Theorem frrlem5e 32114

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 5542	. . . 4 ⊢ dom ∪ 𝐶 = ∪ 𝑧 ∈ 𝐶 dom 𝑧
2	1	eleq2i 2884	. . 3 ⊢ (𝑋 ∈ dom ∪ 𝐶 ↔ 𝑋 ∈ ∪ 𝑧 ∈ 𝐶 dom 𝑧)
3		eliun 4723	. . 3 ⊢ (𝑋 ∈ ∪ 𝑧 ∈ 𝐶 dom 𝑧 ↔ ∃𝑧 ∈ 𝐶 𝑋 ∈ dom 𝑧)
4	2, 3	bitri 266	. 2 ⊢ (𝑋 ∈ dom ∪ 𝐶 ↔ ∃𝑧 ∈ 𝐶 𝑋 ∈ dom 𝑧)
5		ssel2 3800	. . . . 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 32106	. . . . . . 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 2926	. . . . . 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		predeq3 5904	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑋 → Pred(𝑅, 𝐴, 𝑡) = Pred(𝑅, 𝐴, 𝑋))
10	9	sseq1d 3836	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑋 → (Pred(𝑅, 𝐴, 𝑡) ⊆ 𝑤 ↔ Pred(𝑅, 𝐴, 𝑋) ⊆ 𝑤))
11	10	rspccv 3506	. . . . . . . . . 10 ⊢ (∀𝑡 ∈ 𝑤 Pred(𝑅, 𝐴, 𝑡) ⊆ 𝑤 → (𝑋 ∈ 𝑤 → Pred(𝑅, 𝐴, 𝑋) ⊆ 𝑤))
12	11	ad2antlr 709	. . . . . . . . 9 ⊢ (((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))
13		fndm 6204	. . . . . . . . . . 11 ⊢ (𝑧 Fn 𝑤 → dom 𝑧 = 𝑤)
14	13	eleq2d 2878	. . . . . . . . . 10 ⊢ (𝑧 Fn 𝑤 → (𝑋 ∈ dom 𝑧 ↔ 𝑋 ∈ 𝑤))
15	13	sseq2d 3837	. . . . . . . . . 10 ⊢ (𝑧 Fn 𝑤 → (Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑧 ↔ Pred(𝑅, 𝐴, 𝑋) ⊆ 𝑤))
16	14, 15	imbi12d 335	. . . . . . . . 9 ⊢ (𝑧 Fn 𝑤 → ((𝑋 ∈ dom 𝑧 → Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑧) ↔ (𝑋 ∈ 𝑤 → Pred(𝑅, 𝐴, 𝑋) ⊆ 𝑤)))
17	12, 16	syl5ibr 237	. . . . . . . 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)))
18	17	3impib 1137	. . . . . . 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))
19	18	exlimiv 2021	. . . . . 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))
20	8, 19	sylbi 208	. . . . 5 ⊢ (𝑧 ∈ 𝐵 → (𝑋 ∈ dom 𝑧 → Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑧))
21	5, 20	syl 17	. . . 4 ⊢ ((𝐶 ⊆ 𝐵 ∧ 𝑧 ∈ 𝐶) → (𝑋 ∈ dom 𝑧 → Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑧))
22		dmeq 5532	. . . . . . . . . 10 ⊢ (𝑤 = 𝑧 → dom 𝑤 = dom 𝑧)
23	22	sseq2d 3837	. . . . . . . . 9 ⊢ (𝑤 = 𝑧 → (Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑤 ↔ Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑧))
24	23	rspcev 3509	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝐶 ∧ Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑧) → ∃𝑤 ∈ 𝐶 Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑤)
25		ssiun 4761	. . . . . . . 8 ⊢ (∃𝑤 ∈ 𝐶 Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑤 → Pred(𝑅, 𝐴, 𝑋) ⊆ ∪ 𝑤 ∈ 𝐶 dom 𝑤)
26	24, 25	syl 17	. . . . . . 7 ⊢ ((𝑧 ∈ 𝐶 ∧ Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑧) → Pred(𝑅, 𝐴, 𝑋) ⊆ ∪ 𝑤 ∈ 𝐶 dom 𝑤)
27		dmuni 5542	. . . . . . 7 ⊢ dom ∪ 𝐶 = ∪ 𝑤 ∈ 𝐶 dom 𝑤
28	26, 27	syl6sseqr 3856	. . . . . 6 ⊢ ((𝑧 ∈ 𝐶 ∧ Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑧) → Pred(𝑅, 𝐴, 𝑋) ⊆ dom ∪ 𝐶)
29	28	ex 399	. . . . 5 ⊢ (𝑧 ∈ 𝐶 → (Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑧 → Pred(𝑅, 𝐴, 𝑋) ⊆ dom ∪ 𝐶))
30	29	adantl 469	. . . 4 ⊢ ((𝐶 ⊆ 𝐵 ∧ 𝑧 ∈ 𝐶) → (Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑧 → Pred(𝑅, 𝐴, 𝑋) ⊆ dom ∪ 𝐶))
31	21, 30	syld 47	. . 3 ⊢ ((𝐶 ⊆ 𝐵 ∧ 𝑧 ∈ 𝐶) → (𝑋 ∈ dom 𝑧 → Pred(𝑅, 𝐴, 𝑋) ⊆ dom ∪ 𝐶))
32	31	rexlimdva 3226	. 2 ⊢ (𝐶 ⊆ 𝐵 → (∃𝑧 ∈ 𝐶 𝑋 ∈ dom 𝑧 → Pred(𝑅, 𝐴, 𝑋) ⊆ dom ∪ 𝐶))
33	4, 32	syl5bi 233	1 ⊢ (𝐶 ⊆ 𝐵 → (𝑋 ∈ dom ∪ 𝐶 → Pred(𝑅, 𝐴, 𝑋) ⊆ dom ∪ 𝐶))

Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1100   = wceq 1637  ∃wex 1859   ∈ wcel 2157  {cab 2799  ∀wral 3103  ∃wrex 3104   ⊆ wss 3776  ∪ cuni 4637  ∪ ciun 4719   Fr wfr 5274   Se wse 5275  dom cdm 5318   ↾ cres 5320  Predcpred 5899   Fn wfn 6099  ‘cfv 6104  (class class class)co 6877

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1877  ax-4 1894  ax-5 2001  ax-6 2069  ax-7 2105  ax-9 2166  ax-10 2186  ax-11 2202  ax-12 2215  ax-13 2422  ax-ext 2791

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 866  df-3an 1102  df-tru 1641  df-ex 1860  df-nf 1864  df-sb 2062  df-clab 2800  df-cleq 2806  df-clel 2809  df-nfc 2944  df-ral 3108  df-rex 3109  df-rab 3112  df-v 3400  df-dif 3779  df-un 3781  df-in 3783  df-ss 3790  df-nul 4124  df-if 4287  df-sn 4378  df-pr 4380  df-op 4384  df-uni 4638  df-iun 4721  df-br 4852  df-opab 4914  df-xp 5324  df-rel 5325  df-cnv 5326  df-co 5327  df-dm 5328  df-rn 5329  df-res 5330  df-ima 5331  df-pred 5900  df-iota 6067  df-fun 6106  df-fn 6107  df-fv 6112  df-ov 6880

This theorem is referenced by: (None)