Theorem bnj1110 32249

Description: Technical lemma for bnj69 32277. 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 32050	. . . . . . . 8 ⊢ ∃𝑗((𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷) → (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗))
3		bnj219 31998	. . . . . . . . . . 11 ⊢ (𝑖 = suc 𝑗 → 𝑗 E 𝑖)
4	3	adantl 484	. . . . . . . . . 10 ⊢ ((𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗) → 𝑗 E 𝑖)
5	4	ancli 551	. . . . . . . . 9 ⊢ ((𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗) → ((𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗) ∧ 𝑗 E 𝑖))
6		df-3an 1085	. . . . . . . . 9 ⊢ ((𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖) ↔ ((𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗) ∧ 𝑗 E 𝑖))
7	5, 6	sylibr 236	. . . . . . . 8 ⊢ ((𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗) → (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖))
8	2, 7	bnj1023 32047	. . . . . . 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 32070	. . . . . . . . . . 11 ⊢ (𝜒 → 𝑛 ∈ 𝐷)
11	10	3ad2ant3 1131	. . . . . . . . . 10 ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒) → 𝑛 ∈ 𝐷)
12		bnj1110.19	. . . . . . . . . . 11 ⊢ (𝜑0 ↔ (𝑖 ∈ 𝑛 ∧ 𝜎 ∧ 𝑓 ∈ 𝐾 ∧ 𝑖 ∈ dom 𝑓))
13	12	bnj1232 32070	. . . . . . . . . 10 ⊢ (𝜑0 → 𝑖 ∈ 𝑛)
14	11, 13	anim12ci 615	. . . . . . . . 9 ⊢ (((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0) → (𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷))
15	14	anim2i 618	. . . . . . . 8 ⊢ ((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → (𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷)))
16		3anass 1091	. . . . . . . 8 ⊢ ((𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷) ↔ (𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷)))
17	15, 16	sylibr 236	. . . . . . 7 ⊢ ((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → (𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷))
18	8, 17	bnj1101 32051	. . . . . 6 ⊢ ∃𝑗((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖))
19		3simpb 1145	. . . . . . . . 9 ⊢ ((𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖) → (𝑗 ∈ 𝑛 ∧ 𝑗 E 𝑖))
20	12	bnj1235 32071	. . . . . . . . . . 11 ⊢ (𝜑0 → 𝜎)
21	20	ad2antll 727	. . . . . . . . . 10 ⊢ ((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → 𝜎)
22		bnj1110.18	. . . . . . . . . 10 ⊢ (𝜎 ↔ ((𝑗 ∈ 𝑛 ∧ 𝑗 E 𝑖) → 𝜂′))
23	21, 22	sylib 220	. . . . . . . . 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		pm3.43 476	. . . . . . 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 685	. . . . . 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 31988	. . . . 5 ⊢ ∃𝑗((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → ((𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖) ∧ 𝜂′))
29	12	bnj1247 32075	. . . . . . 7 ⊢ (𝜑0 → 𝑓 ∈ 𝐾)
30	29	ad2antll 727	. . . . . 6 ⊢ ((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → 𝑓 ∈ 𝐾)
31		pm3.43i 475	. . . . . 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 31988	. . . 4 ⊢ ∃𝑗((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → (𝑓 ∈ 𝐾 ∧ ((𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖) ∧ 𝜂′)))
34		fndm 6450	. . . . . . . . 9 ⊢ (𝑓 Fn 𝑛 → dom 𝑓 = 𝑛)
35	9, 34	bnj770 32029	. . . . . . . 8 ⊢ (𝜒 → dom 𝑓 = 𝑛)
36	35	3ad2ant3 1131	. . . . . . 7 ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒) → dom 𝑓 = 𝑛)
37	36	ad2antrl 726	. . . . . 6 ⊢ ((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → dom 𝑓 = 𝑛)
38	37	eleq2d 2898	. . . . 5 ⊢ ((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → (𝑗 ∈ dom 𝑓 ↔ 𝑗 ∈ 𝑛))
39		pm3.43i 475	. . . . 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 31988	. . 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 31974	. . . . . 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 31967	. . . . . 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 279	. . . . 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 338	. . . 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 1844	. . 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 233	. 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 1132	. . . 4 ⊢ ((𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖) → 𝑗 ∈ 𝑛)
49	48	bnj706 32020	. . 3 ⊢ (((𝑗 ∈ dom 𝑓 ↔ 𝑗 ∈ 𝑛) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖) ∧ 𝑓 ∈ 𝐾 ∧ 𝜂′) → 𝑗 ∈ 𝑛)
50		simp2 1133	. . . 4 ⊢ ((𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖) → 𝑖 = suc 𝑗)
51	50	bnj706 32020	. . 3 ⊢ (((𝑗 ∈ dom 𝑓 ↔ 𝑗 ∈ 𝑛) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖) ∧ 𝑓 ∈ 𝐾 ∧ 𝜂′) → 𝑖 = suc 𝑗)
52		bnj258 31973	. . . . 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 499	. . . 4 ⊢ (((𝑗 ∈ dom 𝑓 ↔ 𝑗 ∈ 𝑛) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖) ∧ 𝑓 ∈ 𝐾 ∧ 𝜂′) → 𝑓 ∈ 𝐾)
54		bnj642 32014	. . . . 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 259	. . . 4 ⊢ (((𝑗 ∈ dom 𝑓 ↔ 𝑗 ∈ 𝑛) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖) ∧ 𝑓 ∈ 𝐾 ∧ 𝜂′) → 𝑗 ∈ dom 𝑓)
56		bnj645 32016	. . . . 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 220	. . . 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 697	. . 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 1124	. 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 32047	1 ⊢ ∃𝑗((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ (𝑓‘𝑗) ⊆ 𝐵))

Syntax hints:   → wi 4   ↔ wb 208   ∧ wa 398   ∧ w3a 1083   = wceq 1533  ∃wex 1776   ∈ wcel 2110   ≠ wne 3016   ∖ cdif 3933   ⊆ wss 3936  ∅c0 4291  {csn 4561   class class class wbr 5059   E cep 5459  dom cdm 5550  suc csuc 6188   Fn wfn 6345  ‘cfv 6350  ωcom 7574   ∧ w-bnj17 31951

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793  ax-sep 5196  ax-nul 5203  ax-pr 5322  ax-un 7455

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-mo 2618  df-eu 2650  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3497  df-sbc 3773  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-pss 3954  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4562  df-pr 4564  df-tp 4566  df-op 4568  df-uni 4833  df-br 5060  df-opab 5122  df-tr 5166  df-eprel 5460  df-po 5469  df-so 5470  df-fr 5509  df-we 5511  df-ord 6189  df-on 6190  df-lim 6191  df-suc 6192  df-fn 6353  df-om 7575  df-bnj17 31952

This theorem is referenced by:  bnj1118  32251