Theorem frrlem5e 31032

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 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))))}

Assertion

Ref	Expression
frrlem5e	⊢ (𝐶 ⊆ 𝐵 → (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.

Step	Hyp	Ref	Expression
1		dmuni 5256	. . . 4 ⊢ dom ∪ 𝐶 = ∪ 𝑧 ∈ 𝐶 dom 𝑧
2	1	eleq2i 2680	. . 3 ⊢ (𝑋 ∈ dom ∪ 𝐶 ↔ 𝑋 ∈ ∪ 𝑧 ∈ 𝐶 dom 𝑧)
3		eliun 4460	. . 3 ⊢ (𝑋 ∈ ∪ 𝑧 ∈ 𝐶 dom 𝑧 ↔ ∃𝑧 ∈ 𝐶 𝑋 ∈ dom 𝑧)
4	2, 3	bitri 263	. 2 ⊢ (𝑋 ∈ dom ∪ 𝐶 ↔ ∃𝑧 ∈ 𝐶 𝑋 ∈ dom 𝑧)
5		ssel2 3563	. . . . 5 ⊢ ((𝐶 ⊆ 𝐵 ∧ 𝑧 ∈ 𝐶) → 𝑧 ∈ 𝐵)
6		frrlem5.3	. . . . . . . 8 ⊢ 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦)))))}
7	6	frrlem1 31024	. . . . . . 7 ⊢ 𝐵 = {𝑧 ∣ ∃𝑤(𝑧 Fn 𝑤 ∧ (𝑤 ⊆ 𝐴 ∧ ∀𝑡 ∈ 𝑤 Pred(𝑅, 𝐴, 𝑡) ⊆ 𝑤 ∧ ∀𝑡 ∈ 𝑤 (𝑧‘𝑡) = (𝑡𝐺(𝑧 ↾ Pred(𝑅, 𝐴, 𝑡)))))}
8	7	abeq2i 2722	. . . . . 6 ⊢ (𝑧 ∈ 𝐵 ↔ ∃𝑤(𝑧 Fn 𝑤 ∧ (𝑤 ⊆ 𝐴 ∧ ∀𝑡 ∈ 𝑤 Pred(𝑅, 𝐴, 𝑡) ⊆ 𝑤 ∧ ∀𝑡 ∈ 𝑤 (𝑧‘𝑡) = (𝑡𝐺(𝑧 ↾ Pred(𝑅, 𝐴, 𝑡))))))
9		fndm 5904	. . . . . . . . 9 ⊢ (𝑧 Fn 𝑤 → dom 𝑧 = 𝑤)
10		predeq3 5601	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑋 → Pred(𝑅, 𝐴, 𝑡) = Pred(𝑅, 𝐴, 𝑋))
11	10	sseq1d 3595	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑋 → (Pred(𝑅, 𝐴, 𝑡) ⊆ 𝑤 ↔ Pred(𝑅, 𝐴, 𝑋) ⊆ 𝑤))
12	11	rspccv 3279	. . . . . . . . . . 11 ⊢ (∀𝑡 ∈ 𝑤 Pred(𝑅, 𝐴, 𝑡) ⊆ 𝑤 → (𝑋 ∈ 𝑤 → Pred(𝑅, 𝐴, 𝑋) ⊆ 𝑤))
13	12	3ad2ant2 1076	. . . . . . . . . 10 ⊢ ((𝑤 ⊆ 𝐴 ∧ ∀𝑡 ∈ 𝑤 Pred(𝑅, 𝐴, 𝑡) ⊆ 𝑤 ∧ ∀𝑡 ∈ 𝑤 (𝑧‘𝑡) = (𝑡𝐺(𝑧 ↾ Pred(𝑅, 𝐴, 𝑡)))) → (𝑋 ∈ 𝑤 → Pred(𝑅, 𝐴, 𝑋) ⊆ 𝑤))
14		eleq2 2677	. . . . . . . . . . 11 ⊢ (dom 𝑧 = 𝑤 → (𝑋 ∈ dom 𝑧 ↔ 𝑋 ∈ 𝑤))
15		sseq2 3590	. . . . . . . . . . 11 ⊢ (dom 𝑧 = 𝑤 → (Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑧 ↔ Pred(𝑅, 𝐴, 𝑋) ⊆ 𝑤))
16	14, 15	imbi12d 333	. . . . . . . . . 10 ⊢ (dom 𝑧 = 𝑤 → ((𝑋 ∈ dom 𝑧 → Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑧) ↔ (𝑋 ∈ 𝑤 → Pred(𝑅, 𝐴, 𝑋) ⊆ 𝑤)))
17	13, 16	syl5ibr 235	. . . . . . . . 9 ⊢ (dom 𝑧 = 𝑤 → ((𝑤 ⊆ 𝐴 ∧ ∀𝑡 ∈ 𝑤 Pred(𝑅, 𝐴, 𝑡) ⊆ 𝑤 ∧ ∀𝑡 ∈ 𝑤 (𝑧‘𝑡) = (𝑡𝐺(𝑧 ↾ Pred(𝑅, 𝐴, 𝑡)))) → (𝑋 ∈ dom 𝑧 → Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑧)))
18	9, 17	syl 17	. . . . . . . 8 ⊢ (𝑧 Fn 𝑤 → ((𝑤 ⊆ 𝐴 ∧ ∀𝑡 ∈ 𝑤 Pred(𝑅, 𝐴, 𝑡) ⊆ 𝑤 ∧ ∀𝑡 ∈ 𝑤 (𝑧‘𝑡) = (𝑡𝐺(𝑧 ↾ Pred(𝑅, 𝐴, 𝑡)))) → (𝑋 ∈ dom 𝑧 → Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑧)))
19	18	imp 444	. . . . . . 7 ⊢ ((𝑧 Fn 𝑤 ∧ (𝑤 ⊆ 𝐴 ∧ ∀𝑡 ∈ 𝑤 Pred(𝑅, 𝐴, 𝑡) ⊆ 𝑤 ∧ ∀𝑡 ∈ 𝑤 (𝑧‘𝑡) = (𝑡𝐺(𝑧 ↾ Pred(𝑅, 𝐴, 𝑡))))) → (𝑋 ∈ dom 𝑧 → Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑧))
20	19	exlimiv 1845	. . . . . 6 ⊢ (∃𝑤(𝑧 Fn 𝑤 ∧ (𝑤 ⊆ 𝐴 ∧ ∀𝑡 ∈ 𝑤 Pred(𝑅, 𝐴, 𝑡) ⊆ 𝑤 ∧ ∀𝑡 ∈ 𝑤 (𝑧‘𝑡) = (𝑡𝐺(𝑧 ↾ Pred(𝑅, 𝐴, 𝑡))))) → (𝑋 ∈ dom 𝑧 → Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑧))
21	8, 20	sylbi 206	. . . . 5 ⊢ (𝑧 ∈ 𝐵 → (𝑋 ∈ dom 𝑧 → Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑧))
22	5, 21	syl 17	. . . 4 ⊢ ((𝐶 ⊆ 𝐵 ∧ 𝑧 ∈ 𝐶) → (𝑋 ∈ dom 𝑧 → Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑧))
23		dmeq 5246	. . . . . . . . . 10 ⊢ (𝑤 = 𝑧 → dom 𝑤 = dom 𝑧)
24	23	sseq2d 3596	. . . . . . . . 9 ⊢ (𝑤 = 𝑧 → (Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑤 ↔ Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑧))
25	24	rspcev 3282	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝐶 ∧ Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑧) → ∃𝑤 ∈ 𝐶 Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑤)
26		ssiun 4498	. . . . . . . 8 ⊢ (∃𝑤 ∈ 𝐶 Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑤 → Pred(𝑅, 𝐴, 𝑋) ⊆ ∪ 𝑤 ∈ 𝐶 dom 𝑤)
27	25, 26	syl 17	. . . . . . 7 ⊢ ((𝑧 ∈ 𝐶 ∧ Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑧) → Pred(𝑅, 𝐴, 𝑋) ⊆ ∪ 𝑤 ∈ 𝐶 dom 𝑤)
28		dmuni 5256	. . . . . . 7 ⊢ dom ∪ 𝐶 = ∪ 𝑤 ∈ 𝐶 dom 𝑤
29	27, 28	syl6sseqr 3615	. . . . . 6 ⊢ ((𝑧 ∈ 𝐶 ∧ Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑧) → Pred(𝑅, 𝐴, 𝑋) ⊆ dom ∪ 𝐶)
30	29	ex 449	. . . . 5 ⊢ (𝑧 ∈ 𝐶 → (Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑧 → Pred(𝑅, 𝐴, 𝑋) ⊆ dom ∪ 𝐶))
31	30	adantl 481	. . . 4 ⊢ ((𝐶 ⊆ 𝐵 ∧ 𝑧 ∈ 𝐶) → (Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑧 → Pred(𝑅, 𝐴, 𝑋) ⊆ dom ∪ 𝐶))
32	22, 31	syld 46	. . 3 ⊢ ((𝐶 ⊆ 𝐵 ∧ 𝑧 ∈ 𝐶) → (𝑋 ∈ dom 𝑧 → Pred(𝑅, 𝐴, 𝑋) ⊆ dom ∪ 𝐶))
33	32	rexlimdva 3013	. 2 ⊢ (𝐶 ⊆ 𝐵 → (∃𝑧 ∈ 𝐶 𝑋 ∈ dom 𝑧 → Pred(𝑅, 𝐴, 𝑋) ⊆ dom ∪ 𝐶))
34	4, 33	syl5bi 231	1 ⊢ (𝐶 ⊆ 𝐵 → (𝑋 ∈ dom ∪ 𝐶 → Pred(𝑅, 𝐴, 𝑋) ⊆ dom ∪ 𝐶))

Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977  {cab 2596  ∀wral 2896  ∃wrex 2897   ⊆ wss 3540  ∪ cuni 4372  ∪ ciun 4455   Fr wfr 4994   Se wse 4995  dom cdm 5038   ↾ cres 5040  Predcpred 5596   Fn wfn 5799  ‘cfv 5804  (class class class)co 6549

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-iota 5768  df-fun 5806  df-fn 5807  df-fv 5812  df-ov 6552

This theorem is referenced by: (None)