . . . 13 𝑏(𝑏 ∈ V ↦ 𝑤𝑏 Pred(𝑅, 𝐴, 𝑤))
1817, 15nfrdg 8364 . . . . . . . . . . . 12 𝑏rec((𝑏 ∈ V ↦ 𝑤𝑏 Pred(𝑅, 𝐴, 𝑤)), Pred(𝑅, 𝐴, 𝑋))
192, 18nfcxfr 2902 . . . . . . . . . . 11 𝑏𝐹
2019, 16nffv 6856 . . . . . . . . . 10 𝑏(𝐹𝑛)
21 nfcv 2904 . . . . . . . . . 10 𝑏Pred(𝑅, 𝐴, 𝑡)
2220, 21nfiun 4988 . . . . . . . . 9 𝑏 𝑡 ∈ (𝐹𝑛)Pred(𝑅, 𝐴, 𝑡)
23 predeq3 6261 . . . . . . . . . . 11 (𝑤 = 𝑡 → Pred(𝑅, 𝐴, 𝑤) = Pred(𝑅, 𝐴, 𝑡))
2423cbviunv 5004 . . . . . . . . . 10 𝑤𝑏 Pred(𝑅, 𝐴, 𝑤) = 𝑡𝑏 Pred(𝑅, 𝐴, 𝑡)
25 iuneq1 4974 . . . . . . . . . 10 (𝑏 = (𝐹𝑛) → 𝑡𝑏 Pred(𝑅, 𝐴, 𝑡) = 𝑡 ∈ (𝐹𝑛)Pred(𝑅, 𝐴, 𝑡))
2624, 25eqtrid 2785 . . . . . . . . 9 (𝑏 = (𝐹𝑛) → 𝑤𝑏 Pred(𝑅, 𝐴, 𝑤) = 𝑡 ∈ (𝐹𝑛)Pred(𝑅, 𝐴, 𝑡))
2715, 16, 22, 2, 26rdgsucmptf 8378 . . . . . . . 8 ((𝑛 ∈ On ∧ 𝑡 ∈ (𝐹𝑛)Pred(𝑅, 𝐴, 𝑡) ∈ V) → (𝐹‘suc 𝑛) = 𝑡 ∈ (𝐹𝑛)Pred(𝑅, 𝐴, 𝑡))
28 iunss 5009 . . . . . . . . 9 ( 𝑡 ∈ (𝐹𝑛)Pred(𝑅, 𝐴, 𝑡) ⊆ 𝐴 ↔ ∀𝑡 ∈ (𝐹𝑛)Pred(𝑅, 𝐴, 𝑡) ⊆ 𝐴)
29 predss 6265 . . . . . . . . . 10 Pred(𝑅, 𝐴, 𝑡) ⊆ 𝐴
3029a1i 11 . . . . . . . . 9 (𝑡 ∈ (𝐹𝑛) → Pred(𝑅, 𝐴, 𝑡) ⊆ 𝐴)
3128, 30mprgbir 3068 . . . . . . . 8 𝑡 ∈ (𝐹𝑛)Pred(𝑅, 𝐴, 𝑡) ⊆ 𝐴
3227, 31eqsstrdi 4002 . . . . . . 7 ((𝑛 ∈ On ∧ 𝑡 ∈ (𝐹𝑛)Pred(𝑅, 𝐴, 𝑡) ∈ V) → (𝐹‘suc 𝑛) ⊆ 𝐴)
3314, 32sylan 581 . . . . . 6 ((𝑛 ∈ ω ∧ 𝑡 ∈ (𝐹𝑛)Pred(𝑅, 𝐴, 𝑡) ∈ V) → (𝐹‘suc 𝑛) ⊆ 𝐴)
3415, 16, 22, 2, 26rdgsucmptnf 8379 . . . . . . . 8 𝑡 ∈ (𝐹𝑛)Pred(𝑅, 𝐴, 𝑡) ∈ V → (𝐹‘suc 𝑛) = ∅)
3534, 10eqsstrdi 4002 . . . . . . 7 𝑡 ∈ (𝐹𝑛)Pred(𝑅, 𝐴, 𝑡) ∈ V → (𝐹‘suc 𝑛) ⊆ 𝐴)
3635adantl 483 . . . . . 6 ((𝑛 ∈ ω ∧ ¬ 𝑡 ∈ (𝐹𝑛)Pred(𝑅, 𝐴, 𝑡) ∈ V) → (𝐹‘suc 𝑛) ⊆ 𝐴)
3733, 36pm2.61dan 812 . . . . 5 (𝑛 ∈ ω → (𝐹‘suc 𝑛) ⊆ 𝐴)
38 fveq2 6846 . . . . . 6 (𝑁 = suc 𝑛 → (𝐹𝑁) = (𝐹‘suc 𝑛))
3938sseq1d 3979 . . . . 5 (𝑁 = suc 𝑛 → ((𝐹𝑁) ⊆ 𝐴 ↔ (𝐹‘suc 𝑛) ⊆ 𝐴))
4037, 39syl5ibrcom 247 . . . 4 (𝑛 ∈ ω → (𝑁 = suc 𝑛 → (𝐹𝑁) ⊆ 𝐴))
4140rexlimiv 3142 . . 3 (∃𝑛 ∈ ω 𝑁 = suc 𝑛 → (𝐹𝑁) ⊆ 𝐴)
4213, 41jaoi 856 . 2 ((𝑁 = ∅ ∨ ∃𝑛 ∈ ω 𝑁 = suc 𝑛) → (𝐹𝑁) ⊆ 𝐴)
431, 42syl 17 1 (𝑁 ∈ ω → (𝐹𝑁) ⊆ 𝐴)
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 397  wo 846   = wceq 1542  wcel 2107  wrex 3070  Vcvv 3447  wss 3914  c0 4286   ciun 4958  cmpt 5192  Predcpred 6256  Oncon0 6321  suc csuc 6323  cfv 6500  ωcom 7806  reccrdg 8359
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5246  ax-sep 5260  ax-nul 5267  ax-pr 5388  ax-un 7676
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3353  df-rab 3407  df-v 3449  df-sbc 3744  df-csb 3860  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-pss 3933  df-nul 4287  df-if 4491  df-pw 4566  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4870  df-iun 4960  df-br 5110  df-opab 5172  df-mpt 5193  df-tr 5227  df-id 5535  df-eprel 5541  df-po 5549  df-so 5550  df-fr 5592  df-we 5594  df-xp 5643  df-rel 5644  df-cnv 5645  df-co 5646  df-dm 5647  df-rn 5648  df-res 5649  df-ima 5650  df-pred 6257  df-ord 6324  df-on 6325  df-lim 6326  df-suc 6327  df-iota 6452  df-fun 6502  df-fn 6503  df-f 6504  df-f1 6505  df-fo 6506  df-f1o 6507  df-fv 6508  df-ov 7364  df-om 7807  df-2nd 7926  df-frecs 8216  df-wrecs 8247  df-recs 8321  df-rdg 8360
This theorem is referenced by:  ttrclselem2  9670
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