Theorem bnj517 32765

Description: Technical lemma for bnj518 32766. 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 6756	. . . . . 6 ⊢ (𝑚 = ∅ → (𝐹‘𝑚) = (𝐹‘∅))
2		simpl2 1190	. . . . . . 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 2801	. . . . 5 ⊢ ((((𝑁 ∈ ω ∧ 𝜑 ∧ 𝜓) ∧ 𝑚 ∈ 𝑁) ∧ 𝑚 = ∅) → (𝐹‘𝑚) = pred(𝑋, 𝐴, 𝑅))
6		bnj213 32762	. . . . 5 ⊢ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐴
7	5, 6	eqsstrdi 3971	. . . 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 3184	. . . . . . . . . 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 6765	. . . . . . . . . . . . . 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 32762	. . . . . . . . . . . . . . . . 17 ⊢ pred(𝑦, 𝐴, 𝑅) ⊆ 𝐴
14	13	rgenw 3075	. . . . . . . . . . . . . . . 16 ⊢ ∀𝑦 ∈ (𝐹‘𝑖) pred(𝑦, 𝐴, 𝑅) ⊆ 𝐴
15		iunss 4971	. . . . . . . . . . . . . . . 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 3942	. . . . . . . . . . . . . . 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 406	. . . . . . . . . . 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 3210	. . . . . . . . . 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 412	. . . . . . . 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 1133	. . . . 5 ⊢ ((𝑁 ∈ ω ∧ 𝜑 ∧ 𝜓) → (𝑚 ∈ 𝑁 → (∃𝑖 ∈ ω 𝑚 = suc 𝑖 → (𝐹‘𝑚) ⊆ 𝐴)))
28	27	imp31 417	. . . 4 ⊢ ((((𝑁 ∈ ω ∧ 𝜑 ∧ 𝜓) ∧ 𝑚 ∈ 𝑁) ∧ ∃𝑖 ∈ ω 𝑚 = suc 𝑖) → (𝐹‘𝑚) ⊆ 𝐴)
29		simpr 484	. . . . . 6 ⊢ (((𝑁 ∈ ω ∧ 𝜑 ∧ 𝜓) ∧ 𝑚 ∈ 𝑁) → 𝑚 ∈ 𝑁)
30		simpl1 1189	. . . . . 6 ⊢ (((𝑁 ∈ ω ∧ 𝜑 ∧ 𝜓) ∧ 𝑚 ∈ 𝑁) → 𝑁 ∈ ω)
31		elnn 7698	. . . . . 6 ⊢ ((𝑚 ∈ 𝑁 ∧ 𝑁 ∈ ω) → 𝑚 ∈ ω)
32	29, 30, 31	syl2anc 583	. . . . 5 ⊢ (((𝑁 ∈ ω ∧ 𝜑 ∧ 𝜓) ∧ 𝑚 ∈ 𝑁) → 𝑚 ∈ ω)
33		nn0suc 7716	. . . . 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 955	. . 3 ⊢ (((𝑁 ∈ ω ∧ 𝜑 ∧ 𝜓) ∧ 𝑚 ∈ 𝑁) → (𝐹‘𝑚) ⊆ 𝐴)
36	35	ralrimiva 3107	. 2 ⊢ ((𝑁 ∈ ω ∧ 𝜑 ∧ 𝜓) → ∀𝑚 ∈ 𝑁 (𝐹‘𝑚) ⊆ 𝐴)
37		fveq2 6756	. . . 4 ⊢ (𝑚 = 𝑛 → (𝐹‘𝑚) = (𝐹‘𝑛))
38	37	sseq1d 3948	. . 3 ⊢ (𝑚 = 𝑛 → ((𝐹‘𝑚) ⊆ 𝐴 ↔ (𝐹‘𝑛) ⊆ 𝐴))
39	38	cbvralvw 3372	. 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 395   ∨ wo 843   ∧ w3a 1085   = wceq 1539   ∈ wcel 2108  ∀wral 3063  ∃wrex 3064   ⊆ wss 3883  ∅c0 4253  ∪ ciun 4921  suc csuc 6253  ‘cfv 6418  ωcom 7687   predc-bnj14 32567

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-10 2139  ax-11 2156  ax-12 2173  ax-ext 2709  ax-sep 5218  ax-nul 5225  ax-pr 5347  ax-un 7566

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1784  df-nf 1788  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-nfc 2888  df-ne 2943  df-ral 3068  df-rex 3069  df-rab 3072  df-v 3424  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-pss 3902  df-nul 4254  df-if 4457  df-pw 4532  df-sn 4559  df-pr 4561  df-tp 4563  df-op 4565  df-uni 4837  df-iun 4923  df-br 5071  df-opab 5133  df-tr 5188  df-eprel 5486  df-po 5494  df-so 5495  df-fr 5535  df-we 5537  df-ord 6254  df-on 6255  df-lim 6256  df-suc 6257  df-iota 6376  df-fv 6426  df-om 7688  df-bnj14 32568

This theorem is referenced by:  bnj518  32766