Theorem bnj1110 32962

Description: Technical lemma for bnj69 32990. 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 32763	. . . . . . . 8 ⊢ ∃𝑗((𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷) → (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗))
3		bnj219 32712	. . . . . . . . . . 11 ⊢ (𝑖 = suc 𝑗 → 𝑗 E 𝑖)
4	3	adantl 482	. . . . . . . . . 10 ⊢ ((𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗) → 𝑗 E 𝑖)
5	4	ancli 549	. . . . . . . . 9 ⊢ ((𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗) → ((𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗) ∧ 𝑗 E 𝑖))
6		df-3an 1088	. . . . . . . . 9 ⊢ ((𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖) ↔ ((𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗) ∧ 𝑗 E 𝑖))
7	5, 6	sylibr 233	. . . . . . . 8 ⊢ ((𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗) → (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖))
8	2, 7	bnj1023 32760	. . . . . . 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 32783	. . . . . . . . . . 11 ⊢ (𝜒 → 𝑛 ∈ 𝐷)
11	10	3ad2ant3 1134	. . . . . . . . . 10 ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒) → 𝑛 ∈ 𝐷)
12		bnj1110.19	. . . . . . . . . . 11 ⊢ (𝜑0 ↔ (𝑖 ∈ 𝑛 ∧ 𝜎 ∧ 𝑓 ∈ 𝐾 ∧ 𝑖 ∈ dom 𝑓))
13	12	bnj1232 32783	. . . . . . . . . 10 ⊢ (𝜑0 → 𝑖 ∈ 𝑛)
14	11, 13	anim12ci 614	. . . . . . . . 9 ⊢ (((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0) → (𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷))
15	14	anim2i 617	. . . . . . . 8 ⊢ ((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → (𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷)))
16		3anass 1094	. . . . . . . 8 ⊢ ((𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷) ↔ (𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷)))
17	15, 16	sylibr 233	. . . . . . 7 ⊢ ((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → (𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷))
18	8, 17	bnj1101 32764	. . . . . 6 ⊢ ∃𝑗((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖))
19		3simpb 1148	. . . . . . . . 9 ⊢ ((𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖) → (𝑗 ∈ 𝑛 ∧ 𝑗 E 𝑖))
20	12	bnj1235 32784	. . . . . . . . . . 11 ⊢ (𝜑0 → 𝜎)
21	20	ad2antll 726	. . . . . . . . . 10 ⊢ ((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → 𝜎)
22		bnj1110.18	. . . . . . . . . 10 ⊢ (𝜎 ↔ ((𝑗 ∈ 𝑛 ∧ 𝑗 E 𝑖) → 𝜂′))
23	21, 22	sylib 217	. . . . . . . . 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		pm3.43 474	. . . . . . 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 684	. . . . . 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 32702	. . . . 5 ⊢ ∃𝑗((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → ((𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖) ∧ 𝜂′))
29	12	bnj1247 32788	. . . . . . 7 ⊢ (𝜑0 → 𝑓 ∈ 𝐾)
30	29	ad2antll 726	. . . . . 6 ⊢ ((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → 𝑓 ∈ 𝐾)
31		pm3.43i 473	. . . . . 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 32702	. . . 4 ⊢ ∃𝑗((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → (𝑓 ∈ 𝐾 ∧ ((𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖) ∧ 𝜂′)))
34		fndm 6536	. . . . . . . . 9 ⊢ (𝑓 Fn 𝑛 → dom 𝑓 = 𝑛)
35	9, 34	bnj770 32743	. . . . . . . 8 ⊢ (𝜒 → dom 𝑓 = 𝑛)
36	35	3ad2ant3 1134	. . . . . . 7 ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒) → dom 𝑓 = 𝑛)
37	36	ad2antrl 725	. . . . . 6 ⊢ ((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → dom 𝑓 = 𝑛)
38	37	eleq2d 2824	. . . . 5 ⊢ ((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → (𝑗 ∈ dom 𝑓 ↔ 𝑗 ∈ 𝑛))
39		pm3.43i 473	. . . . 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 32702	. . 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 32688	. . . . . 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 32681	. . . . . 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 276	. . . . 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 336	. . . 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 1850	. . 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 230	. 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 1135	. . . 4 ⊢ ((𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖) → 𝑗 ∈ 𝑛)
49	48	bnj706 32734	. . 3 ⊢ (((𝑗 ∈ dom 𝑓 ↔ 𝑗 ∈ 𝑛) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖) ∧ 𝑓 ∈ 𝐾 ∧ 𝜂′) → 𝑗 ∈ 𝑛)
50		simp2 1136	. . . 4 ⊢ ((𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖) → 𝑖 = suc 𝑗)
51	50	bnj706 32734	. . 3 ⊢ (((𝑗 ∈ dom 𝑓 ↔ 𝑗 ∈ 𝑛) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖) ∧ 𝑓 ∈ 𝐾 ∧ 𝜂′) → 𝑖 = suc 𝑗)
52		bnj258 32687	. . . . 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 497	. . . 4 ⊢ (((𝑗 ∈ dom 𝑓 ↔ 𝑗 ∈ 𝑛) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖) ∧ 𝑓 ∈ 𝐾 ∧ 𝜂′) → 𝑓 ∈ 𝐾)
54		bnj642 32728	. . . . 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 256	. . . 4 ⊢ (((𝑗 ∈ dom 𝑓 ↔ 𝑗 ∈ 𝑛) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖) ∧ 𝑓 ∈ 𝐾 ∧ 𝜂′) → 𝑗 ∈ dom 𝑓)
56		bnj645 32730	. . . . 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 217	. . . 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 696	. . 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 1127	. 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 32760	1 ⊢ ∃𝑗((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ (𝑓‘𝑗) ⊆ 𝐵))

Syntax hints:   → wi 4   ↔ wb 205   ∧ wa 396   ∧ w3a 1086   = wceq 1539  ∃wex 1782   ∈ wcel 2106   ≠ wne 2943   ∖ cdif 3884   ⊆ wss 3887  ∅c0 4256  {csn 4561   class class class wbr 5074   E cep 5494  dom cdm 5589  suc csuc 6268   Fn wfn 6428  ‘cfv 6433  ωcom 7712   ∧ w-bnj17 32665

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-11 2154  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-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  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-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-fn 6436  df-om 7713  df-bnj17 32666

This theorem is referenced by:  bnj1118  32964