Theorem cfcoflem 9851

Description: Lemma for cfcof 9853, 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 9837	. . 3 ⊢ (𝐵 ∈ On → ∃𝑔(𝑔:(cf‘𝐵)–1-1→𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑧 ∈ (cf‘𝐵)𝑦 ⊆ (𝑔‘𝑧)))
2		f1f 6593	. . . . . 6 ⊢ (𝑔:(cf‘𝐵)–1-1→𝐵 → 𝑔:(cf‘𝐵)⟶𝐵)
3		fco 6547	. . . . . . . . . . . . 13 ⊢ ((𝑓:𝐵⟶𝐴 ∧ 𝑔:(cf‘𝐵)⟶𝐵) → (𝑓 ∘ 𝑔):(cf‘𝐵)⟶𝐴)
4	3	adantlr 715	. . . . . . . . . . . 12 ⊢ (((𝑓:𝐵⟶𝐴 ∧ Smo 𝑓) ∧ 𝑔:(cf‘𝐵)⟶𝐵) → (𝑓 ∘ 𝑔):(cf‘𝐵)⟶𝐴)
5		r19.29 3166	. . . . . . . . . . . . . . . 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)))
6		ffvelrn 6880	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑔:(cf‘𝐵)⟶𝐵 ∧ 𝑧 ∈ (cf‘𝐵)) → (𝑔‘𝑧) ∈ 𝐵)
7		ffn 6523	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑓:𝐵⟶𝐴 → 𝑓 Fn 𝐵)
8		smoword 8081	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (((𝑓 Fn 𝐵 ∧ Smo 𝑓) ∧ (𝑦 ∈ 𝐵 ∧ (𝑔‘𝑧) ∈ 𝐵)) → (𝑦 ⊆ (𝑔‘𝑧) ↔ (𝑓‘𝑦) ⊆ (𝑓‘(𝑔‘𝑧))))
9	8	biimpd 232	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (((𝑓 Fn 𝐵 ∧ Smo 𝑓) ∧ (𝑦 ∈ 𝐵 ∧ (𝑔‘𝑧) ∈ 𝐵)) → (𝑦 ⊆ (𝑔‘𝑧) → (𝑓‘𝑦) ⊆ (𝑓‘(𝑔‘𝑧))))
10	9	exp32 424	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝑓 Fn 𝐵 ∧ Smo 𝑓) → (𝑦 ∈ 𝐵 → ((𝑔‘𝑧) ∈ 𝐵 → (𝑦 ⊆ (𝑔‘𝑧) → (𝑓‘𝑦) ⊆ (𝑓‘(𝑔‘𝑧))))))
11	7, 10	sylan 583	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑓:𝐵⟶𝐴 ∧ Smo 𝑓) → (𝑦 ∈ 𝐵 → ((𝑔‘𝑧) ∈ 𝐵 → (𝑦 ⊆ (𝑔‘𝑧) → (𝑓‘𝑦) ⊆ (𝑓‘(𝑔‘𝑧))))))
12	6, 11	syl7 74	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑓:𝐵⟶𝐴 ∧ Smo 𝑓) → (𝑦 ∈ 𝐵 → ((𝑔:(cf‘𝐵)⟶𝐵 ∧ 𝑧 ∈ (cf‘𝐵)) → (𝑦 ⊆ (𝑔‘𝑧) → (𝑓‘𝑦) ⊆ (𝑓‘(𝑔‘𝑧))))))
13	12	com23 86	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑓:𝐵⟶𝐴 ∧ Smo 𝑓) → ((𝑔:(cf‘𝐵)⟶𝐵 ∧ 𝑧 ∈ (cf‘𝐵)) → (𝑦 ∈ 𝐵 → (𝑦 ⊆ (𝑔‘𝑧) → (𝑓‘𝑦) ⊆ (𝑓‘(𝑔‘𝑧))))))
14	13	expdimp 456	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ (((𝑓:𝐵⟶𝐴 ∧ Smo 𝑓) ∧ 𝑔:(cf‘𝐵)⟶𝐵) → (𝑧 ∈ (cf‘𝐵) → (𝑦 ∈ 𝐵 → (𝑦 ⊆ (𝑔‘𝑧) → (𝑓‘𝑦) ⊆ (𝑓‘(𝑔‘𝑧))))))
15	14	3imp2 1351	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝑓:𝐵⟶𝐴 ∧ Smo 𝑓) ∧ 𝑔:(cf‘𝐵)⟶𝐵) ∧ (𝑧 ∈ (cf‘𝐵) ∧ 𝑦 ∈ 𝐵 ∧ 𝑦 ⊆ (𝑔‘𝑧))) → (𝑓‘𝑦) ⊆ (𝑓‘(𝑔‘𝑧)))
16		sstr2 3894	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑥 ⊆ (𝑓‘𝑦) → ((𝑓‘𝑦) ⊆ (𝑓‘(𝑔‘𝑧)) → 𝑥 ⊆ (𝑓‘(𝑔‘𝑧))))
17	15, 16	syl5com 31	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
18		fvco3 6788	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑔:(cf‘𝐵)⟶𝐵 ∧ 𝑧 ∈ (cf‘𝐵)) → ((𝑓 ∘ 𝑔)‘𝑧) = (𝑓‘(𝑔‘𝑧)))
19	18	sseq2d 3919	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑔:(cf‘𝐵)⟶𝐵 ∧ 𝑧 ∈ (cf‘𝐵)) → (𝑥 ⊆ ((𝑓 ∘ 𝑔)‘𝑧) ↔ 𝑥 ⊆ (𝑓‘(𝑔‘𝑧))))
20	19	adantll 714	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝑓:𝐵⟶𝐴 ∧ Smo 𝑓) ∧ 𝑔:(cf‘𝐵)⟶𝐵) ∧ 𝑧 ∈ (cf‘𝐵)) → (𝑥 ⊆ ((𝑓 ∘ 𝑔)‘𝑧) ↔ 𝑥 ⊆ (𝑓‘(𝑔‘𝑧))))
21	20	3ad2antr1 1190	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ⊢ ((((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))))
22	17, 21	sylibrd 262	. . . . . . . . . . . . . . . . . . . . . . . . . . . 28 ⊢ ((((𝑓:𝐵⟶𝐴 ∧ Smo 𝑓) ∧ 𝑔:(cf‘𝐵)⟶𝐵) ∧ (𝑧 ∈ (cf‘𝐵) ∧ 𝑦 ∈ 𝐵 ∧ 𝑦 ⊆ (𝑔‘𝑧))) → (𝑥 ⊆ (𝑓‘𝑦) → 𝑥 ⊆ ((𝑓 ∘ 𝑔)‘𝑧)))
23	22	expcom 417	. . . . . . . . . . . . . . . . . . . . . . . . . . 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))))
24	23	3expia 1123	. . . . . . . . . . . . . . . . . . . . . . . . . 26 ⊢ ((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	com4t 93	. . . . . . . . . . . . . . . . . . . . . . . . 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)))))
26	25	imp 410	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((((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	expcomd 420	. . . . . . . . . . . . . . . . . . . . . . 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)))))
28	27	imp31 421	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((((((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	reximdva 3183	. . . . . . . . . . . . . . . . . . . . 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)))
30	29	exp31 423	. . . . . . . . . . . . . . . . . . . 20 ⊢ (((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	com34 91	. . . . . . . . . . . . . . . . . . 19 ⊢ (((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)))))
32	31	impcomd 415	. . . . . . . . . . . . . . . . . 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))))
33	32	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))))
34	33	rexlimdv 3192	. . . . . . . . . . . . . . . 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)))
35	5, 34	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)))
36	35	expdimp 456	. . . . . . . . . . . . . 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)))
37	36	ralimdv 3091	. . . . . . . . . . . . 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)))
38	37	impr 458	. . . . . . . . . . . 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))
39		vex 3402	. . . . . . . . . . . . . 14 ⊢ 𝑓 ∈ V
40		vex 3402	. . . . . . . . . . . . . 14 ⊢ 𝑔 ∈ V
41	39, 40	coex 7686	. . . . . . . . . . . . 13 ⊢ (𝑓 ∘ 𝑔) ∈ V
42		feq1 6504	. . . . . . . . . . . . . 14 ⊢ (ℎ = (𝑓 ∘ 𝑔) → (ℎ:(cf‘𝐵)⟶𝐴 ↔ (𝑓 ∘ 𝑔):(cf‘𝐵)⟶𝐴))
43		fveq1 6694	. . . . . . . . . . . . . . . . 17 ⊢ (ℎ = (𝑓 ∘ 𝑔) → (ℎ‘𝑧) = ((𝑓 ∘ 𝑔)‘𝑧))
44	43	sseq2d 3919	. . . . . . . . . . . . . . . 16 ⊢ (ℎ = (𝑓 ∘ 𝑔) → (𝑥 ⊆ (ℎ‘𝑧) ↔ 𝑥 ⊆ ((𝑓 ∘ 𝑔)‘𝑧)))
45	44	rexbidv 3206	. . . . . . . . . . . . . . 15 ⊢ (ℎ = (𝑓 ∘ 𝑔) → (∃𝑧 ∈ (cf‘𝐵)𝑥 ⊆ (ℎ‘𝑧) ↔ ∃𝑧 ∈ (cf‘𝐵)𝑥 ⊆ ((𝑓 ∘ 𝑔)‘𝑧)))
46	45	ralbidv 3108	. . . . . . . . . . . . . 14 ⊢ (ℎ = (𝑓 ∘ 𝑔) → (∀𝑥 ∈ 𝐴 ∃𝑧 ∈ (cf‘𝐵)𝑥 ⊆ (ℎ‘𝑧) ↔ ∀𝑥 ∈ 𝐴 ∃𝑧 ∈ (cf‘𝐵)𝑥 ⊆ ((𝑓 ∘ 𝑔)‘𝑧)))
47	42, 46	anbi12d 634	. . . . . . . . . . . . 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))))
48	41, 47	spcev 3511	. . . . . . . . . . . 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)))
49	4, 38, 48	syl2an2r 685	. . . . . . . . . . 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)))
50	49	exp43 440	. . . . . . . . . 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))))))
51	50	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))))))
52	51	3impia 1119	. . . . . . . 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)))))
53	52	exlimiv 1938	. . . . . . 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)))))
54	53	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)))))
55	2, 54	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)))))
56	55	imp 410	. . . 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))))
57	56	exlimiv 1938	. . 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))))
58	1, 57	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))))
59		cfon 9834	. . 3 ⊢ (cf‘𝐵) ∈ On
60		cfflb 9838	. . 3 ⊢ ((𝐴 ∈ On ∧ (cf‘𝐵) ∈ On) → (∃ℎ(ℎ:(cf‘𝐵)⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃𝑧 ∈ (cf‘𝐵)𝑥 ⊆ (ℎ‘𝑧)) → (cf‘𝐴) ⊆ (cf‘𝐵)))
61	59, 60	mpan2 691	. 2 ⊢ (𝐴 ∈ On → (∃ℎ(ℎ:(cf‘𝐵)⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃𝑧 ∈ (cf‘𝐵)𝑥 ⊆ (ℎ‘𝑧)) → (cf‘𝐴) ⊆ (cf‘𝐵)))
62	58, 61	sylan9r 512	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 209   ∧ wa 399   ∧ w3a 1089   = wceq 1543  ∃wex 1787   ∈ wcel 2112  ∀wral 3051  ∃wrex 3052   ⊆ wss 3853   ∘ ccom 5540  Oncon0 6191   Fn wfn 6353  ⟶wf 6354  –1-1→wf1 6355  ‘cfv 6358  Smo wsmo 8060  cfccf 9518

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2018  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2160  ax-12 2177  ax-ext 2708  ax-rep 5164  ax-sep 5177  ax-nul 5184  ax-pow 5243  ax-pr 5307  ax-un 7501

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 848  df-3or 1090  df-3an 1091  df-tru 1546  df-fal 1556  df-ex 1788  df-nf 1792  df-sb 2073  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2728  df-clel 2809  df-nfc 2879  df-ne 2933  df-ral 3056  df-rex 3057  df-reu 3058  df-rmo 3059  df-rab 3060  df-v 3400  df-sbc 3684  df-csb 3799  df-dif 3856  df-un 3858  df-in 3860  df-ss 3870  df-pss 3872  df-nul 4224  df-if 4426  df-pw 4501  df-sn 4528  df-pr 4530  df-tp 4532  df-op 4534  df-uni 4806  df-int 4846  df-iun 4892  df-br 5040  df-opab 5102  df-mpt 5121  df-tr 5147  df-id 5440  df-eprel 5445  df-po 5453  df-so 5454  df-fr 5494  df-se 5495  df-we 5496  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-pred 6140  df-ord 6194  df-on 6195  df-suc 6197  df-iota 6316  df-fun 6360  df-fn 6361  df-f 6362  df-f1 6363  df-fo 6364  df-f1o 6365  df-fv 6366  df-isom 6367  df-riota 7148  df-ov 7194  df-oprab 7195  df-mpo 7196  df-1st 7739  df-2nd 7740  df-wrecs 8025  df-smo 8061  df-recs 8086  df-er 8369  df-map 8488  df-en 8605  df-dom 8606  df-sdom 8607  df-card 9520  df-cf 9522  df-acn 9523

This theorem is referenced by:  cfcof  9853  cfidm  9854