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Theorem frrlem5e 31913
Description: Lemma for founded recursion. The domain of the union of a subset of 𝐵 is closed under predecessors. (Contributed by Paul Chapman, 1-May-2012.)
Hypotheses
Ref Expression
frrlem5.1 𝑅 Fr 𝐴
frrlem5.2 𝑅 Se 𝐴
frrlem5.3 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
Assertion
Ref Expression
frrlem5e (𝐶𝐵 → (𝑋 ∈ dom 𝐶 → Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝐶))
Distinct variable groups:   𝐴,𝑓,𝑥,𝑦   𝑓,𝐺,𝑥,𝑦   𝑅,𝑓,𝑥,𝑦   𝑥,𝐵
Allowed substitution hints:   𝐵(𝑦,𝑓)   𝐶(𝑥,𝑦,𝑓)   𝑋(𝑥,𝑦,𝑓)

Proof of Theorem frrlem5e
Dummy variables 𝑧 𝑡 𝑤 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 dmuni 5366 . . . 4 dom 𝐶 = 𝑧𝐶 dom 𝑧
21eleq2i 2722 . . 3 (𝑋 ∈ dom 𝐶𝑋 𝑧𝐶 dom 𝑧)
3 eliun 4556 . . 3 (𝑋 𝑧𝐶 dom 𝑧 ↔ ∃𝑧𝐶 𝑋 ∈ dom 𝑧)
42, 3bitri 264 . 2 (𝑋 ∈ dom 𝐶 ↔ ∃𝑧𝐶 𝑋 ∈ dom 𝑧)
5 ssel2 3631 . . . . 5 ((𝐶𝐵𝑧𝐶) → 𝑧𝐵)
6 frrlem5.3 . . . . . . . 8 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥𝐴 ∧ ∀𝑦𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦𝑥 (𝑓𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
76frrlem1 31905 . . . . . . 7 𝐵 = {𝑧 ∣ ∃𝑤(𝑧 Fn 𝑤 ∧ (𝑤𝐴 ∧ ∀𝑡𝑤 Pred(𝑅, 𝐴, 𝑡) ⊆ 𝑤) ∧ ∀𝑡𝑤 (𝑧𝑡) = (𝑡𝐺(𝑧 ↾ Pred(𝑅, 𝐴, 𝑡))))}
87abeq2i 2764 . . . . . 6 (𝑧𝐵 ↔ ∃𝑤(𝑧 Fn 𝑤 ∧ (𝑤𝐴 ∧ ∀𝑡𝑤 Pred(𝑅, 𝐴, 𝑡) ⊆ 𝑤) ∧ ∀𝑡𝑤 (𝑧𝑡) = (𝑡𝐺(𝑧 ↾ Pred(𝑅, 𝐴, 𝑡)))))
9 predeq3 5722 . . . . . . . . . . . 12 (𝑡 = 𝑋 → Pred(𝑅, 𝐴, 𝑡) = Pred(𝑅, 𝐴, 𝑋))
109sseq1d 3665 . . . . . . . . . . 11 (𝑡 = 𝑋 → (Pred(𝑅, 𝐴, 𝑡) ⊆ 𝑤 ↔ Pred(𝑅, 𝐴, 𝑋) ⊆ 𝑤))
1110rspccv 3337 . . . . . . . . . 10 (∀𝑡𝑤 Pred(𝑅, 𝐴, 𝑡) ⊆ 𝑤 → (𝑋𝑤 → Pred(𝑅, 𝐴, 𝑋) ⊆ 𝑤))
1211ad2antlr 763 . . . . . . . . 9 (((𝑤𝐴 ∧ ∀𝑡𝑤 Pred(𝑅, 𝐴, 𝑡) ⊆ 𝑤) ∧ ∀𝑡𝑤 (𝑧𝑡) = (𝑡𝐺(𝑧 ↾ Pred(𝑅, 𝐴, 𝑡)))) → (𝑋𝑤 → Pred(𝑅, 𝐴, 𝑋) ⊆ 𝑤))
13 fndm 6028 . . . . . . . . . . 11 (𝑧 Fn 𝑤 → dom 𝑧 = 𝑤)
1413eleq2d 2716 . . . . . . . . . 10 (𝑧 Fn 𝑤 → (𝑋 ∈ dom 𝑧𝑋𝑤))
1513sseq2d 3666 . . . . . . . . . 10 (𝑧 Fn 𝑤 → (Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑧 ↔ Pred(𝑅, 𝐴, 𝑋) ⊆ 𝑤))
1614, 15imbi12d 333 . . . . . . . . 9 (𝑧 Fn 𝑤 → ((𝑋 ∈ dom 𝑧 → Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑧) ↔ (𝑋𝑤 → Pred(𝑅, 𝐴, 𝑋) ⊆ 𝑤)))
1712, 16syl5ibr 236 . . . . . . . 8 (𝑧 Fn 𝑤 → (((𝑤𝐴 ∧ ∀𝑡𝑤 Pred(𝑅, 𝐴, 𝑡) ⊆ 𝑤) ∧ ∀𝑡𝑤 (𝑧𝑡) = (𝑡𝐺(𝑧 ↾ Pred(𝑅, 𝐴, 𝑡)))) → (𝑋 ∈ dom 𝑧 → Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑧)))
18173impib 1281 . . . . . . 7 ((𝑧 Fn 𝑤 ∧ (𝑤𝐴 ∧ ∀𝑡𝑤 Pred(𝑅, 𝐴, 𝑡) ⊆ 𝑤) ∧ ∀𝑡𝑤 (𝑧𝑡) = (𝑡𝐺(𝑧 ↾ Pred(𝑅, 𝐴, 𝑡)))) → (𝑋 ∈ dom 𝑧 → Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑧))
1918exlimiv 1898 . . . . . 6 (∃𝑤(𝑧 Fn 𝑤 ∧ (𝑤𝐴 ∧ ∀𝑡𝑤 Pred(𝑅, 𝐴, 𝑡) ⊆ 𝑤) ∧ ∀𝑡𝑤 (𝑧𝑡) = (𝑡𝐺(𝑧 ↾ Pred(𝑅, 𝐴, 𝑡)))) → (𝑋 ∈ dom 𝑧 → Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑧))
208, 19sylbi 207 . . . . 5 (𝑧𝐵 → (𝑋 ∈ dom 𝑧 → Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑧))
215, 20syl 17 . . . 4 ((𝐶𝐵𝑧𝐶) → (𝑋 ∈ dom 𝑧 → Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑧))
22 dmeq 5356 . . . . . . . . . 10 (𝑤 = 𝑧 → dom 𝑤 = dom 𝑧)
2322sseq2d 3666 . . . . . . . . 9 (𝑤 = 𝑧 → (Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑤 ↔ Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑧))
2423rspcev 3340 . . . . . . . 8 ((𝑧𝐶 ∧ Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑧) → ∃𝑤𝐶 Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑤)
25 ssiun 4594 . . . . . . . 8 (∃𝑤𝐶 Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑤 → Pred(𝑅, 𝐴, 𝑋) ⊆ 𝑤𝐶 dom 𝑤)
2624, 25syl 17 . . . . . . 7 ((𝑧𝐶 ∧ Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑧) → Pred(𝑅, 𝐴, 𝑋) ⊆ 𝑤𝐶 dom 𝑤)
27 dmuni 5366 . . . . . . 7 dom 𝐶 = 𝑤𝐶 dom 𝑤
2826, 27syl6sseqr 3685 . . . . . 6 ((𝑧𝐶 ∧ Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑧) → Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝐶)
2928ex 449 . . . . 5 (𝑧𝐶 → (Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑧 → Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝐶))
3029adantl 481 . . . 4 ((𝐶𝐵𝑧𝐶) → (Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑧 → Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝐶))
3121, 30syld 47 . . 3 ((𝐶𝐵𝑧𝐶) → (𝑋 ∈ dom 𝑧 → Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝐶))
3231rexlimdva 3060 . 2 (𝐶𝐵 → (∃𝑧𝐶 𝑋 ∈ dom 𝑧 → Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝐶))
334, 32syl5bi 232 1 (𝐶𝐵 → (𝑋 ∈ dom 𝐶 → Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 383  w3a 1054   = wceq 1523  wex 1744  wcel 2030  {cab 2637  wral 2941  wrex 2942  wss 3607   cuni 4468   ciun 4552   Fr wfr 5099   Se wse 5100  dom cdm 5143  cres 5145  Predcpred 5717   Fn wfn 5921  cfv 5926  (class class class)co 6690
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-iun 4554  df-br 4686  df-opab 4746  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-pred 5718  df-iota 5889  df-fun 5928  df-fn 5929  df-fv 5934  df-ov 6693
This theorem is referenced by: (None)
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