Theorem bnj517 32865

Description: Technical lemma for bnj518 32866. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (Proof shortened by Mario Carneiro, 22-Dec-2016.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj517.1	⊢ (𝜑 ↔ (𝐹‘∅) = pred(𝑋, 𝐴, 𝑅))
bnj517.2	⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑁 → (𝐹‘suc 𝑖) = ∪ 𝑦 ∈ (𝐹‘𝑖) pred(𝑦, 𝐴, 𝑅)))

Assertion

Ref	Expression
bnj517	⊢ ((𝑁 ∈ ω ∧ 𝜑 ∧ 𝜓) → ∀𝑛 ∈ 𝑁 (𝐹‘𝑛) ⊆ 𝐴)

Distinct variable groups:   𝑖,𝑛,𝑦,𝐴   𝑖,𝐹,𝑛   𝑖,𝑁,𝑛

Allowed substitution hints:   𝜑(𝑦,𝑖,𝑛)   𝜓(𝑦,𝑖,𝑛)   𝑅(𝑦,𝑖,𝑛)   𝐹(𝑦)   𝑁(𝑦)   𝑋(𝑦,𝑖,𝑛)

Proof of Theorem bnj517

Dummy variable 𝑚 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6774	. . . . . 6 ⊢ (𝑚 = ∅ → (𝐹‘𝑚) = (𝐹‘∅))
2		simpl2 1191	. . . . . . 7 ⊢ (((𝑁 ∈ ω ∧ 𝜑 ∧ 𝜓) ∧ 𝑚 ∈ 𝑁) → 𝜑)
3		bnj517.1	. . . . . . 7 ⊢ (𝜑 ↔ (𝐹‘∅) = pred(𝑋, 𝐴, 𝑅))
4	2, 3	sylib 217	. . . . . 6 ⊢ (((𝑁 ∈ ω ∧ 𝜑 ∧ 𝜓) ∧ 𝑚 ∈ 𝑁) → (𝐹‘∅) = pred(𝑋, 𝐴, 𝑅))
5	1, 4	sylan9eqr 2800	. . . . 5 ⊢ ((((𝑁 ∈ ω ∧ 𝜑 ∧ 𝜓) ∧ 𝑚 ∈ 𝑁) ∧ 𝑚 = ∅) → (𝐹‘𝑚) = pred(𝑋, 𝐴, 𝑅))
6		bnj213 32862	. . . . 5 ⊢ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐴
7	5, 6	eqsstrdi 3975	. . . 4 ⊢ ((((𝑁 ∈ ω ∧ 𝜑 ∧ 𝜓) ∧ 𝑚 ∈ 𝑁) ∧ 𝑚 = ∅) → (𝐹‘𝑚) ⊆ 𝐴)
8		bnj517.2	. . . . . . 7 ⊢ (𝜓 ↔ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑁 → (𝐹‘suc 𝑖) = ∪ 𝑦 ∈ (𝐹‘𝑖) pred(𝑦, 𝐴, 𝑅)))
9		r19.29r 3185	. . . . . . . . . 10 ⊢ ((∃𝑖 ∈ ω 𝑚 = suc 𝑖 ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑁 → (𝐹‘suc 𝑖) = ∪ 𝑦 ∈ (𝐹‘𝑖) pred(𝑦, 𝐴, 𝑅))) → ∃𝑖 ∈ ω (𝑚 = suc 𝑖 ∧ (suc 𝑖 ∈ 𝑁 → (𝐹‘suc 𝑖) = ∪ 𝑦 ∈ (𝐹‘𝑖) pred(𝑦, 𝐴, 𝑅))))
10		eleq1 2826	. . . . . . . . . . . . . 14 ⊢ (𝑚 = suc 𝑖 → (𝑚 ∈ 𝑁 ↔ suc 𝑖 ∈ 𝑁))
11	10	biimpd 228	. . . . . . . . . . . . 13 ⊢ (𝑚 = suc 𝑖 → (𝑚 ∈ 𝑁 → suc 𝑖 ∈ 𝑁))
12		fveqeq2 6783	. . . . . . . . . . . . . 14 ⊢ (𝑚 = suc 𝑖 → ((𝐹‘𝑚) = ∪ 𝑦 ∈ (𝐹‘𝑖) pred(𝑦, 𝐴, 𝑅) ↔ (𝐹‘suc 𝑖) = ∪ 𝑦 ∈ (𝐹‘𝑖) pred(𝑦, 𝐴, 𝑅)))
13		bnj213 32862	. . . . . . . . . . . . . . . . 17 ⊢ pred(𝑦, 𝐴, 𝑅) ⊆ 𝐴
14	13	rgenw 3076	. . . . . . . . . . . . . . . 16 ⊢ ∀𝑦 ∈ (𝐹‘𝑖) pred(𝑦, 𝐴, 𝑅) ⊆ 𝐴
15		iunss 4975	. . . . . . . . . . . . . . . 16 ⊢ (∪ 𝑦 ∈ (𝐹‘𝑖) pred(𝑦, 𝐴, 𝑅) ⊆ 𝐴 ↔ ∀𝑦 ∈ (𝐹‘𝑖) pred(𝑦, 𝐴, 𝑅) ⊆ 𝐴)
16	14, 15	mpbir 230	. . . . . . . . . . . . . . 15 ⊢ ∪ 𝑦 ∈ (𝐹‘𝑖) pred(𝑦, 𝐴, 𝑅) ⊆ 𝐴
17		sseq1 3946	. . . . . . . . . . . . . . 15 ⊢ ((𝐹‘𝑚) = ∪ 𝑦 ∈ (𝐹‘𝑖) pred(𝑦, 𝐴, 𝑅) → ((𝐹‘𝑚) ⊆ 𝐴 ↔ ∪ 𝑦 ∈ (𝐹‘𝑖) pred(𝑦, 𝐴, 𝑅) ⊆ 𝐴))
18	16, 17	mpbiri 257	. . . . . . . . . . . . . 14 ⊢ ((𝐹‘𝑚) = ∪ 𝑦 ∈ (𝐹‘𝑖) pred(𝑦, 𝐴, 𝑅) → (𝐹‘𝑚) ⊆ 𝐴)
19	12, 18	syl6bir 253	. . . . . . . . . . . . 13 ⊢ (𝑚 = suc 𝑖 → ((𝐹‘suc 𝑖) = ∪ 𝑦 ∈ (𝐹‘𝑖) pred(𝑦, 𝐴, 𝑅) → (𝐹‘𝑚) ⊆ 𝐴))
20	11, 19	imim12d 81	. . . . . . . . . . . 12 ⊢ (𝑚 = suc 𝑖 → ((suc 𝑖 ∈ 𝑁 → (𝐹‘suc 𝑖) = ∪ 𝑦 ∈ (𝐹‘𝑖) pred(𝑦, 𝐴, 𝑅)) → (𝑚 ∈ 𝑁 → (𝐹‘𝑚) ⊆ 𝐴)))
21	20	imp 407	. . . . . . . . . . 11 ⊢ ((𝑚 = suc 𝑖 ∧ (suc 𝑖 ∈ 𝑁 → (𝐹‘suc 𝑖) = ∪ 𝑦 ∈ (𝐹‘𝑖) pred(𝑦, 𝐴, 𝑅))) → (𝑚 ∈ 𝑁 → (𝐹‘𝑚) ⊆ 𝐴))
22	21	rexlimivw 3211	. . . . . . . . . 10 ⊢ (∃𝑖 ∈ ω (𝑚 = suc 𝑖 ∧ (suc 𝑖 ∈ 𝑁 → (𝐹‘suc 𝑖) = ∪ 𝑦 ∈ (𝐹‘𝑖) pred(𝑦, 𝐴, 𝑅))) → (𝑚 ∈ 𝑁 → (𝐹‘𝑚) ⊆ 𝐴))
23	9, 22	syl 17	. . . . . . . . 9 ⊢ ((∃𝑖 ∈ ω 𝑚 = suc 𝑖 ∧ ∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑁 → (𝐹‘suc 𝑖) = ∪ 𝑦 ∈ (𝐹‘𝑖) pred(𝑦, 𝐴, 𝑅))) → (𝑚 ∈ 𝑁 → (𝐹‘𝑚) ⊆ 𝐴))
24	23	ex 413	. . . . . . . 8 ⊢ (∃𝑖 ∈ ω 𝑚 = suc 𝑖 → (∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑁 → (𝐹‘suc 𝑖) = ∪ 𝑦 ∈ (𝐹‘𝑖) pred(𝑦, 𝐴, 𝑅)) → (𝑚 ∈ 𝑁 → (𝐹‘𝑚) ⊆ 𝐴)))
25	24	com3l 89	. . . . . . 7 ⊢ (∀𝑖 ∈ ω (suc 𝑖 ∈ 𝑁 → (𝐹‘suc 𝑖) = ∪ 𝑦 ∈ (𝐹‘𝑖) pred(𝑦, 𝐴, 𝑅)) → (𝑚 ∈ 𝑁 → (∃𝑖 ∈ ω 𝑚 = suc 𝑖 → (𝐹‘𝑚) ⊆ 𝐴)))
26	8, 25	sylbi 216	. . . . . 6 ⊢ (𝜓 → (𝑚 ∈ 𝑁 → (∃𝑖 ∈ ω 𝑚 = suc 𝑖 → (𝐹‘𝑚) ⊆ 𝐴)))
27	26	3ad2ant3 1134	. . . . 5 ⊢ ((𝑁 ∈ ω ∧ 𝜑 ∧ 𝜓) → (𝑚 ∈ 𝑁 → (∃𝑖 ∈ ω 𝑚 = suc 𝑖 → (𝐹‘𝑚) ⊆ 𝐴)))
28	27	imp31 418	. . . 4 ⊢ ((((𝑁 ∈ ω ∧ 𝜑 ∧ 𝜓) ∧ 𝑚 ∈ 𝑁) ∧ ∃𝑖 ∈ ω 𝑚 = suc 𝑖) → (𝐹‘𝑚) ⊆ 𝐴)
29		simpr 485	. . . . . 6 ⊢ (((𝑁 ∈ ω ∧ 𝜑 ∧ 𝜓) ∧ 𝑚 ∈ 𝑁) → 𝑚 ∈ 𝑁)
30		simpl1 1190	. . . . . 6 ⊢ (((𝑁 ∈ ω ∧ 𝜑 ∧ 𝜓) ∧ 𝑚 ∈ 𝑁) → 𝑁 ∈ ω)
31		elnn 7723	. . . . . 6 ⊢ ((𝑚 ∈ 𝑁 ∧ 𝑁 ∈ ω) → 𝑚 ∈ ω)
32	29, 30, 31	syl2anc 584	. . . . 5 ⊢ (((𝑁 ∈ ω ∧ 𝜑 ∧ 𝜓) ∧ 𝑚 ∈ 𝑁) → 𝑚 ∈ ω)
33		nn0suc 7742	. . . . 5 ⊢ (𝑚 ∈ ω → (𝑚 = ∅ ∨ ∃𝑖 ∈ ω 𝑚 = suc 𝑖))
34	32, 33	syl 17	. . . 4 ⊢ (((𝑁 ∈ ω ∧ 𝜑 ∧ 𝜓) ∧ 𝑚 ∈ 𝑁) → (𝑚 = ∅ ∨ ∃𝑖 ∈ ω 𝑚 = suc 𝑖))
35	7, 28, 34	mpjaodan 956	. . 3 ⊢ (((𝑁 ∈ ω ∧ 𝜑 ∧ 𝜓) ∧ 𝑚 ∈ 𝑁) → (𝐹‘𝑚) ⊆ 𝐴)
36	35	ralrimiva 3103	. 2 ⊢ ((𝑁 ∈ ω ∧ 𝜑 ∧ 𝜓) → ∀𝑚 ∈ 𝑁 (𝐹‘𝑚) ⊆ 𝐴)
37		fveq2 6774	. . . 4 ⊢ (𝑚 = 𝑛 → (𝐹‘𝑚) = (𝐹‘𝑛))
38	37	sseq1d 3952	. . 3 ⊢ (𝑚 = 𝑛 → ((𝐹‘𝑚) ⊆ 𝐴 ↔ (𝐹‘𝑛) ⊆ 𝐴))
39	38	cbvralvw 3383	. 2 ⊢ (∀𝑚 ∈ 𝑁 (𝐹‘𝑚) ⊆ 𝐴 ↔ ∀𝑛 ∈ 𝑁 (𝐹‘𝑛) ⊆ 𝐴)
40	36, 39	sylib 217	1 ⊢ ((𝑁 ∈ ω ∧ 𝜑 ∧ 𝜓) → ∀𝑛 ∈ 𝑁 (𝐹‘𝑛) ⊆ 𝐴)

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 844   ∧ w3a 1086   = wceq 1539   ∈ wcel 2106  ∀wral 3064  ∃wrex 3065   ⊆ wss 3887  ∅c0 4256  ∪ ciun 4924  suc csuc 6268  ‘cfv 6433  ωcom 7712   predc-bnj14 32667

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352  ax-un 7588

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-nf 1787  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-nfc 2889  df-ne 2944  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-pss 3906  df-nul 4257  df-if 4460  df-pw 4535  df-sn 4562  df-pr 4564  df-op 4568  df-uni 4840  df-iun 4926  df-br 5075  df-opab 5137  df-tr 5192  df-eprel 5495  df-po 5503  df-so 5504  df-fr 5544  df-we 5546  df-ord 6269  df-on 6270  df-lim 6271  df-suc 6272  df-iota 6391  df-fv 6441  df-om 7713  df-bnj14 32668

This theorem is referenced by:  bnj518  32866