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Theorem frrlem5e 32125
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 5472 . . . 4 dom 𝐶 = 𝑧𝐶 dom 𝑧
21eleq2i 2842 . . 3 (𝑋 ∈ dom 𝐶𝑋 𝑧𝐶 dom 𝑧)
3 eliun 4658 . . 3 (𝑋 𝑧𝐶 dom 𝑧 ↔ ∃𝑧𝐶 𝑋 ∈ dom 𝑧)
42, 3bitri 264 . 2 (𝑋 ∈ dom 𝐶 ↔ ∃𝑧𝐶 𝑋 ∈ dom 𝑧)
5 ssel2 3747 . . . . 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 32117 . . . . . . 7 𝐵 = {𝑧 ∣ ∃𝑤(𝑧 Fn 𝑤 ∧ (𝑤𝐴 ∧ ∀𝑡𝑤 Pred(𝑅, 𝐴, 𝑡) ⊆ 𝑤) ∧ ∀𝑡𝑤 (𝑧𝑡) = (𝑡𝐺(𝑧 ↾ Pred(𝑅, 𝐴, 𝑡))))}
87abeq2i 2884 . . . . . 6 (𝑧𝐵 ↔ ∃𝑤(𝑧 Fn 𝑤 ∧ (𝑤𝐴 ∧ ∀𝑡𝑤 Pred(𝑅, 𝐴, 𝑡) ⊆ 𝑤) ∧ ∀𝑡𝑤 (𝑧𝑡) = (𝑡𝐺(𝑧 ↾ Pred(𝑅, 𝐴, 𝑡)))))
9 predeq3 5827 . . . . . . . . . . . 12 (𝑡 = 𝑋 → Pred(𝑅, 𝐴, 𝑡) = Pred(𝑅, 𝐴, 𝑋))
109sseq1d 3781 . . . . . . . . . . 11 (𝑡 = 𝑋 → (Pred(𝑅, 𝐴, 𝑡) ⊆ 𝑤 ↔ Pred(𝑅, 𝐴, 𝑋) ⊆ 𝑤))
1110rspccv 3457 . . . . . . . . . 10 (∀𝑡𝑤 Pred(𝑅, 𝐴, 𝑡) ⊆ 𝑤 → (𝑋𝑤 → Pred(𝑅, 𝐴, 𝑋) ⊆ 𝑤))
1211ad2antlr 706 . . . . . . . . 9 (((𝑤𝐴 ∧ ∀𝑡𝑤 Pred(𝑅, 𝐴, 𝑡) ⊆ 𝑤) ∧ ∀𝑡𝑤 (𝑧𝑡) = (𝑡𝐺(𝑧 ↾ Pred(𝑅, 𝐴, 𝑡)))) → (𝑋𝑤 → Pred(𝑅, 𝐴, 𝑋) ⊆ 𝑤))
13 fndm 6130 . . . . . . . . . . 11 (𝑧 Fn 𝑤 → dom 𝑧 = 𝑤)
1413eleq2d 2836 . . . . . . . . . 10 (𝑧 Fn 𝑤 → (𝑋 ∈ dom 𝑧𝑋𝑤))
1513sseq2d 3782 . . . . . . . . . 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 1108 . . . . . . 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 2010 . . . . . 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 5462 . . . . . . . . . 10 (𝑤 = 𝑧 → dom 𝑤 = dom 𝑧)
2322sseq2d 3782 . . . . . . . . 9 (𝑤 = 𝑧 → (Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑤 ↔ Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑧))
2423rspcev 3460 . . . . . . . 8 ((𝑧𝐶 ∧ Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑧) → ∃𝑤𝐶 Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑤)
25 ssiun 4696 . . . . . . . 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 5472 . . . . . . 7 dom 𝐶 = 𝑤𝐶 dom 𝑤
2826, 27syl6sseqr 3801 . . . . . 6 ((𝑧𝐶 ∧ Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑧) → Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝐶)
2928ex 397 . . . . 5 (𝑧𝐶 → (Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑧 → Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝐶))
3029adantl 467 . . . 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 3179 . 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 382  w3a 1071   = wceq 1631  wex 1852  wcel 2145  {cab 2757  wral 3061  wrex 3062  wss 3723   cuni 4574   ciun 4654   Fr wfr 5205   Se wse 5206  dom cdm 5249  cres 5251  Predcpred 5822   Fn wfn 6026  cfv 6031  (class class class)co 6793
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-iota 5994  df-fun 6033  df-fn 6034  df-fv 6039  df-ov 6796
This theorem is referenced by: (None)
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