Theorem bnj1110 35612

Description: Technical lemma for bnj69 35640. 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 35414	. . . . . . . 8 ⊢ ∃𝑗((𝑖 ≠ ∅ ∧ 𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷) → (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗))
3		bnj219 35364	. . . . . . . . . . 11 ⊢ (𝑖 = suc 𝑗 → 𝑗 E 𝑖)
4	3	adantl 487	. . . . . . . . . 10 ⊢ ((𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗) → 𝑗 E 𝑖)
5	4	ancli 558	. . . . . . . . 9 ⊢ ((𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗) → ((𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗) ∧ 𝑗 E 𝑖))
6		df-3an 1105	. . . . . . . . 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 35411	. . . . . . 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 35433	. . . . . . . . . . 11 ⊢ (𝜒 → 𝑛 ∈ 𝐷)
11	10	3ad2ant3 1153	. . . . . . . . . 10 ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒) → 𝑛 ∈ 𝐷)
12		bnj1110.19	. . . . . . . . . . 11 ⊢ (𝜑0 ↔ (𝑖 ∈ 𝑛 ∧ 𝜎 ∧ 𝑓 ∈ 𝐾 ∧ 𝑖 ∈ dom 𝑓))
13	12	bnj1232 35433	. . . . . . . . . 10 ⊢ (𝜑0 → 𝑖 ∈ 𝑛)
14	11, 13	anim12ci 626	. . . . . . . . 9 ⊢ (((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0) → (𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷))
15	14	anim2i 629	. . . . . . . 8 ⊢ ((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → (𝑖 ≠ ∅ ∧ (𝑖 ∈ 𝑛 ∧ 𝑛 ∈ 𝐷)))
16		3anass 1111	. . . . . . . 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 35415	. . . . . 6 ⊢ ∃𝑗((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖))
19		3simpb 1167	. . . . . . . . 9 ⊢ ((𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖) → (𝑗 ∈ 𝑛 ∧ 𝑗 E 𝑖))
20	12	bnj1235 35434	. . . . . . . . . . 11 ⊢ (𝜑0 → 𝜎)
21	20	ad2antll 742	. . . . . . . . . 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 35	. . . . . . . 8 ⊢ ((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → ((𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖) → 𝜂′))
25	24	a2i 15	. . . . . . 7 ⊢ (((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖)) → ((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → 𝜂′))
26		pm3.43 479	. . . . . . 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 700	. . . . . 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 35354	. . . . 5 ⊢ ∃𝑗((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → ((𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖) ∧ 𝜂′))
29	12	bnj1247 35438	. . . . . . 7 ⊢ (𝜑0 → 𝑓 ∈ 𝐾)
30	29	ad2antll 742	. . . . . 6 ⊢ ((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → 𝑓 ∈ 𝐾)
31		pm3.43i 478	. . . . . 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 35354	. . . 4 ⊢ ∃𝑗((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → (𝑓 ∈ 𝐾 ∧ ((𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖) ∧ 𝜂′)))
34		fndm 6642	. . . . . . . . 9 ⊢ (𝑓 Fn 𝑛 → dom 𝑓 = 𝑛)
35	9, 34	bnj770 35394	. . . . . . . 8 ⊢ (𝜒 → dom 𝑓 = 𝑛)
36	35	3ad2ant3 1153	. . . . . . 7 ⊢ ((𝜃 ∧ 𝜏 ∧ 𝜒) → dom 𝑓 = 𝑛)
37	36	ad2antrl 741	. . . . . 6 ⊢ ((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → dom 𝑓 = 𝑛)
38	37	eleq2d 2846	. . . . 5 ⊢ ((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → (𝑗 ∈ dom 𝑓 ↔ 𝑗 ∈ 𝑛))
39		pm3.43i 478	. . . . 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 35354	. . 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 35340	. . . . . 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 35333	. . . . . 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 1880	. . 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 1154	. . . 4 ⊢ ((𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖) → 𝑗 ∈ 𝑛)
49	48	bnj706 35385	. . 3 ⊢ (((𝑗 ∈ dom 𝑓 ↔ 𝑗 ∈ 𝑛) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖) ∧ 𝑓 ∈ 𝐾 ∧ 𝜂′) → 𝑗 ∈ 𝑛)
50		simp2 1155	. . . 4 ⊢ ((𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖) → 𝑖 = suc 𝑗)
51	50	bnj706 35385	. . 3 ⊢ (((𝑗 ∈ dom 𝑓 ↔ 𝑗 ∈ 𝑛) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖) ∧ 𝑓 ∈ 𝐾 ∧ 𝜂′) → 𝑖 = suc 𝑗)
52		bnj258 35339	. . . . 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 503	. . . 4 ⊢ (((𝑗 ∈ dom 𝑓 ↔ 𝑗 ∈ 𝑛) ∧ (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ 𝑗 E 𝑖) ∧ 𝑓 ∈ 𝐾 ∧ 𝜂′) → 𝑓 ∈ 𝐾)
54		bnj642 35379	. . . . 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 35381	. . . . 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 712	. . 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 1146	. 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 35411	1 ⊢ ∃𝑗((𝑖 ≠ ∅ ∧ ((𝜃 ∧ 𝜏 ∧ 𝜒) ∧ 𝜑0)) → (𝑗 ∈ 𝑛 ∧ 𝑖 = suc 𝑗 ∧ (𝑓‘𝑗) ⊆ 𝐵))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1569  ∃wex 1811   ∈ wcel 2144   ≠ wne 2955   ⊆ wss 3895   ∖ cdif 4045  ∅c0 4278  {csn 4583   class class class wbr 5102   E cep 5549  dom cdm 5650  suc csuc 6364   Fn wfn 6533  ‘cfv 6538  ωcom 7877   ∧ w-bnj17 35317

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-tr 5212  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-fn 6541  df-om 7878  df-bnj17 35318

This theorem is used by:  bnj1118  35614