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Theorem coftr 10287

Description: If there is a cofinal map from 𝐵 to 𝐴 and another from 𝐶 to 𝐴, then there is also a cofinal map from 𝐶 to 𝐵. Proposition 11.9 of [TakeutiZaring] p. 102. A limited form of transitivity for the "cof" relation. This is really a lemma for cfcof 10288. (Contributed by Mario Carneiro, 16-Mar-2013.)

**Hypothesis**
Ref	Expression
coftr.1	⊢ 𝐻 = (𝑡 ∈ 𝐶 ↦ ∩ {𝑛 ∈ 𝐵 ∣ (𝑔‘𝑡) ⊆ (𝑓‘𝑛)})

**Assertion**
Ref	Expression
coftr	⊢ (∃𝑓(𝑓:𝐵⟶𝐴 ∧ Smo 𝑓 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ (𝑓‘𝑦)) → (∃𝑔(𝑔:𝐶⟶𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐶 𝑧 ⊆ (𝑔‘𝑤)) → ∃ℎ(ℎ:𝐶⟶𝐵 ∧ ∀𝑠 ∈ 𝐵 ∃𝑤 ∈ 𝐶 𝑠 ⊆ (ℎ‘𝑤))))

Distinct variable groups: 𝐴,𝑓,𝑔,𝑠,𝑤,𝑥 𝑧,𝐴,𝑓,𝑔,𝑠,𝑤 𝐵,𝑓,𝑔,ℎ,𝑠,𝑤 𝐵,𝑛,𝑡,𝑓,𝑔,𝑤 𝑥,𝐵,𝑦,𝑓,𝑔,𝑠,𝑤 𝐶,𝑓,𝑔,ℎ,𝑠,𝑤 𝑡,𝐶 𝑧,𝐶 ℎ,𝐻,𝑠,𝑤 𝑦,𝑛 Allowed substitution hints: 𝐴(𝑦,𝑡,ℎ,𝑛) 𝐵(𝑧) 𝐶(𝑥,𝑦,𝑛) 𝐻(𝑥,𝑦,𝑧,𝑡,𝑓,𝑔,𝑛)

**Proof of Theorem coftr**
Step	Hyp	Ref	Expression
1		fdm 6715	. . . . . . . 8 ⊢ (𝑔:𝐶⟶𝐴 → dom 𝑔 = 𝐶)
2		vex 3463	. . . . . . . . 9 ⊢ 𝑔 ∈ V
3	2	dmex 7905	. . . . . . . 8 ⊢ dom 𝑔 ∈ V
4	1, 3	eqeltrrdi 2843	. . . . . . 7 ⊢ (𝑔:𝐶⟶𝐴 → 𝐶 ∈ V)
5		coftr.1	. . . . . . . . 9 ⊢ 𝐻 = (𝑡 ∈ 𝐶 ↦ ∩ {𝑛 ∈ 𝐵 ∣ (𝑔‘𝑡) ⊆ (𝑓‘𝑛)})
6		fveq2 6876	. . . . . . . . . . . . 13 ⊢ (𝑡 = 𝑤 → (𝑔‘𝑡) = (𝑔‘𝑤))
7	6	sseq1d 3990	. . . . . . . . . . . 12 ⊢ (𝑡 = 𝑤 → ((𝑔‘𝑡) ⊆ (𝑓‘𝑛) ↔ (𝑔‘𝑤) ⊆ (𝑓‘𝑛)))
8	7	rabbidv 3423	. . . . . . . . . . 11 ⊢ (𝑡 = 𝑤 → {𝑛 ∈ 𝐵 ∣ (𝑔‘𝑡) ⊆ (𝑓‘𝑛)} = {𝑛 ∈ 𝐵 ∣ (𝑔‘𝑤) ⊆ (𝑓‘𝑛)})
9	8	inteqd 4927	. . . . . . . . . 10 ⊢ (𝑡 = 𝑤 → ∩ {𝑛 ∈ 𝐵 ∣ (𝑔‘𝑡) ⊆ (𝑓‘𝑛)} = ∩ {𝑛 ∈ 𝐵 ∣ (𝑔‘𝑤) ⊆ (𝑓‘𝑛)})
10	9	cbvmptv 5225	. . . . . . . . 9 ⊢ (𝑡 ∈ 𝐶 ↦ ∩ {𝑛 ∈ 𝐵 ∣ (𝑔‘𝑡) ⊆ (𝑓‘𝑛)}) = (𝑤 ∈ 𝐶 ↦ ∩ {𝑛 ∈ 𝐵 ∣ (𝑔‘𝑤) ⊆ (𝑓‘𝑛)})
11	5, 10	eqtri 2758	. . . . . . . 8 ⊢ 𝐻 = (𝑤 ∈ 𝐶 ↦ ∩ {𝑛 ∈ 𝐵 ∣ (𝑔‘𝑤) ⊆ (𝑓‘𝑛)})
12		mptexg 7213	. . . . . . . 8 ⊢ (𝐶 ∈ V → (𝑤 ∈ 𝐶 ↦ ∩ {𝑛 ∈ 𝐵 ∣ (𝑔‘𝑤) ⊆ (𝑓‘𝑛)}) ∈ V)
13	11, 12	eqeltrid 2838	. . . . . . 7 ⊢ (𝐶 ∈ V → 𝐻 ∈ V)
14	4, 13	syl 17	. . . . . 6 ⊢ (𝑔:𝐶⟶𝐴 → 𝐻 ∈ V)
15	14	ad2antrl 728	. . . . 5 ⊢ (((𝑓:𝐵⟶𝐴 ∧ Smo 𝑓 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ (𝑓‘𝑦)) ∧ (𝑔:𝐶⟶𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐶 𝑧 ⊆ (𝑔‘𝑤))) → 𝐻 ∈ V)
16		ffn 6706	. . . . . . . . 9 ⊢ (𝑓:𝐵⟶𝐴 → 𝑓 Fn 𝐵)
17		smodm2 8369	. . . . . . . . 9 ⊢ ((𝑓 Fn 𝐵 ∧ Smo 𝑓) → Ord 𝐵)
18	16, 17	sylan 580	. . . . . . . 8 ⊢ ((𝑓:𝐵⟶𝐴 ∧ Smo 𝑓) → Ord 𝐵)
19	18	3adant3 1132	. . . . . . 7 ⊢ ((𝑓:𝐵⟶𝐴 ∧ Smo 𝑓 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ (𝑓‘𝑦)) → Ord 𝐵)
20	19	adantr 480	. . . . . 6 ⊢ (((𝑓:𝐵⟶𝐴 ∧ Smo 𝑓 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ (𝑓‘𝑦)) ∧ (𝑔:𝐶⟶𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐶 𝑧 ⊆ (𝑔‘𝑤))) → Ord 𝐵)
21		simpl3 1194	. . . . . 6 ⊢ (((𝑓:𝐵⟶𝐴 ∧ Smo 𝑓 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ (𝑓‘𝑦)) ∧ (𝑔:𝐶⟶𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐶 𝑧 ⊆ (𝑔‘𝑤))) → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ (𝑓‘𝑦))
22		simprl 770	. . . . . 6 ⊢ (((𝑓:𝐵⟶𝐴 ∧ Smo 𝑓 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ (𝑓‘𝑦)) ∧ (𝑔:𝐶⟶𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐶 𝑧 ⊆ (𝑔‘𝑤))) → 𝑔:𝐶⟶𝐴)
23		simpl1 1192	. . . . . . . 8 ⊢ (((Ord 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ (𝑓‘𝑦) ∧ 𝑔:𝐶⟶𝐴) ∧ 𝑤 ∈ 𝐶) → Ord 𝐵)
24		simpl2 1193	. . . . . . . . 9 ⊢ (((Ord 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ (𝑓‘𝑦) ∧ 𝑔:𝐶⟶𝐴) ∧ 𝑤 ∈ 𝐶) → ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ (𝑓‘𝑦))
25		ffvelcdm 7071	. . . . . . . . . 10 ⊢ ((𝑔:𝐶⟶𝐴 ∧ 𝑤 ∈ 𝐶) → (𝑔‘𝑤) ∈ 𝐴)
26	25	3ad2antl3 1188	. . . . . . . . 9 ⊢ (((Ord 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ (𝑓‘𝑦) ∧ 𝑔:𝐶⟶𝐴) ∧ 𝑤 ∈ 𝐶) → (𝑔‘𝑤) ∈ 𝐴)
27		sseq1 3984	. . . . . . . . . . 11 ⊢ (𝑥 = (𝑔‘𝑤) → (𝑥 ⊆ (𝑓‘𝑦) ↔ (𝑔‘𝑤) ⊆ (𝑓‘𝑦)))
28	27	rexbidv 3164	. . . . . . . . . 10 ⊢ (𝑥 = (𝑔‘𝑤) → (∃𝑦 ∈ 𝐵 𝑥 ⊆ (𝑓‘𝑦) ↔ ∃𝑦 ∈ 𝐵 (𝑔‘𝑤) ⊆ (𝑓‘𝑦)))
29	28	rspccv 3598	. . . . . . . . 9 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ (𝑓‘𝑦) → ((𝑔‘𝑤) ∈ 𝐴 → ∃𝑦 ∈ 𝐵 (𝑔‘𝑤) ⊆ (𝑓‘𝑦)))
30	24, 26, 29	sylc 65	. . . . . . . 8 ⊢ (((Ord 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ (𝑓‘𝑦) ∧ 𝑔:𝐶⟶𝐴) ∧ 𝑤 ∈ 𝐶) → ∃𝑦 ∈ 𝐵 (𝑔‘𝑤) ⊆ (𝑓‘𝑦))
31		ssrab2 4055	. . . . . . . . . . . . 13 ⊢ {𝑛 ∈ 𝐵 ∣ (𝑔‘𝑤) ⊆ (𝑓‘𝑛)} ⊆ 𝐵
32		ordsson 7777	. . . . . . . . . . . . 13 ⊢ (Ord 𝐵 → 𝐵 ⊆ On)
33	31, 32	sstrid 3970	. . . . . . . . . . . 12 ⊢ (Ord 𝐵 → {𝑛 ∈ 𝐵 ∣ (𝑔‘𝑤) ⊆ (𝑓‘𝑛)} ⊆ On)
34		fveq2 6876	. . . . . . . . . . . . . . 15 ⊢ (𝑛 = 𝑦 → (𝑓‘𝑛) = (𝑓‘𝑦))
35	34	sseq2d 3991	. . . . . . . . . . . . . 14 ⊢ (𝑛 = 𝑦 → ((𝑔‘𝑤) ⊆ (𝑓‘𝑛) ↔ (𝑔‘𝑤) ⊆ (𝑓‘𝑦)))
36	35	rspcev 3601	. . . . . . . . . . . . 13 ⊢ ((𝑦 ∈ 𝐵 ∧ (𝑔‘𝑤) ⊆ (𝑓‘𝑦)) → ∃𝑛 ∈ 𝐵 (𝑔‘𝑤) ⊆ (𝑓‘𝑛))
37		rabn0 4364	. . . . . . . . . . . . 13 ⊢ ({𝑛 ∈ 𝐵 ∣ (𝑔‘𝑤) ⊆ (𝑓‘𝑛)} ≠ ∅ ↔ ∃𝑛 ∈ 𝐵 (𝑔‘𝑤) ⊆ (𝑓‘𝑛))
38	36, 37	sylibr 234	. . . . . . . . . . . 12 ⊢ ((𝑦 ∈ 𝐵 ∧ (𝑔‘𝑤) ⊆ (𝑓‘𝑦)) → {𝑛 ∈ 𝐵 ∣ (𝑔‘𝑤) ⊆ (𝑓‘𝑛)} ≠ ∅)
39		oninton 7789	. . . . . . . . . . . 12 ⊢ (({𝑛 ∈ 𝐵 ∣ (𝑔‘𝑤) ⊆ (𝑓‘𝑛)} ⊆ On ∧ {𝑛 ∈ 𝐵 ∣ (𝑔‘𝑤) ⊆ (𝑓‘𝑛)} ≠ ∅) → ∩ {𝑛 ∈ 𝐵 ∣ (𝑔‘𝑤) ⊆ (𝑓‘𝑛)} ∈ On)
40	33, 38, 39	syl2an 596	. . . . . . . . . . 11 ⊢ ((Ord 𝐵 ∧ (𝑦 ∈ 𝐵 ∧ (𝑔‘𝑤) ⊆ (𝑓‘𝑦))) → ∩ {𝑛 ∈ 𝐵 ∣ (𝑔‘𝑤) ⊆ (𝑓‘𝑛)} ∈ On)
41		eloni 6362	. . . . . . . . . . 11 ⊢ (∩ {𝑛 ∈ 𝐵 ∣ (𝑔‘𝑤) ⊆ (𝑓‘𝑛)} ∈ On → Ord ∩ {𝑛 ∈ 𝐵 ∣ (𝑔‘𝑤) ⊆ (𝑓‘𝑛)})
42	40, 41	syl 17	. . . . . . . . . 10 ⊢ ((Ord 𝐵 ∧ (𝑦 ∈ 𝐵 ∧ (𝑔‘𝑤) ⊆ (𝑓‘𝑦))) → Ord ∩ {𝑛 ∈ 𝐵 ∣ (𝑔‘𝑤) ⊆ (𝑓‘𝑛)})
43		simpl 482	. . . . . . . . . 10 ⊢ ((Ord 𝐵 ∧ (𝑦 ∈ 𝐵 ∧ (𝑔‘𝑤) ⊆ (𝑓‘𝑦))) → Ord 𝐵)
44	35	intminss 4950	. . . . . . . . . . 11 ⊢ ((𝑦 ∈ 𝐵 ∧ (𝑔‘𝑤) ⊆ (𝑓‘𝑦)) → ∩ {𝑛 ∈ 𝐵 ∣ (𝑔‘𝑤) ⊆ (𝑓‘𝑛)} ⊆ 𝑦)
45	44	adantl 481	. . . . . . . . . 10 ⊢ ((Ord 𝐵 ∧ (𝑦 ∈ 𝐵 ∧ (𝑔‘𝑤) ⊆ (𝑓‘𝑦))) → ∩ {𝑛 ∈ 𝐵 ∣ (𝑔‘𝑤) ⊆ (𝑓‘𝑛)} ⊆ 𝑦)
46		simprl 770	. . . . . . . . . 10 ⊢ ((Ord 𝐵 ∧ (𝑦 ∈ 𝐵 ∧ (𝑔‘𝑤) ⊆ (𝑓‘𝑦))) → 𝑦 ∈ 𝐵)
47		ordtr2 6397	. . . . . . . . . . 11 ⊢ ((Ord ∩ {𝑛 ∈ 𝐵 ∣ (𝑔‘𝑤) ⊆ (𝑓‘𝑛)} ∧ Ord 𝐵) → ((∩ {𝑛 ∈ 𝐵 ∣ (𝑔‘𝑤) ⊆ (𝑓‘𝑛)} ⊆ 𝑦 ∧ 𝑦 ∈ 𝐵) → ∩ {𝑛 ∈ 𝐵 ∣ (𝑔‘𝑤) ⊆ (𝑓‘𝑛)} ∈ 𝐵))
48	47	imp 406	. . . . . . . . . 10 ⊢ (((Ord ∩ {𝑛 ∈ 𝐵 ∣ (𝑔‘𝑤) ⊆ (𝑓‘𝑛)} ∧ Ord 𝐵) ∧ (∩ {𝑛 ∈ 𝐵 ∣ (𝑔‘𝑤) ⊆ (𝑓‘𝑛)} ⊆ 𝑦 ∧ 𝑦 ∈ 𝐵)) → ∩ {𝑛 ∈ 𝐵 ∣ (𝑔‘𝑤) ⊆ (𝑓‘𝑛)} ∈ 𝐵)
49	42, 43, 45, 46, 48	syl22anc 838	. . . . . . . . 9 ⊢ ((Ord 𝐵 ∧ (𝑦 ∈ 𝐵 ∧ (𝑔‘𝑤) ⊆ (𝑓‘𝑦))) → ∩ {𝑛 ∈ 𝐵 ∣ (𝑔‘𝑤) ⊆ (𝑓‘𝑛)} ∈ 𝐵)
50	49	rexlimdvaa 3142	. . . . . . . 8 ⊢ (Ord 𝐵 → (∃𝑦 ∈ 𝐵 (𝑔‘𝑤) ⊆ (𝑓‘𝑦) → ∩ {𝑛 ∈ 𝐵 ∣ (𝑔‘𝑤) ⊆ (𝑓‘𝑛)} ∈ 𝐵))
51	23, 30, 50	sylc 65	. . . . . . 7 ⊢ (((Ord 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ (𝑓‘𝑦) ∧ 𝑔:𝐶⟶𝐴) ∧ 𝑤 ∈ 𝐶) → ∩ {𝑛 ∈ 𝐵 ∣ (𝑔‘𝑤) ⊆ (𝑓‘𝑛)} ∈ 𝐵)
52	51, 11	fmptd 7104	. . . . . 6 ⊢ ((Ord 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ (𝑓‘𝑦) ∧ 𝑔:𝐶⟶𝐴) → 𝐻:𝐶⟶𝐵)
53	20, 21, 22, 52	syl3anc 1373	. . . . 5 ⊢ (((𝑓:𝐵⟶𝐴 ∧ Smo 𝑓 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ (𝑓‘𝑦)) ∧ (𝑔:𝐶⟶𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐶 𝑧 ⊆ (𝑔‘𝑤))) → 𝐻:𝐶⟶𝐵)
54		simprr 772	. . . . . . . 8 ⊢ (((𝑓:𝐵⟶𝐴 ∧ Smo 𝑓 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ (𝑓‘𝑦)) ∧ (𝑔:𝐶⟶𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐶 𝑧 ⊆ (𝑔‘𝑤))) → ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐶 𝑧 ⊆ (𝑔‘𝑤))
55		simpl1 1192	. . . . . . . 8 ⊢ (((𝑓:𝐵⟶𝐴 ∧ Smo 𝑓 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ (𝑓‘𝑦)) ∧ (𝑔:𝐶⟶𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐶 𝑧 ⊆ (𝑔‘𝑤))) → 𝑓:𝐵⟶𝐴)
56		ffvelcdm 7071	. . . . . . . . . 10 ⊢ ((𝑓:𝐵⟶𝐴 ∧ 𝑠 ∈ 𝐵) → (𝑓‘𝑠) ∈ 𝐴)
57		sseq1 3984	. . . . . . . . . . . 12 ⊢ (𝑧 = (𝑓‘𝑠) → (𝑧 ⊆ (𝑔‘𝑤) ↔ (𝑓‘𝑠) ⊆ (𝑔‘𝑤)))
58	57	rexbidv 3164	. . . . . . . . . . 11 ⊢ (𝑧 = (𝑓‘𝑠) → (∃𝑤 ∈ 𝐶 𝑧 ⊆ (𝑔‘𝑤) ↔ ∃𝑤 ∈ 𝐶 (𝑓‘𝑠) ⊆ (𝑔‘𝑤)))
59	58	rspccv 3598	. . . . . . . . . 10 ⊢ (∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐶 𝑧 ⊆ (𝑔‘𝑤) → ((𝑓‘𝑠) ∈ 𝐴 → ∃𝑤 ∈ 𝐶 (𝑓‘𝑠) ⊆ (𝑔‘𝑤)))
60	56, 59	syl5 34	. . . . . . . . 9 ⊢ (∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐶 𝑧 ⊆ (𝑔‘𝑤) → ((𝑓:𝐵⟶𝐴 ∧ 𝑠 ∈ 𝐵) → ∃𝑤 ∈ 𝐶 (𝑓‘𝑠) ⊆ (𝑔‘𝑤)))
61	60	expdimp 452	. . . . . . . 8 ⊢ ((∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐶 𝑧 ⊆ (𝑔‘𝑤) ∧ 𝑓:𝐵⟶𝐴) → (𝑠 ∈ 𝐵 → ∃𝑤 ∈ 𝐶 (𝑓‘𝑠) ⊆ (𝑔‘𝑤)))
62	54, 55, 61	syl2anc 584	. . . . . . 7 ⊢ (((𝑓:𝐵⟶𝐴 ∧ Smo 𝑓 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ (𝑓‘𝑦)) ∧ (𝑔:𝐶⟶𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐶 𝑧 ⊆ (𝑔‘𝑤))) → (𝑠 ∈ 𝐵 → ∃𝑤 ∈ 𝐶 (𝑓‘𝑠) ⊆ (𝑔‘𝑤)))
63	55, 16	syl 17	. . . . . . . 8 ⊢ (((𝑓:𝐵⟶𝐴 ∧ Smo 𝑓 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ (𝑓‘𝑦)) ∧ (𝑔:𝐶⟶𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐶 𝑧 ⊆ (𝑔‘𝑤))) → 𝑓 Fn 𝐵)
64		simpl2 1193	. . . . . . . 8 ⊢ (((𝑓:𝐵⟶𝐴 ∧ Smo 𝑓 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ (𝑓‘𝑦)) ∧ (𝑔:𝐶⟶𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐶 𝑧 ⊆ (𝑔‘𝑤))) → Smo 𝑓)
65		simpr 484	. . . . . . . . . . . . . . . 16 ⊢ (((Ord 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ (𝑓‘𝑦) ∧ 𝑔:𝐶⟶𝐴) ∧ 𝑤 ∈ 𝐶) → 𝑤 ∈ 𝐶)
66	65, 51	jca 511	. . . . . . . . . . . . . . 15 ⊢ (((Ord 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ (𝑓‘𝑦) ∧ 𝑔:𝐶⟶𝐴) ∧ 𝑤 ∈ 𝐶) → (𝑤 ∈ 𝐶 ∧ ∩ {𝑛 ∈ 𝐵 ∣ (𝑔‘𝑤) ⊆ (𝑓‘𝑛)} ∈ 𝐵))
67	35	elrab 3671	. . . . . . . . . . . . . . . . . . 19 ⊢ (𝑦 ∈ {𝑛 ∈ 𝐵 ∣ (𝑔‘𝑤) ⊆ (𝑓‘𝑛)} ↔ (𝑦 ∈ 𝐵 ∧ (𝑔‘𝑤) ⊆ (𝑓‘𝑦)))
68		sstr2 3965	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((𝑓‘𝑠) ⊆ (𝑔‘𝑤) → ((𝑔‘𝑤) ⊆ (𝑓‘𝑦) → (𝑓‘𝑠) ⊆ (𝑓‘𝑦)))
69		smoword 8380	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ (((𝑓 Fn 𝐵 ∧ Smo 𝑓) ∧ (𝑠 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → (𝑠 ⊆ 𝑦 ↔ (𝑓‘𝑠) ⊆ (𝑓‘𝑦)))
70	69	biimprd 248	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (((𝑓 Fn 𝐵 ∧ Smo 𝑓) ∧ (𝑠 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑓‘𝑠) ⊆ (𝑓‘𝑦) → 𝑠 ⊆ 𝑦))
71	68, 70	syl9r 78	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((𝑓 Fn 𝐵 ∧ Smo 𝑓) ∧ (𝑠 ∈ 𝐵 ∧ 𝑦 ∈ 𝐵)) → ((𝑓‘𝑠) ⊆ (𝑔‘𝑤) → ((𝑔‘𝑤) ⊆ (𝑓‘𝑦) → 𝑠 ⊆ 𝑦)))
72	71	expr 456	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑓 Fn 𝐵 ∧ Smo 𝑓) ∧ 𝑠 ∈ 𝐵) → (𝑦 ∈ 𝐵 → ((𝑓‘𝑠) ⊆ (𝑔‘𝑤) → ((𝑔‘𝑤) ⊆ (𝑓‘𝑦) → 𝑠 ⊆ 𝑦))))
73	72	com23 86	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑓 Fn 𝐵 ∧ Smo 𝑓) ∧ 𝑠 ∈ 𝐵) → ((𝑓‘𝑠) ⊆ (𝑔‘𝑤) → (𝑦 ∈ 𝐵 → ((𝑔‘𝑤) ⊆ (𝑓‘𝑦) → 𝑠 ⊆ 𝑦))))
74	73	imp4b 421	. . . . . . . . . . . . . . . . . . 19 ⊢ ((((𝑓 Fn 𝐵 ∧ Smo 𝑓) ∧ 𝑠 ∈ 𝐵) ∧ (𝑓‘𝑠) ⊆ (𝑔‘𝑤)) → ((𝑦 ∈ 𝐵 ∧ (𝑔‘𝑤) ⊆ (𝑓‘𝑦)) → 𝑠 ⊆ 𝑦))
75	67, 74	biimtrid 242	. . . . . . . . . . . . . . . . . 18 ⊢ ((((𝑓 Fn 𝐵 ∧ Smo 𝑓) ∧ 𝑠 ∈ 𝐵) ∧ (𝑓‘𝑠) ⊆ (𝑔‘𝑤)) → (𝑦 ∈ {𝑛 ∈ 𝐵 ∣ (𝑔‘𝑤) ⊆ (𝑓‘𝑛)} → 𝑠 ⊆ 𝑦))
76	75	ralrimiv 3131	. . . . . . . . . . . . . . . . 17 ⊢ ((((𝑓 Fn 𝐵 ∧ Smo 𝑓) ∧ 𝑠 ∈ 𝐵) ∧ (𝑓‘𝑠) ⊆ (𝑔‘𝑤)) → ∀𝑦 ∈ {𝑛 ∈ 𝐵 ∣ (𝑔‘𝑤) ⊆ (𝑓‘𝑛)}𝑠 ⊆ 𝑦)
77		ssint 4940	. . . . . . . . . . . . . . . . 17 ⊢ (𝑠 ⊆ ∩ {𝑛 ∈ 𝐵 ∣ (𝑔‘𝑤) ⊆ (𝑓‘𝑛)} ↔ ∀𝑦 ∈ {𝑛 ∈ 𝐵 ∣ (𝑔‘𝑤) ⊆ (𝑓‘𝑛)}𝑠 ⊆ 𝑦)
78	76, 77	sylibr 234	. . . . . . . . . . . . . . . 16 ⊢ ((((𝑓 Fn 𝐵 ∧ Smo 𝑓) ∧ 𝑠 ∈ 𝐵) ∧ (𝑓‘𝑠) ⊆ (𝑔‘𝑤)) → 𝑠 ⊆ ∩ {𝑛 ∈ 𝐵 ∣ (𝑔‘𝑤) ⊆ (𝑓‘𝑛)})
79	9, 5	fvmptg 6984	. . . . . . . . . . . . . . . . 17 ⊢ ((𝑤 ∈ 𝐶 ∧ ∩ {𝑛 ∈ 𝐵 ∣ (𝑔‘𝑤) ⊆ (𝑓‘𝑛)} ∈ 𝐵) → (𝐻‘𝑤) = ∩ {𝑛 ∈ 𝐵 ∣ (𝑔‘𝑤) ⊆ (𝑓‘𝑛)})
80	79	sseq2d 3991	. . . . . . . . . . . . . . . 16 ⊢ ((𝑤 ∈ 𝐶 ∧ ∩ {𝑛 ∈ 𝐵 ∣ (𝑔‘𝑤) ⊆ (𝑓‘𝑛)} ∈ 𝐵) → (𝑠 ⊆ (𝐻‘𝑤) ↔ 𝑠 ⊆ ∩ {𝑛 ∈ 𝐵 ∣ (𝑔‘𝑤) ⊆ (𝑓‘𝑛)}))
81	78, 80	syl5ibrcom 247	. . . . . . . . . . . . . . 15 ⊢ ((((𝑓 Fn 𝐵 ∧ Smo 𝑓) ∧ 𝑠 ∈ 𝐵) ∧ (𝑓‘𝑠) ⊆ (𝑔‘𝑤)) → ((𝑤 ∈ 𝐶 ∧ ∩ {𝑛 ∈ 𝐵 ∣ (𝑔‘𝑤) ⊆ (𝑓‘𝑛)} ∈ 𝐵) → 𝑠 ⊆ (𝐻‘𝑤)))
82	66, 81	syl5 34	. . . . . . . . . . . . . 14 ⊢ ((((𝑓 Fn 𝐵 ∧ Smo 𝑓) ∧ 𝑠 ∈ 𝐵) ∧ (𝑓‘𝑠) ⊆ (𝑔‘𝑤)) → (((Ord 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ (𝑓‘𝑦) ∧ 𝑔:𝐶⟶𝐴) ∧ 𝑤 ∈ 𝐶) → 𝑠 ⊆ (𝐻‘𝑤)))
83	82	ex 412	. . . . . . . . . . . . 13 ⊢ (((𝑓 Fn 𝐵 ∧ Smo 𝑓) ∧ 𝑠 ∈ 𝐵) → ((𝑓‘𝑠) ⊆ (𝑔‘𝑤) → (((Ord 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ (𝑓‘𝑦) ∧ 𝑔:𝐶⟶𝐴) ∧ 𝑤 ∈ 𝐶) → 𝑠 ⊆ (𝐻‘𝑤))))
84	83	com23 86	. . . . . . . . . . . 12 ⊢ (((𝑓 Fn 𝐵 ∧ Smo 𝑓) ∧ 𝑠 ∈ 𝐵) → (((Ord 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ (𝑓‘𝑦) ∧ 𝑔:𝐶⟶𝐴) ∧ 𝑤 ∈ 𝐶) → ((𝑓‘𝑠) ⊆ (𝑔‘𝑤) → 𝑠 ⊆ (𝐻‘𝑤))))
85	84	expdimp 452	. . . . . . . . . . 11 ⊢ ((((𝑓 Fn 𝐵 ∧ Smo 𝑓) ∧ 𝑠 ∈ 𝐵) ∧ (Ord 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ (𝑓‘𝑦) ∧ 𝑔:𝐶⟶𝐴)) → (𝑤 ∈ 𝐶 → ((𝑓‘𝑠) ⊆ (𝑔‘𝑤) → 𝑠 ⊆ (𝐻‘𝑤))))
86	85	reximdvai 3151	. . . . . . . . . 10 ⊢ ((((𝑓 Fn 𝐵 ∧ Smo 𝑓) ∧ 𝑠 ∈ 𝐵) ∧ (Ord 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ (𝑓‘𝑦) ∧ 𝑔:𝐶⟶𝐴)) → (∃𝑤 ∈ 𝐶 (𝑓‘𝑠) ⊆ (𝑔‘𝑤) → ∃𝑤 ∈ 𝐶 𝑠 ⊆ (𝐻‘𝑤)))
87	86	ancoms 458	. . . . . . . . 9 ⊢ (((Ord 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ (𝑓‘𝑦) ∧ 𝑔:𝐶⟶𝐴) ∧ ((𝑓 Fn 𝐵 ∧ Smo 𝑓) ∧ 𝑠 ∈ 𝐵)) → (∃𝑤 ∈ 𝐶 (𝑓‘𝑠) ⊆ (𝑔‘𝑤) → ∃𝑤 ∈ 𝐶 𝑠 ⊆ (𝐻‘𝑤)))
88	87	expr 456	. . . . . . . 8 ⊢ (((Ord 𝐵 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ (𝑓‘𝑦) ∧ 𝑔:𝐶⟶𝐴) ∧ (𝑓 Fn 𝐵 ∧ Smo 𝑓)) → (𝑠 ∈ 𝐵 → (∃𝑤 ∈ 𝐶 (𝑓‘𝑠) ⊆ (𝑔‘𝑤) → ∃𝑤 ∈ 𝐶 𝑠 ⊆ (𝐻‘𝑤))))
89	20, 21, 22, 63, 64, 88	syl32anc 1380	. . . . . . 7 ⊢ (((𝑓:𝐵⟶𝐴 ∧ Smo 𝑓 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ (𝑓‘𝑦)) ∧ (𝑔:𝐶⟶𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐶 𝑧 ⊆ (𝑔‘𝑤))) → (𝑠 ∈ 𝐵 → (∃𝑤 ∈ 𝐶 (𝑓‘𝑠) ⊆ (𝑔‘𝑤) → ∃𝑤 ∈ 𝐶 𝑠 ⊆ (𝐻‘𝑤))))
90	62, 89	mpdd 43	. . . . . 6 ⊢ (((𝑓:𝐵⟶𝐴 ∧ Smo 𝑓 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ (𝑓‘𝑦)) ∧ (𝑔:𝐶⟶𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐶 𝑧 ⊆ (𝑔‘𝑤))) → (𝑠 ∈ 𝐵 → ∃𝑤 ∈ 𝐶 𝑠 ⊆ (𝐻‘𝑤)))
91	90	ralrimiv 3131	. . . . 5 ⊢ (((𝑓:𝐵⟶𝐴 ∧ Smo 𝑓 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ (𝑓‘𝑦)) ∧ (𝑔:𝐶⟶𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐶 𝑧 ⊆ (𝑔‘𝑤))) → ∀𝑠 ∈ 𝐵 ∃𝑤 ∈ 𝐶 𝑠 ⊆ (𝐻‘𝑤))
92		feq1 6686	. . . . . . . 8 ⊢ (ℎ = 𝐻 → (ℎ:𝐶⟶𝐵 ↔ 𝐻:𝐶⟶𝐵))
93		fveq1 6875	. . . . . . . . . . 11 ⊢ (ℎ = 𝐻 → (ℎ‘𝑤) = (𝐻‘𝑤))
94	93	sseq2d 3991	. . . . . . . . . 10 ⊢ (ℎ = 𝐻 → (𝑠 ⊆ (ℎ‘𝑤) ↔ 𝑠 ⊆ (𝐻‘𝑤)))
95	94	rexbidv 3164	. . . . . . . . 9 ⊢ (ℎ = 𝐻 → (∃𝑤 ∈ 𝐶 𝑠 ⊆ (ℎ‘𝑤) ↔ ∃𝑤 ∈ 𝐶 𝑠 ⊆ (𝐻‘𝑤)))
96	95	ralbidv 3163	. . . . . . . 8 ⊢ (ℎ = 𝐻 → (∀𝑠 ∈ 𝐵 ∃𝑤 ∈ 𝐶 𝑠 ⊆ (ℎ‘𝑤) ↔ ∀𝑠 ∈ 𝐵 ∃𝑤 ∈ 𝐶 𝑠 ⊆ (𝐻‘𝑤)))
97	92, 96	anbi12d 632	. . . . . . 7 ⊢ (ℎ = 𝐻 → ((ℎ:𝐶⟶𝐵 ∧ ∀𝑠 ∈ 𝐵 ∃𝑤 ∈ 𝐶 𝑠 ⊆ (ℎ‘𝑤)) ↔ (𝐻:𝐶⟶𝐵 ∧ ∀𝑠 ∈ 𝐵 ∃𝑤 ∈ 𝐶 𝑠 ⊆ (𝐻‘𝑤))))
98	97	spcegv 3576	. . . . . 6 ⊢ (𝐻 ∈ V → ((𝐻:𝐶⟶𝐵 ∧ ∀𝑠 ∈ 𝐵 ∃𝑤 ∈ 𝐶 𝑠 ⊆ (𝐻‘𝑤)) → ∃ℎ(ℎ:𝐶⟶𝐵 ∧ ∀𝑠 ∈ 𝐵 ∃𝑤 ∈ 𝐶 𝑠 ⊆ (ℎ‘𝑤))))
99	98	3impib 1116	. . . . 5 ⊢ ((𝐻 ∈ V ∧ 𝐻:𝐶⟶𝐵 ∧ ∀𝑠 ∈ 𝐵 ∃𝑤 ∈ 𝐶 𝑠 ⊆ (𝐻‘𝑤)) → ∃ℎ(ℎ:𝐶⟶𝐵 ∧ ∀𝑠 ∈ 𝐵 ∃𝑤 ∈ 𝐶 𝑠 ⊆ (ℎ‘𝑤)))
100	15, 53, 91, 99	syl3anc 1373	. . . 4 ⊢ (((𝑓:𝐵⟶𝐴 ∧ Smo 𝑓 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ (𝑓‘𝑦)) ∧ (𝑔:𝐶⟶𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐶 𝑧 ⊆ (𝑔‘𝑤))) → ∃ℎ(ℎ:𝐶⟶𝐵 ∧ ∀𝑠 ∈ 𝐵 ∃𝑤 ∈ 𝐶 𝑠 ⊆ (ℎ‘𝑤)))
101	100	ex 412	. . 3 ⊢ ((𝑓:𝐵⟶𝐴 ∧ Smo 𝑓 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ (𝑓‘𝑦)) → ((𝑔:𝐶⟶𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐶 𝑧 ⊆ (𝑔‘𝑤)) → ∃ℎ(ℎ:𝐶⟶𝐵 ∧ ∀𝑠 ∈ 𝐵 ∃𝑤 ∈ 𝐶 𝑠 ⊆ (ℎ‘𝑤))))
102	101	exlimdv 1933	. 2 ⊢ ((𝑓:𝐵⟶𝐴 ∧ Smo 𝑓 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ (𝑓‘𝑦)) → (∃𝑔(𝑔:𝐶⟶𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐶 𝑧 ⊆ (𝑔‘𝑤)) → ∃ℎ(ℎ:𝐶⟶𝐵 ∧ ∀𝑠 ∈ 𝐵 ∃𝑤 ∈ 𝐶 𝑠 ⊆ (ℎ‘𝑤))))
103	102	exlimiv 1930	1 ⊢ (∃𝑓(𝑓:𝐵⟶𝐴 ∧ Smo 𝑓 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ (𝑓‘𝑦)) → (∃𝑔(𝑔:𝐶⟶𝐴 ∧ ∀𝑧 ∈ 𝐴 ∃𝑤 ∈ 𝐶 𝑧 ⊆ (𝑔‘𝑤)) → ∃ℎ(ℎ:𝐶⟶𝐵 ∧ ∀𝑠 ∈ 𝐵 ∃𝑤 ∈ 𝐶 𝑠 ⊆ (ℎ‘𝑤))))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 = wceq 1540 ∃wex 1779 ∈ wcel 2108 ≠ wne 2932 ∀wral 3051 ∃wrex 3060 {crab 3415 Vcvv 3459 ⊆ wss 3926 ∅c0 4308 ∩ cint 4922 ↦ cmpt 5201 dom cdm 5654 Ord word 6351 Oncon0 6352 Fn wfn 6526 ⟶wf 6527 ‘cfv 6531 Smo wsmo 8359

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-rep 5249 ax-sep 5266 ax-nul 5276 ax-pr 5402 ax-un 7729

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-ral 3052 df-rex 3061 df-reu 3360 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-pss 3946 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-op 4608 df-uni 4884 df-int 4923 df-iun 4969 df-br 5120 df-opab 5182 df-mpt 5202 df-tr 5230 df-id 5548 df-eprel 5553 df-po 5561 df-so 5562 df-fr 5606 df-we 5608 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-ord 6355 df-on 6356 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-smo 8360

This theorem is referenced by: cfcof 10288

W3C validator