Theorem bnj517 34511

Description: Technical lemma for bnj518 34512. 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 6892	. . . . . 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 2790	. . . . 5 ⊢ ((((𝑁 ∈ ω ∧ 𝜑 ∧ 𝜓) ∧ 𝑚 ∈ 𝑁) ∧ 𝑚 = ∅) → (𝐹‘𝑚) = pred(𝑋, 𝐴, 𝑅))
6		bnj213 34508	. . . . 5 ⊢ pred(𝑋, 𝐴, 𝑅) ⊆ 𝐴
7	5, 6	eqsstrdi 4033	. . . 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 3112	. . . . . . . . . 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 2817	. . . . . . . . . . . . . 14 ⊢ (𝑚 = suc 𝑖 → (𝑚 ∈ 𝑁 ↔ suc 𝑖 ∈ 𝑁))
11	10	biimpd 228	. . . . . . . . . . . . 13 ⊢ (𝑚 = suc 𝑖 → (𝑚 ∈ 𝑁 → suc 𝑖 ∈ 𝑁))
12		fveqeq2 6901	. . . . . . . . . . . . . 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 34508	. . . . . . . . . . . . . . . . 17 ⊢ pred(𝑦, 𝐴, 𝑅) ⊆ 𝐴
14	13	rgenw 3061	. . . . . . . . . . . . . . . 16 ⊢ ∀𝑦 ∈ (𝐹‘𝑖) pred(𝑦, 𝐴, 𝑅) ⊆ 𝐴
15		iunss 5043	. . . . . . . . . . . . . . . 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 4004	. . . . . . . . . . . . . . 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 258	. . . . . . . . . . . . . 14 ⊢ ((𝐹‘𝑚) = ∪ 𝑦 ∈ (𝐹‘𝑖) pred(𝑦, 𝐴, 𝑅) → (𝐹‘𝑚) ⊆ 𝐴)
19	12, 18	syl6bir 254	. . . . . . . . . . . . 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 3147	. . . . . . . . . 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 7876	. . . . . 6 ⊢ ((𝑚 ∈ 𝑁 ∧ 𝑁 ∈ ω) → 𝑚 ∈ ω)
32	29, 30, 31	syl2anc 583	. . . . 5 ⊢ (((𝑁 ∈ ω ∧ 𝜑 ∧ 𝜓) ∧ 𝑚 ∈ 𝑁) → 𝑚 ∈ ω)
33		nn0suc 7896	. . . . 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 957	. . 3 ⊢ (((𝑁 ∈ ω ∧ 𝜑 ∧ 𝜓) ∧ 𝑚 ∈ 𝑁) → (𝐹‘𝑚) ⊆ 𝐴)
36	35	ralrimiva 3142	. 2 ⊢ ((𝑁 ∈ ω ∧ 𝜑 ∧ 𝜓) → ∀𝑚 ∈ 𝑁 (𝐹‘𝑚) ⊆ 𝐴)
37		fveq2 6892	. . . 4 ⊢ (𝑚 = 𝑛 → (𝐹‘𝑚) = (𝐹‘𝑛))
38	37	sseq1d 4010	. . 3 ⊢ (𝑚 = 𝑛 → ((𝐹‘𝑚) ⊆ 𝐴 ↔ (𝐹‘𝑛) ⊆ 𝐴))
39	38	cbvralvw 3230	. 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 846   ∧ w3a 1085   = wceq 1534   ∈ wcel 2099  ∀wral 3057  ∃wrex 3066   ⊆ wss 3945  ∅c0 4319  ∪ ciun 4992  suc csuc 6366  ‘cfv 6543  ωcom 7865   predc-bnj14 34314

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2699  ax-sep 5294  ax-nul 5301  ax-pr 5424  ax-un 7735

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-nfc 2881  df-ne 2937  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-pss 3964  df-nul 4320  df-if 4526  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4905  df-iun 4994  df-br 5144  df-opab 5206  df-tr 5261  df-eprel 5577  df-po 5585  df-so 5586  df-fr 5628  df-we 5630  df-ord 6367  df-on 6368  df-lim 6369  df-suc 6370  df-iota 6495  df-fv 6551  df-om 7866  df-bnj14 34315

This theorem is referenced by:  bnj518  34512