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 ∧ (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 5472	. . . 4 ⊢ dom ∪ 𝐶 = ∪ 𝑧 ∈ 𝐶 dom 𝑧
2	1	eleq2i 2842	. . 3 ⊢ (𝑋 ∈ dom ∪ 𝐶 ↔ 𝑋 ∈ ∪ 𝑧 ∈ 𝐶 dom 𝑧)
3		eliun 4658	. . 3 ⊢ (𝑋 ∈ ∪ 𝑧 ∈ 𝐶 dom 𝑧 ↔ ∃𝑧 ∈ 𝐶 𝑋 ∈ dom 𝑧)
4	2, 3	bitri 264	. 2 ⊢ (𝑋 ∈ dom ∪ 𝐶 ↔ ∃𝑧 ∈ 𝐶 𝑋 ∈ dom 𝑧)
5		ssel2 3747	. . . . 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 32117	. . . . . . 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 2884	. . . . . 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 5827	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑋 → Pred(𝑅, 𝐴, 𝑡) = Pred(𝑅, 𝐴, 𝑋))
10	9	sseq1d 3781	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑋 → (Pred(𝑅, 𝐴, 𝑡) ⊆ 𝑤 ↔ Pred(𝑅, 𝐴, 𝑋) ⊆ 𝑤))
11	10	rspccv 3457	. . . . . . . . . 10 ⊢ (∀𝑡 ∈ 𝑤 Pred(𝑅, 𝐴, 𝑡) ⊆ 𝑤 → (𝑋 ∈ 𝑤 → Pred(𝑅, 𝐴, 𝑋) ⊆ 𝑤))
12	11	ad2antlr 706	. . . . . . . . 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 6130	. . . . . . . . . . 11 ⊢ (𝑧 Fn 𝑤 → dom 𝑧 = 𝑤)
14	13	eleq2d 2836	. . . . . . . . . 10 ⊢ (𝑧 Fn 𝑤 → (𝑋 ∈ dom 𝑧 ↔ 𝑋 ∈ 𝑤))
15	13	sseq2d 3782	. . . . . . . . . 10 ⊢ (𝑧 Fn 𝑤 → (Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑧 ↔ Pred(𝑅, 𝐴, 𝑋) ⊆ 𝑤))
16	14, 15	imbi12d 333	. . . . . . . . 9 ⊢ (𝑧 Fn 𝑤 → ((𝑋 ∈ dom 𝑧 → Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑧) ↔ (𝑋 ∈ 𝑤 → Pred(𝑅, 𝐴, 𝑋) ⊆ 𝑤)))
17	12, 16	syl5ibr 236	. . . . . . . 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 1108	. . . . . . 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 2010	. . . . . 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 207	. . . . 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 5462	. . . . . . . . . 10 ⊢ (𝑤 = 𝑧 → dom 𝑤 = dom 𝑧)
23	22	sseq2d 3782	. . . . . . . . 9 ⊢ (𝑤 = 𝑧 → (Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑤 ↔ Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑧))
24	23	rspcev 3460	. . . . . . . 8 ⊢ ((𝑧 ∈ 𝐶 ∧ Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑧) → ∃𝑤 ∈ 𝐶 Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑤)
25		ssiun 4696	. . . . . . . 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 5472	. . . . . . 7 ⊢ dom ∪ 𝐶 = ∪ 𝑤 ∈ 𝐶 dom 𝑤
28	26, 27	syl6sseqr 3801	. . . . . 6 ⊢ ((𝑧 ∈ 𝐶 ∧ Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑧) → Pred(𝑅, 𝐴, 𝑋) ⊆ dom ∪ 𝐶)
29	28	ex 397	. . . . 5 ⊢ (𝑧 ∈ 𝐶 → (Pred(𝑅, 𝐴, 𝑋) ⊆ dom 𝑧 → Pred(𝑅, 𝐴, 𝑋) ⊆ dom ∪ 𝐶))
30	29	adantl 467	. . . 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 3179	. 2 ⊢ (𝐶 ⊆ 𝐵 → (∃𝑧 ∈ 𝐶 𝑋 ∈ dom 𝑧 → Pred(𝑅, 𝐴, 𝑋) ⊆ dom ∪ 𝐶))
33	4, 32	syl5bi 232	1 ⊢ (𝐶 ⊆ 𝐵 → (𝑋 ∈ dom ∪ 𝐶 → Pred(𝑅, 𝐴, 𝑋) ⊆ dom ∪ 𝐶))

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)