Theorem cfcoflem 9349

Description: Lemma for cfcof 9351, showing subset relation in one direction. (Contributed by Mario Carneiro, 9-Mar-2013.) (Revised by Mario Carneiro, 26-Dec-2014.)

Assertion

Ref	Expression
cfcoflem	⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∃𝑓(𝑓:𝐵⟶𝐴 ∧ Smo 𝑓 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ (𝑓‘𝑦)) → (cf‘𝐴) ⊆ (cf‘𝐵)))

Distinct variable groups:   𝐴,𝑓,𝑥,𝑦   𝐵,𝑓,𝑥,𝑦

Proof of Theorem cfcoflem

Dummy variables 𝑔 ℎ 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		cff1 9335	. . 3 ⊢ (𝐵 ∈ On → ∃𝑔(𝑔:(cf‘𝐵)–1-1→𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑧 ∈ (cf‘𝐵)𝑦 ⊆ (𝑔‘𝑧)))
2		f1f 6285	. . . . . 6 ⊢ (𝑔:(cf‘𝐵)–1-1→𝐵 → 𝑔:(cf‘𝐵)⟶𝐵)
3		fco 6242	. . . . . . . . . . . . . 14 ⊢ ((𝑓:𝐵⟶𝐴 ∧ 𝑔:(cf‘𝐵)⟶𝐵) → (𝑓 ∘ 𝑔):(cf‘𝐵)⟶𝐴)
4	3	adantlr 706	. . . . . . . . . . . . 13 ⊢ (((𝑓:𝐵⟶𝐴 ∧ Smo 𝑓) ∧ 𝑔:(cf‘𝐵)⟶𝐵) → (𝑓 ∘ 𝑔):(cf‘𝐵)⟶𝐴)
5	4	adantr 472	. . . . . . . . . . . 12 ⊢ ((((𝑓:𝐵⟶𝐴 ∧ Smo 𝑓) ∧ 𝑔:(cf‘𝐵)⟶𝐵) ∧ (∀𝑦 ∈ 𝐵 ∃𝑧 ∈ (cf‘𝐵)𝑦 ⊆ (𝑔‘𝑧) ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ (𝑓‘𝑦))) → (𝑓 ∘ 𝑔):(cf‘𝐵)⟶𝐴)
6		r19.29 3219	. . . . . . . . . . . . . . . 16 ⊢ ((∀𝑦 ∈ 𝐵 ∃𝑧 ∈ (cf‘𝐵)𝑦 ⊆ (𝑔‘𝑧) ∧ ∃𝑦 ∈ 𝐵 𝑥 ⊆ (𝑓‘𝑦)) → ∃𝑦 ∈ 𝐵 (∃𝑧 ∈ (cf‘𝐵)𝑦 ⊆ (𝑔‘𝑧) ∧ 𝑥 ⊆ (𝑓‘𝑦)))
7		ffvelrn 6549	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝑔:(cf‘𝐵)⟶𝐵 ∧ 𝑧 ∈ (cf‘𝐵)) → (𝑔‘𝑧) ∈ 𝐵)
8		ffn 6225	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑓:𝐵⟶𝐴 → 𝑓 Fn 𝐵)
9		smoword 7669	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (((𝑓 Fn 𝐵 ∧ Smo 𝑓) ∧ (𝑦 ∈ 𝐵 ∧ (𝑔‘𝑧) ∈ 𝐵)) → (𝑦 ⊆ (𝑔‘𝑧) ↔ (𝑓‘𝑦) ⊆ (𝑓‘(𝑔‘𝑧))))
10	9	biimpd 220	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (((𝑓 Fn 𝐵 ∧ Smo 𝑓) ∧ (𝑦 ∈ 𝐵 ∧ (𝑔‘𝑧) ∈ 𝐵)) → (𝑦 ⊆ (𝑔‘𝑧) → (𝑓‘𝑦) ⊆ (𝑓‘(𝑔‘𝑧))))
11	10	exp32 411	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝑓 Fn 𝐵 ∧ Smo 𝑓) → (𝑦 ∈ 𝐵 → ((𝑔‘𝑧) ∈ 𝐵 → (𝑦 ⊆ (𝑔‘𝑧) → (𝑓‘𝑦) ⊆ (𝑓‘(𝑔‘𝑧))))))
12	8, 11	sylan 575	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝑓:𝐵⟶𝐴 ∧ Smo 𝑓) → (𝑦 ∈ 𝐵 → ((𝑔‘𝑧) ∈ 𝐵 → (𝑦 ⊆ (𝑔‘𝑧) → (𝑓‘𝑦) ⊆ (𝑓‘(𝑔‘𝑧))))))
13	7, 12	syl7 74	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑓:𝐵⟶𝐴 ∧ Smo 𝑓) → (𝑦 ∈ 𝐵 → ((𝑔:(cf‘𝐵)⟶𝐵 ∧ 𝑧 ∈ (cf‘𝐵)) → (𝑦 ⊆ (𝑔‘𝑧) → (𝑓‘𝑦) ⊆ (𝑓‘(𝑔‘𝑧))))))
14	13	com23 86	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑓:𝐵⟶𝐴 ∧ Smo 𝑓) → ((𝑔:(cf‘𝐵)⟶𝐵 ∧ 𝑧 ∈ (cf‘𝐵)) → (𝑦 ∈ 𝐵 → (𝑦 ⊆ (𝑔‘𝑧) → (𝑓‘𝑦) ⊆ (𝑓‘(𝑔‘𝑧))))))
15	14	expdimp 444	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ (((𝑓:𝐵⟶𝐴 ∧ Smo 𝑓) ∧ 𝑔:(cf‘𝐵)⟶𝐵) → (𝑧 ∈ (cf‘𝐵) → (𝑦 ∈ 𝐵 → (𝑦 ⊆ (𝑔‘𝑧) → (𝑓‘𝑦) ⊆ (𝑓‘(𝑔‘𝑧))))))
16	15	3imp2 1458	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝑓:𝐵⟶𝐴 ∧ Smo 𝑓) ∧ 𝑔:(cf‘𝐵)⟶𝐵) ∧ (𝑧 ∈ (cf‘𝐵) ∧ 𝑦 ∈ 𝐵 ∧ 𝑦 ⊆ (𝑔‘𝑧))) → (𝑓‘𝑦) ⊆ (𝑓‘(𝑔‘𝑧)))
17		sstr2 3770	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (𝑥 ⊆ (𝑓‘𝑦) → ((𝑓‘𝑦) ⊆ (𝑓‘(𝑔‘𝑧)) → 𝑥 ⊆ (𝑓‘(𝑔‘𝑧))))
18	16, 17	syl5com 31	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝑓:𝐵⟶𝐴 ∧ Smo 𝑓) ∧ 𝑔:(cf‘𝐵)⟶𝐵) ∧ (𝑧 ∈ (cf‘𝐵) ∧ 𝑦 ∈ 𝐵 ∧ 𝑦 ⊆ (𝑔‘𝑧))) → (𝑥 ⊆ (𝑓‘𝑦) → 𝑥 ⊆ (𝑓‘(𝑔‘𝑧))))
19		fvco3 6466	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑔:(cf‘𝐵)⟶𝐵 ∧ 𝑧 ∈ (cf‘𝐵)) → ((𝑓 ∘ 𝑔)‘𝑧) = (𝑓‘(𝑔‘𝑧)))
20	19	sseq2d 3795	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑔:(cf‘𝐵)⟶𝐵 ∧ 𝑧 ∈ (cf‘𝐵)) → (𝑥 ⊆ ((𝑓 ∘ 𝑔)‘𝑧) ↔ 𝑥 ⊆ (𝑓‘(𝑔‘𝑧))))
21	20	adantll 705	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝑓:𝐵⟶𝐴 ∧ Smo 𝑓) ∧ 𝑔:(cf‘𝐵)⟶𝐵) ∧ 𝑧 ∈ (cf‘𝐵)) → (𝑥 ⊆ ((𝑓 ∘ 𝑔)‘𝑧) ↔ 𝑥 ⊆ (𝑓‘(𝑔‘𝑧))))
22	21	3ad2antr1 1239	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝑓:𝐵⟶𝐴 ∧ Smo 𝑓) ∧ 𝑔:(cf‘𝐵)⟶𝐵) ∧ (𝑧 ∈ (cf‘𝐵) ∧ 𝑦 ∈ 𝐵 ∧ 𝑦 ⊆ (𝑔‘𝑧))) → (𝑥 ⊆ ((𝑓 ∘ 𝑔)‘𝑧) ↔ 𝑥 ⊆ (𝑓‘(𝑔‘𝑧))))
23	18, 22	sylibrd 250	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((𝑓:𝐵⟶𝐴 ∧ Smo 𝑓) ∧ 𝑔:(cf‘𝐵)⟶𝐵) ∧ (𝑧 ∈ (cf‘𝐵) ∧ 𝑦 ∈ 𝐵 ∧ 𝑦 ⊆ (𝑔‘𝑧))) → (𝑥 ⊆ (𝑓‘𝑦) → 𝑥 ⊆ ((𝑓 ∘ 𝑔)‘𝑧)))
24	23	expcom 402	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((𝑧 ∈ (cf‘𝐵) ∧ 𝑦 ∈ 𝐵 ∧ 𝑦 ⊆ (𝑔‘𝑧)) → (((𝑓:𝐵⟶𝐴 ∧ Smo 𝑓) ∧ 𝑔:(cf‘𝐵)⟶𝐵) → (𝑥 ⊆ (𝑓‘𝑦) → 𝑥 ⊆ ((𝑓 ∘ 𝑔)‘𝑧))))
25	24	3expia 1150	. . . . . . . . . . . . . . . . . . . . . . . . . . 27 ⊢ ((𝑧 ∈ (cf‘𝐵) ∧ 𝑦 ∈ 𝐵) → (𝑦 ⊆ (𝑔‘𝑧) → (((𝑓:𝐵⟶𝐴 ∧ Smo 𝑓) ∧ 𝑔:(cf‘𝐵)⟶𝐵) → (𝑥 ⊆ (𝑓‘𝑦) → 𝑥 ⊆ ((𝑓 ∘ 𝑔)‘𝑧)))))
26	25	com4t 93	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ (((𝑓:𝐵⟶𝐴 ∧ Smo 𝑓) ∧ 𝑔:(cf‘𝐵)⟶𝐵) → (𝑥 ⊆ (𝑓‘𝑦) → ((𝑧 ∈ (cf‘𝐵) ∧ 𝑦 ∈ 𝐵) → (𝑦 ⊆ (𝑔‘𝑧) → 𝑥 ⊆ ((𝑓 ∘ 𝑔)‘𝑧)))))
27	26	imp 395	. . . . . . . . . . . . . . . . . . . . . . . . 25 ⊢ ((((𝑓:𝐵⟶𝐴 ∧ Smo 𝑓) ∧ 𝑔:(cf‘𝐵)⟶𝐵) ∧ 𝑥 ⊆ (𝑓‘𝑦)) → ((𝑧 ∈ (cf‘𝐵) ∧ 𝑦 ∈ 𝐵) → (𝑦 ⊆ (𝑔‘𝑧) → 𝑥 ⊆ ((𝑓 ∘ 𝑔)‘𝑧))))
28	27	expcomd 406	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((𝑓:𝐵⟶𝐴 ∧ Smo 𝑓) ∧ 𝑔:(cf‘𝐵)⟶𝐵) ∧ 𝑥 ⊆ (𝑓‘𝑦)) → (𝑦 ∈ 𝐵 → (𝑧 ∈ (cf‘𝐵) → (𝑦 ⊆ (𝑔‘𝑧) → 𝑥 ⊆ ((𝑓 ∘ 𝑔)‘𝑧)))))
29	28	imp31 408	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ ((((((𝑓:𝐵⟶𝐴 ∧ Smo 𝑓) ∧ 𝑔:(cf‘𝐵)⟶𝐵) ∧ 𝑥 ⊆ (𝑓‘𝑦)) ∧ 𝑦 ∈ 𝐵) ∧ 𝑧 ∈ (cf‘𝐵)) → (𝑦 ⊆ (𝑔‘𝑧) → 𝑥 ⊆ ((𝑓 ∘ 𝑔)‘𝑧)))
30	29	reximdva 3163	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ (((((𝑓:𝐵⟶𝐴 ∧ Smo 𝑓) ∧ 𝑔:(cf‘𝐵)⟶𝐵) ∧ 𝑥 ⊆ (𝑓‘𝑦)) ∧ 𝑦 ∈ 𝐵) → (∃𝑧 ∈ (cf‘𝐵)𝑦 ⊆ (𝑔‘𝑧) → ∃𝑧 ∈ (cf‘𝐵)𝑥 ⊆ ((𝑓 ∘ 𝑔)‘𝑧)))
31	30	exp31 410	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝑓:𝐵⟶𝐴 ∧ Smo 𝑓) ∧ 𝑔:(cf‘𝐵)⟶𝐵) → (𝑥 ⊆ (𝑓‘𝑦) → (𝑦 ∈ 𝐵 → (∃𝑧 ∈ (cf‘𝐵)𝑦 ⊆ (𝑔‘𝑧) → ∃𝑧 ∈ (cf‘𝐵)𝑥 ⊆ ((𝑓 ∘ 𝑔)‘𝑧)))))
32	31	com34 91	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((𝑓:𝐵⟶𝐴 ∧ Smo 𝑓) ∧ 𝑔:(cf‘𝐵)⟶𝐵) → (𝑥 ⊆ (𝑓‘𝑦) → (∃𝑧 ∈ (cf‘𝐵)𝑦 ⊆ (𝑔‘𝑧) → (𝑦 ∈ 𝐵 → ∃𝑧 ∈ (cf‘𝐵)𝑥 ⊆ ((𝑓 ∘ 𝑔)‘𝑧)))))
33	32	com23 86	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝑓:𝐵⟶𝐴 ∧ Smo 𝑓) ∧ 𝑔:(cf‘𝐵)⟶𝐵) → (∃𝑧 ∈ (cf‘𝐵)𝑦 ⊆ (𝑔‘𝑧) → (𝑥 ⊆ (𝑓‘𝑦) → (𝑦 ∈ 𝐵 → ∃𝑧 ∈ (cf‘𝐵)𝑥 ⊆ ((𝑓 ∘ 𝑔)‘𝑧)))))
34	33	impd 398	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝑓:𝐵⟶𝐴 ∧ Smo 𝑓) ∧ 𝑔:(cf‘𝐵)⟶𝐵) → ((∃𝑧 ∈ (cf‘𝐵)𝑦 ⊆ (𝑔‘𝑧) ∧ 𝑥 ⊆ (𝑓‘𝑦)) → (𝑦 ∈ 𝐵 → ∃𝑧 ∈ (cf‘𝐵)𝑥 ⊆ ((𝑓 ∘ 𝑔)‘𝑧))))
35	34	com23 86	. . . . . . . . . . . . . . . . 17 ⊢ (((𝑓:𝐵⟶𝐴 ∧ Smo 𝑓) ∧ 𝑔:(cf‘𝐵)⟶𝐵) → (𝑦 ∈ 𝐵 → ((∃𝑧 ∈ (cf‘𝐵)𝑦 ⊆ (𝑔‘𝑧) ∧ 𝑥 ⊆ (𝑓‘𝑦)) → ∃𝑧 ∈ (cf‘𝐵)𝑥 ⊆ ((𝑓 ∘ 𝑔)‘𝑧))))
36	35	rexlimdv 3177	. . . . . . . . . . . . . . . 16 ⊢ (((𝑓:𝐵⟶𝐴 ∧ Smo 𝑓) ∧ 𝑔:(cf‘𝐵)⟶𝐵) → (∃𝑦 ∈ 𝐵 (∃𝑧 ∈ (cf‘𝐵)𝑦 ⊆ (𝑔‘𝑧) ∧ 𝑥 ⊆ (𝑓‘𝑦)) → ∃𝑧 ∈ (cf‘𝐵)𝑥 ⊆ ((𝑓 ∘ 𝑔)‘𝑧)))
37	6, 36	syl5 34	. . . . . . . . . . . . . . 15 ⊢ (((𝑓:𝐵⟶𝐴 ∧ Smo 𝑓) ∧ 𝑔:(cf‘𝐵)⟶𝐵) → ((∀𝑦 ∈ 𝐵 ∃𝑧 ∈ (cf‘𝐵)𝑦 ⊆ (𝑔‘𝑧) ∧ ∃𝑦 ∈ 𝐵 𝑥 ⊆ (𝑓‘𝑦)) → ∃𝑧 ∈ (cf‘𝐵)𝑥 ⊆ ((𝑓 ∘ 𝑔)‘𝑧)))
38	37	expdimp 444	. . . . . . . . . . . . . 14 ⊢ ((((𝑓:𝐵⟶𝐴 ∧ Smo 𝑓) ∧ 𝑔:(cf‘𝐵)⟶𝐵) ∧ ∀𝑦 ∈ 𝐵 ∃𝑧 ∈ (cf‘𝐵)𝑦 ⊆ (𝑔‘𝑧)) → (∃𝑦 ∈ 𝐵 𝑥 ⊆ (𝑓‘𝑦) → ∃𝑧 ∈ (cf‘𝐵)𝑥 ⊆ ((𝑓 ∘ 𝑔)‘𝑧)))
39	38	ralimdv 3110	. . . . . . . . . . . . 13 ⊢ ((((𝑓:𝐵⟶𝐴 ∧ Smo 𝑓) ∧ 𝑔:(cf‘𝐵)⟶𝐵) ∧ ∀𝑦 ∈ 𝐵 ∃𝑧 ∈ (cf‘𝐵)𝑦 ⊆ (𝑔‘𝑧)) → (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ (𝑓‘𝑦) → ∀𝑥 ∈ 𝐴 ∃𝑧 ∈ (cf‘𝐵)𝑥 ⊆ ((𝑓 ∘ 𝑔)‘𝑧)))
40	39	impr 446	. . . . . . . . . . . 12 ⊢ ((((𝑓:𝐵⟶𝐴 ∧ Smo 𝑓) ∧ 𝑔:(cf‘𝐵)⟶𝐵) ∧ (∀𝑦 ∈ 𝐵 ∃𝑧 ∈ (cf‘𝐵)𝑦 ⊆ (𝑔‘𝑧) ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ (𝑓‘𝑦))) → ∀𝑥 ∈ 𝐴 ∃𝑧 ∈ (cf‘𝐵)𝑥 ⊆ ((𝑓 ∘ 𝑔)‘𝑧))
41		vex 3353	. . . . . . . . . . . . . 14 ⊢ 𝑓 ∈ V
42		vex 3353	. . . . . . . . . . . . . 14 ⊢ 𝑔 ∈ V
43	41, 42	coex 7318	. . . . . . . . . . . . 13 ⊢ (𝑓 ∘ 𝑔) ∈ V
44		feq1 6206	. . . . . . . . . . . . . 14 ⊢ (ℎ = (𝑓 ∘ 𝑔) → (ℎ:(cf‘𝐵)⟶𝐴 ↔ (𝑓 ∘ 𝑔):(cf‘𝐵)⟶𝐴))
45		fveq1 6376	. . . . . . . . . . . . . . . . 17 ⊢ (ℎ = (𝑓 ∘ 𝑔) → (ℎ‘𝑧) = ((𝑓 ∘ 𝑔)‘𝑧))
46	45	sseq2d 3795	. . . . . . . . . . . . . . . 16 ⊢ (ℎ = (𝑓 ∘ 𝑔) → (𝑥 ⊆ (ℎ‘𝑧) ↔ 𝑥 ⊆ ((𝑓 ∘ 𝑔)‘𝑧)))
47	46	rexbidv 3199	. . . . . . . . . . . . . . 15 ⊢ (ℎ = (𝑓 ∘ 𝑔) → (∃𝑧 ∈ (cf‘𝐵)𝑥 ⊆ (ℎ‘𝑧) ↔ ∃𝑧 ∈ (cf‘𝐵)𝑥 ⊆ ((𝑓 ∘ 𝑔)‘𝑧)))
48	47	ralbidv 3133	. . . . . . . . . . . . . 14 ⊢ (ℎ = (𝑓 ∘ 𝑔) → (∀𝑥 ∈ 𝐴 ∃𝑧 ∈ (cf‘𝐵)𝑥 ⊆ (ℎ‘𝑧) ↔ ∀𝑥 ∈ 𝐴 ∃𝑧 ∈ (cf‘𝐵)𝑥 ⊆ ((𝑓 ∘ 𝑔)‘𝑧)))
49	44, 48	anbi12d 624	. . . . . . . . . . . . 13 ⊢ (ℎ = (𝑓 ∘ 𝑔) → ((ℎ:(cf‘𝐵)⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃𝑧 ∈ (cf‘𝐵)𝑥 ⊆ (ℎ‘𝑧)) ↔ ((𝑓 ∘ 𝑔):(cf‘𝐵)⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃𝑧 ∈ (cf‘𝐵)𝑥 ⊆ ((𝑓 ∘ 𝑔)‘𝑧))))
50	43, 49	spcev 3453	. . . . . . . . . . . 12 ⊢ (((𝑓 ∘ 𝑔):(cf‘𝐵)⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃𝑧 ∈ (cf‘𝐵)𝑥 ⊆ ((𝑓 ∘ 𝑔)‘𝑧)) → ∃ℎ(ℎ:(cf‘𝐵)⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃𝑧 ∈ (cf‘𝐵)𝑥 ⊆ (ℎ‘𝑧)))
51	5, 40, 50	syl2anc 579	. . . . . . . . . . 11 ⊢ ((((𝑓:𝐵⟶𝐴 ∧ Smo 𝑓) ∧ 𝑔:(cf‘𝐵)⟶𝐵) ∧ (∀𝑦 ∈ 𝐵 ∃𝑧 ∈ (cf‘𝐵)𝑦 ⊆ (𝑔‘𝑧) ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ (𝑓‘𝑦))) → ∃ℎ(ℎ:(cf‘𝐵)⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃𝑧 ∈ (cf‘𝐵)𝑥 ⊆ (ℎ‘𝑧)))
52	51	exp43 427	. . . . . . . . . 10 ⊢ ((𝑓:𝐵⟶𝐴 ∧ Smo 𝑓) → (𝑔:(cf‘𝐵)⟶𝐵 → (∀𝑦 ∈ 𝐵 ∃𝑧 ∈ (cf‘𝐵)𝑦 ⊆ (𝑔‘𝑧) → (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ (𝑓‘𝑦) → ∃ℎ(ℎ:(cf‘𝐵)⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃𝑧 ∈ (cf‘𝐵)𝑥 ⊆ (ℎ‘𝑧))))))
53	52	com24 95	. . . . . . . . 9 ⊢ ((𝑓:𝐵⟶𝐴 ∧ Smo 𝑓) → (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ (𝑓‘𝑦) → (∀𝑦 ∈ 𝐵 ∃𝑧 ∈ (cf‘𝐵)𝑦 ⊆ (𝑔‘𝑧) → (𝑔:(cf‘𝐵)⟶𝐵 → ∃ℎ(ℎ:(cf‘𝐵)⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃𝑧 ∈ (cf‘𝐵)𝑥 ⊆ (ℎ‘𝑧))))))
54	53	3impia 1145	. . . . . . . 8 ⊢ ((𝑓:𝐵⟶𝐴 ∧ Smo 𝑓 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ (𝑓‘𝑦)) → (∀𝑦 ∈ 𝐵 ∃𝑧 ∈ (cf‘𝐵)𝑦 ⊆ (𝑔‘𝑧) → (𝑔:(cf‘𝐵)⟶𝐵 → ∃ℎ(ℎ:(cf‘𝐵)⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃𝑧 ∈ (cf‘𝐵)𝑥 ⊆ (ℎ‘𝑧)))))
55	54	exlimiv 2025	. . . . . . 7 ⊢ (∃𝑓(𝑓:𝐵⟶𝐴 ∧ Smo 𝑓 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ (𝑓‘𝑦)) → (∀𝑦 ∈ 𝐵 ∃𝑧 ∈ (cf‘𝐵)𝑦 ⊆ (𝑔‘𝑧) → (𝑔:(cf‘𝐵)⟶𝐵 → ∃ℎ(ℎ:(cf‘𝐵)⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃𝑧 ∈ (cf‘𝐵)𝑥 ⊆ (ℎ‘𝑧)))))
56	55	com13 88	. . . . . 6 ⊢ (𝑔:(cf‘𝐵)⟶𝐵 → (∀𝑦 ∈ 𝐵 ∃𝑧 ∈ (cf‘𝐵)𝑦 ⊆ (𝑔‘𝑧) → (∃𝑓(𝑓:𝐵⟶𝐴 ∧ Smo 𝑓 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ (𝑓‘𝑦)) → ∃ℎ(ℎ:(cf‘𝐵)⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃𝑧 ∈ (cf‘𝐵)𝑥 ⊆ (ℎ‘𝑧)))))
57	2, 56	syl 17	. . . . 5 ⊢ (𝑔:(cf‘𝐵)–1-1→𝐵 → (∀𝑦 ∈ 𝐵 ∃𝑧 ∈ (cf‘𝐵)𝑦 ⊆ (𝑔‘𝑧) → (∃𝑓(𝑓:𝐵⟶𝐴 ∧ Smo 𝑓 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ (𝑓‘𝑦)) → ∃ℎ(ℎ:(cf‘𝐵)⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃𝑧 ∈ (cf‘𝐵)𝑥 ⊆ (ℎ‘𝑧)))))
58	57	imp 395	. . . 4 ⊢ ((𝑔:(cf‘𝐵)–1-1→𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑧 ∈ (cf‘𝐵)𝑦 ⊆ (𝑔‘𝑧)) → (∃𝑓(𝑓:𝐵⟶𝐴 ∧ Smo 𝑓 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ (𝑓‘𝑦)) → ∃ℎ(ℎ:(cf‘𝐵)⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃𝑧 ∈ (cf‘𝐵)𝑥 ⊆ (ℎ‘𝑧))))
59	58	exlimiv 2025	. . 3 ⊢ (∃𝑔(𝑔:(cf‘𝐵)–1-1→𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑧 ∈ (cf‘𝐵)𝑦 ⊆ (𝑔‘𝑧)) → (∃𝑓(𝑓:𝐵⟶𝐴 ∧ Smo 𝑓 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ (𝑓‘𝑦)) → ∃ℎ(ℎ:(cf‘𝐵)⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃𝑧 ∈ (cf‘𝐵)𝑥 ⊆ (ℎ‘𝑧))))
60	1, 59	syl 17	. 2 ⊢ (𝐵 ∈ On → (∃𝑓(𝑓:𝐵⟶𝐴 ∧ Smo 𝑓 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ (𝑓‘𝑦)) → ∃ℎ(ℎ:(cf‘𝐵)⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃𝑧 ∈ (cf‘𝐵)𝑥 ⊆ (ℎ‘𝑧))))
61		cfon 9332	. . 3 ⊢ (cf‘𝐵) ∈ On
62		cfflb 9336	. . 3 ⊢ ((𝐴 ∈ On ∧ (cf‘𝐵) ∈ On) → (∃ℎ(ℎ:(cf‘𝐵)⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃𝑧 ∈ (cf‘𝐵)𝑥 ⊆ (ℎ‘𝑧)) → (cf‘𝐴) ⊆ (cf‘𝐵)))
63	61, 62	mpan2 682	. 2 ⊢ (𝐴 ∈ On → (∃ℎ(ℎ:(cf‘𝐵)⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃𝑧 ∈ (cf‘𝐵)𝑥 ⊆ (ℎ‘𝑧)) → (cf‘𝐴) ⊆ (cf‘𝐵)))
64	60, 63	sylan9r 504	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∃𝑓(𝑓:𝐵⟶𝐴 ∧ Smo 𝑓 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ (𝑓‘𝑦)) → (cf‘𝐴) ⊆ (cf‘𝐵)))

Syntax hints:   → wi 4   ↔ wb 197   ∧ wa 384   ∧ w3a 1107   = wceq 1652  ∃wex 1874   ∈ wcel 2155  ∀wral 3055  ∃wrex 3056   ⊆ wss 3734   ∘ ccom 5283  Oncon0 5910   Fn wfn 6065  ⟶wf 6066  –1-1→wf1 6067  ‘cfv 6070  Smo wsmo 7648  cfccf 9016

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-8 2157  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-13 2352  ax-ext 2743  ax-rep 4932  ax-sep 4943  ax-nul 4951  ax-pow 5003  ax-pr 5064  ax-un 7149

This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-3or 1108  df-3an 1109  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-mo 2565  df-eu 2582  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-ne 2938  df-ral 3060  df-rex 3061  df-reu 3062  df-rmo 3063  df-rab 3064  df-v 3352  df-sbc 3599  df-csb 3694  df-dif 3737  df-un 3739  df-in 3741  df-ss 3748  df-pss 3750  df-nul 4082  df-if 4246  df-pw 4319  df-sn 4337  df-pr 4339  df-tp 4341  df-op 4343  df-uni 4597  df-int 4636  df-iun 4680  df-br 4812  df-opab 4874  df-mpt 4891  df-tr 4914  df-id 5187  df-eprel 5192  df-po 5200  df-so 5201  df-fr 5238  df-se 5239  df-we 5240  df-xp 5285  df-rel 5286  df-cnv 5287  df-co 5288  df-dm 5289  df-rn 5290  df-res 5291  df-ima 5292  df-pred 5867  df-ord 5913  df-on 5914  df-suc 5916  df-iota 6033  df-fun 6072  df-fn 6073  df-f 6074  df-f1 6075  df-fo 6076  df-f1o 6077  df-fv 6078  df-isom 6079  df-riota 6805  df-ov 6847  df-oprab 6848  df-mpt2 6849  df-1st 7368  df-2nd 7369  df-wrecs 7612  df-smo 7649  df-recs 7674  df-er 7949  df-map 8064  df-en 8163  df-dom 8164  df-sdom 8165  df-card 9018  df-cf 9020  df-acn 9021

This theorem is referenced by:  cfcof  9351  cfidm  9352