Theorem bnj1110 32364

Description: Technical lemma for bnj69 32392. 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.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj1110.3	⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
bnj1110.7	⊢ 𝐷 = (ω ∖ {∅})
bnj1110.18	⊢ (𝜎 ↔ ((𝑗 ∈ 𝑛 ∧ 𝑗 E 𝑖) → 𝜂′))
bnj1110.19	⊢ (𝜑0 ↔ (𝑖 ∈ 𝑛 ∧ 𝜎 ∧ 𝑓 ∈ 𝐾 ∧ 𝑖 ∈ dom 𝑓))
bnj1110.26	⊢ (𝜂′ ↔ ((𝑓 ∈ 𝐾 ∧ 𝑗 ∈ dom 𝑓) → (𝑓‘𝑗) ⊆ 𝐵))

Assertion

Ref	Expression
bnj1110	⊢ ∃𝑗((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ (𝑓‘𝑗) ⊆ 𝐵))

Distinct variable groups:   𝐷,𝑗   𝑖,𝑗   𝑗,𝑛

Allowed substitution hints:   𝜑(𝑓,𝑖,𝑗,𝑛)   𝜓(𝑓,𝑖,𝑗,𝑛)   𝜒(𝑓,𝑖,𝑗,𝑛)   𝜃(𝑓,𝑖,𝑗,𝑛)   𝜏(𝑓,𝑖,𝑗,𝑛)   𝜎(𝑓,𝑖,𝑗,𝑛)   𝐵(𝑓,𝑖,𝑗,𝑛)   𝐷(𝑓,𝑖,𝑛)   𝐾(𝑓,𝑖,𝑗,𝑛)   𝜂′(𝑓,𝑖,𝑗,𝑛)   𝜑₀(𝑓,𝑖,𝑗,𝑛)

Proof of Theorem bnj1110

Step	Hyp	Ref	Expression
1		bnj1110.7	. . . . . . . . 9 ⊢ 𝐷 = (ω ∖ {∅})
2	1	bnj1098 32165	. . . . . . . 8 ⊢ ∃𝑗((𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷) → (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗))
3		bnj219 32113	. . . . . . . . . . 11 ⊢ (𝑖 = suc 𝑗 → 𝑗 E 𝑖)
4	3	adantl 485	. . . . . . . . . 10 ⊢ ((𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗) → 𝑗 E 𝑖)
5	4	ancli 552	. . . . . . . . 9 ⊢ ((𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗) → ((𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗) ∧ 𝑗 E 𝑖))
6		df-3an 1086	. . . . . . . . 9 ⊢ ((𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖) ↔ ((𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗) ∧ 𝑗 E 𝑖))
7	5, 6	sylibr 237	. . . . . . . 8 ⊢ ((𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗) → (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖))
8	2, 7	bnj1023 32162	. . . . . . 7 ⊢ ∃𝑗((𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷) → (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖))
9		bnj1110.3	. . . . . . . . . . . 12 ⊢ (𝜒 ↔ (𝑛 ∈ 𝐷 ∧ 𝑓 Fn 𝑛 ∧ 𝜑 ∧ 𝜓))
10	9	bnj1232 32185	. . . . . . . . . . 11 ⊢ (𝜒 → 𝑛 ∈ 𝐷)
11	10	3ad2ant3 1132	. . . . . . . . . 10 ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒) → 𝑛 ∈ 𝐷)
12		bnj1110.19	. . . . . . . . . . 11 ⊢ (𝜑0 ↔ (𝑖 ∈ 𝑛 ∧ 𝜎 ∧ 𝑓 ∈ 𝐾 ∧ 𝑖 ∈ dom 𝑓))
13	12	bnj1232 32185	. . . . . . . . . 10 ⊢ (𝜑0 → 𝑖 ∈ 𝑛)
14	11, 13	anim12ci 616	. . . . . . . . 9 ⊢ (((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0) → (𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷))
15	14	anim2i 619	. . . . . . . 8 ⊢ ((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → (𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷)))
16		3anass 1092	. . . . . . . 8 ⊢ ((𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷) ↔ (𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷)))
17	15, 16	sylibr 237	. . . . . . 7 ⊢ ((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → (𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷))
18	8, 17	bnj1101 32166	. . . . . 6 ⊢ ∃𝑗((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖))
19		3simpb 1146	. . . . . . . . 9 ⊢ ((𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖) → (𝑗 ∈ 𝑛 ∧ 𝑗 E 𝑖))
20	12	bnj1235 32186	. . . . . . . . . . 11 ⊢ (𝜑0 → 𝜎)
21	20	ad2antll 728	. . . . . . . . . 10 ⊢ ((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → 𝜎)
22		bnj1110.18	. . . . . . . . . 10 ⊢ (𝜎 ↔ ((𝑗 ∈ 𝑛 ∧ 𝑗 E 𝑖) → 𝜂′))
23	21, 22	sylib 221	. . . . . . . . 9 ⊢ ((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → ((𝑗 ∈ 𝑛 ∧ 𝑗 E 𝑖) → 𝜂′))
24	19, 23	syl5 34	. . . . . . . 8 ⊢ ((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → ((𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖) → 𝜂′))
25	24	a2i 14	. . . . . . 7 ⊢ (((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖)) → ((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → 𝜂′))
26		pm3.43 477	. . . . . . 7 ⊢ ((((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖)) ∧ ((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → 𝜂′)) → ((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → ((𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖) ∧ 𝜂′)))
27	25, 26	mpdan 686	. . . . . 6 ⊢ (((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖)) → ((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → ((𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖) ∧ 𝜂′)))
28	18, 27	bnj101 32103	. . . . 5 ⊢ ∃𝑗((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → ((𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖) ∧ 𝜂′))
29	12	bnj1247 32190	. . . . . . 7 ⊢ (𝜑0 → 𝑓 ∈ 𝐾)
30	29	ad2antll 728	. . . . . 6 ⊢ ((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → 𝑓 ∈ 𝐾)
31		pm3.43i 476	. . . . . 6 ⊢ (((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → 𝑓 ∈ 𝐾) → (((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → ((𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖) ∧ 𝜂′)) → ((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → (𝑓 ∈ 𝐾 ∧ ((𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖) ∧ 𝜂′)))))
32	30, 31	ax-mp 5	. . . . 5 ⊢ (((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → ((𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖) ∧ 𝜂′)) → ((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → (𝑓 ∈ 𝐾 ∧ ((𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖) ∧ 𝜂′))))
33	28, 32	bnj101 32103	. . . 4 ⊢ ∃𝑗((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → (𝑓 ∈ 𝐾 ∧ ((𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖) ∧ 𝜂′)))
34		fndm 6425	. . . . . . . . 9 ⊢ (𝑓 Fn 𝑛 → dom 𝑓 = 𝑛)
35	9, 34	bnj770 32144	. . . . . . . 8 ⊢ (𝜒 → dom 𝑓 = 𝑛)
36	35	3ad2ant3 1132	. . . . . . 7 ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒) → dom 𝑓 = 𝑛)
37	36	ad2antrl 727	. . . . . 6 ⊢ ((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → dom 𝑓 = 𝑛)
38	37	eleq2d 2875	. . . . 5 ⊢ ((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → (𝑗 ∈ dom 𝑓 ↔ 𝑗 ∈ 𝑛))
39		pm3.43i 476	. . . . 5 ⊢ (((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → (𝑗 ∈ dom 𝑓 ↔ 𝑗 ∈ 𝑛)) → (((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → (𝑓 ∈ 𝐾 ∧ ((𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖) ∧ 𝜂′))) → ((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → ((𝑗 ∈ dom 𝑓 ↔ 𝑗 ∈ 𝑛) ∧ (𝑓 ∈ 𝐾 ∧ ((𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖) ∧ 𝜂′))))))
40	38, 39	ax-mp 5	. . . 4 ⊢ (((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → (𝑓 ∈ 𝐾 ∧ ((𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖) ∧ 𝜂′))) → ((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → ((𝑗 ∈ dom 𝑓 ↔ 𝑗 ∈ 𝑛) ∧ (𝑓 ∈ 𝐾 ∧ ((𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖) ∧ 𝜂′)))))
41	33, 40	bnj101 32103	. . 3 ⊢ ∃𝑗((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → ((𝑗 ∈ dom 𝑓 ↔ 𝑗 ∈ 𝑛) ∧ (𝑓 ∈ 𝐾 ∧ ((𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖) ∧ 𝜂′))))
42		bnj268 32089	. . . . . 6 ⊢ (((𝑗 ∈ dom 𝑓 ↔ 𝑗 ∈ 𝑛) ∧ 𝑓 ∈ 𝐾 ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖) ∧ 𝜂′) ↔ ((𝑗 ∈ dom 𝑓 ↔ 𝑗 ∈ 𝑛) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖) ∧ 𝑓 ∈ 𝐾 ∧ 𝜂′))
43		bnj251 32082	. . . . . 6 ⊢ (((𝑗 ∈ dom 𝑓 ↔ 𝑗 ∈ 𝑛) ∧ 𝑓 ∈ 𝐾 ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖) ∧ 𝜂′) ↔ ((𝑗 ∈ dom 𝑓 ↔ 𝑗 ∈ 𝑛) ∧ (𝑓 ∈ 𝐾 ∧ ((𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖) ∧ 𝜂′))))
44	42, 43	bitr3i 280	. . . . 5 ⊢ (((𝑗 ∈ dom 𝑓 ↔ 𝑗 ∈ 𝑛) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖) ∧ 𝑓 ∈ 𝐾 ∧ 𝜂′) ↔ ((𝑗 ∈ dom 𝑓 ↔ 𝑗 ∈ 𝑛) ∧ (𝑓 ∈ 𝐾 ∧ ((𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖) ∧ 𝜂′))))
45	44	imbi2i 339	. . . 4 ⊢ (((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → ((𝑗 ∈ dom 𝑓 ↔ 𝑗 ∈ 𝑛) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖) ∧ 𝑓 ∈ 𝐾 ∧ 𝜂′)) ↔ ((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → ((𝑗 ∈ dom 𝑓 ↔ 𝑗 ∈ 𝑛) ∧ (𝑓 ∈ 𝐾 ∧ ((𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖) ∧ 𝜂′)))))
46	45	exbii 1849	. . 3 ⊢ (∃𝑗((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → ((𝑗 ∈ dom 𝑓 ↔ 𝑗 ∈ 𝑛) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖) ∧ 𝑓 ∈ 𝐾 ∧ 𝜂′)) ↔ ∃𝑗((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → ((𝑗 ∈ dom 𝑓 ↔ 𝑗 ∈ 𝑛) ∧ (𝑓 ∈ 𝐾 ∧ ((𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖) ∧ 𝜂′)))))
47	41, 46	mpbir 234	. 2 ⊢ ∃𝑗((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → ((𝑗 ∈ dom 𝑓 ↔ 𝑗 ∈ 𝑛) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖) ∧ 𝑓 ∈ 𝐾 ∧ 𝜂′))
48		simp1 1133	. . . 4 ⊢ ((𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖) → 𝑗 ∈ 𝑛)
49	48	bnj706 32135	. . 3 ⊢ (((𝑗 ∈ dom 𝑓 ↔ 𝑗 ∈ 𝑛) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖) ∧ 𝑓 ∈ 𝐾 ∧ 𝜂′) → 𝑗 ∈ 𝑛)
50		simp2 1134	. . . 4 ⊢ ((𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖) → 𝑖 = suc 𝑗)
51	50	bnj706 32135	. . 3 ⊢ (((𝑗 ∈ dom 𝑓 ↔ 𝑗 ∈ 𝑛) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖) ∧ 𝑓 ∈ 𝐾 ∧ 𝜂′) → 𝑖 = suc 𝑗)
52		bnj258 32088	. . . . 5 ⊢ (((𝑗 ∈ dom 𝑓 ↔ 𝑗 ∈ 𝑛) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖) ∧ 𝑓 ∈ 𝐾 ∧ 𝜂′) ↔ (((𝑗 ∈ dom 𝑓 ↔ 𝑗 ∈ 𝑛) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖) ∧ 𝜂′) ∧ 𝑓 ∈ 𝐾))
53	52	simprbi 500	. . . 4 ⊢ (((𝑗 ∈ dom 𝑓 ↔ 𝑗 ∈ 𝑛) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖) ∧ 𝑓 ∈ 𝐾 ∧ 𝜂′) → 𝑓 ∈ 𝐾)
54		bnj642 32129	. . . . 5 ⊢ (((𝑗 ∈ dom 𝑓 ↔ 𝑗 ∈ 𝑛) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖) ∧ 𝑓 ∈ 𝐾 ∧ 𝜂′) → (𝑗 ∈ dom 𝑓 ↔ 𝑗 ∈ 𝑛))
55	49, 54	mpbird 260	. . . 4 ⊢ (((𝑗 ∈ dom 𝑓 ↔ 𝑗 ∈ 𝑛) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖) ∧ 𝑓 ∈ 𝐾 ∧ 𝜂′) → 𝑗 ∈ dom 𝑓)
56		bnj645 32131	. . . . 5 ⊢ (((𝑗 ∈ dom 𝑓 ↔ 𝑗 ∈ 𝑛) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖) ∧ 𝑓 ∈ 𝐾 ∧ 𝜂′) → 𝜂′)
57		bnj1110.26	. . . . 5 ⊢ (𝜂′ ↔ ((𝑓 ∈ 𝐾 ∧ 𝑗 ∈ dom 𝑓) → (𝑓‘𝑗) ⊆ 𝐵))
58	56, 57	sylib 221	. . . 4 ⊢ (((𝑗 ∈ dom 𝑓 ↔ 𝑗 ∈ 𝑛) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖) ∧ 𝑓 ∈ 𝐾 ∧ 𝜂′) → ((𝑓 ∈ 𝐾 ∧ 𝑗 ∈ dom 𝑓) → (𝑓‘𝑗) ⊆ 𝐵))
59	53, 55, 58	mp2and 698	. . 3 ⊢ (((𝑗 ∈ dom 𝑓 ↔ 𝑗 ∈ 𝑛) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖) ∧ 𝑓 ∈ 𝐾 ∧ 𝜂′) → (𝑓‘𝑗) ⊆ 𝐵)
60	49, 51, 59	3jca 1125	. 2 ⊢ (((𝑗 ∈ dom 𝑓 ↔ 𝑗 ∈ 𝑛) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖) ∧ 𝑓 ∈ 𝐾 ∧ 𝜂′) → (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ (𝑓‘𝑗) ⊆ 𝐵))
61	47, 60	bnj1023 32162	1 ⊢ ∃𝑗((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ (𝑓‘𝑗) ⊆ 𝐵))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 399   ∧ w3a 1084   = wceq 1538  ∃wex 1781   ∈ wcel 2111   ≠ wne 2987   ∖ cdif 3878   ⊆ wss 3881  ∅c0 4243  {csn 4525   class class class wbr 5030   E cep 5429  dom cdm 5519  suc csuc 6161   Fn wfn 6319  ‘cfv 6324  ωcom 7560   ∧ w-bnj17 32066

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2770  ax-sep 5167  ax-nul 5174  ax-pr 5295  ax-un 7441

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2598  df-eu 2629  df-clab 2777  df-cleq 2791  df-clel 2870  df-nfc 2938  df-ne 2988  df-ral 3111  df-rex 3112  df-rab 3115  df-v 3443  df-sbc 3721  df-dif 3884  df-un 3886  df-in 3888  df-ss 3898  df-pss 3900  df-nul 4244  df-if 4426  df-pw 4499  df-sn 4526  df-pr 4528  df-tp 4530  df-op 4532  df-uni 4801  df-br 5031  df-opab 5093  df-tr 5137  df-eprel 5430  df-po 5438  df-so 5439  df-fr 5478  df-we 5480  df-ord 6162  df-on 6163  df-lim 6164  df-suc 6165  df-fn 6327  df-om 7561  df-bnj17 32067

This theorem is referenced by:  bnj1118  32366