Theorem cfcoflem 10277

Description: Lemma for cfcof 10279, 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 10263	. . 3 ⊢ (𝐵 ∈ On → ∃𝑔(𝑔:(cf‘𝐵)–1-1→𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑧 ∈ (cf‘𝐵)𝑦 ⊆ (𝑔‘𝑧)))
2		f1f 6772	. . . . . 6 ⊢ (𝑔:(cf‘𝐵)–1-1→𝐵 → 𝑔:(cf‘𝐵)⟶𝐵)
3		fco 6728	. . . . . . . . . . . . 13 ⊢ ((𝑓:𝐵⟶𝐴 ∧ 𝑔:(cf‘𝐵)⟶𝐵) → (𝑓 ∘ 𝑔):(cf‘𝐵)⟶𝐴)
4	3	adantlr 728	. . . . . . . . . . . 12 ⊢ (((𝑓:𝐵⟶𝐴 ∧ Smo 𝑓) ∧ 𝑔:(cf‘𝐵)⟶𝐵) → (𝑓 ∘ 𝑔):(cf‘𝐵)⟶𝐴)
5		r19.29 3125	. . . . . . . . . . . . . . . 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		ffvelcdm 7075	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑔:(cf‘𝐵)⟶𝐵 ∧ 𝑧 ∈ (cf‘𝐵)) → (𝑔‘𝑧) ∈ 𝐵)
7		ffn 6703	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ (𝑓:𝐵⟶𝐴 → 𝑓 Fn 𝐵)
8		smoword 8356	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 426	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝑓 Fn 𝐵 ∧ Smo 𝑓) → (𝑦 ∈ 𝐵 → ((𝑔‘𝑧) ∈ 𝐵 → (𝑦 ⊆ (𝑔‘𝑧) → (𝑓‘𝑦) ⊆ (𝑓‘(𝑔‘𝑧))))))
11	7, 10	sylan 592	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34 ⊢ ((𝑓:𝐵⟶𝐴 ∧ Smo 𝑓) → (𝑦 ∈ 𝐵 → ((𝑔‘𝑧) ∈ 𝐵 → (𝑦 ⊆ (𝑔‘𝑧) → (𝑓‘𝑦) ⊆ (𝑓‘(𝑔‘𝑧))))))
12	6, 11	syl7 75	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 87	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 458	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1368	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3938	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ (𝑥 ⊆ (𝑓‘𝑦) → ((𝑓‘𝑦) ⊆ (𝑓‘(𝑔‘𝑧)) → 𝑥 ⊆ (𝑓‘(𝑔‘𝑧))))
17	15, 16	syl5com 32	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6979	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑔:(cf‘𝐵)⟶𝐵 ∧ 𝑧 ∈ (cf‘𝐵)) → ((𝑓 ∘ 𝑔)‘𝑧) = (𝑓‘(𝑔‘𝑧)))
19	18	sseq2d 3963	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((𝑔:(cf‘𝐵)⟶𝐵 ∧ 𝑧 ∈ (cf‘𝐵)) → (𝑥 ⊆ ((𝑓 ∘ 𝑔)‘𝑧) ↔ 𝑥 ⊆ (𝑓‘(𝑔‘𝑧))))
20	19	adantll 727	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 ⊢ ((((𝑓:𝐵⟶𝐴 ∧ Smo 𝑓) ∧ 𝑔:(cf‘𝐵)⟶𝐵) ∧ 𝑧 ∈ (cf‘𝐵)) → (𝑥 ⊆ ((𝑓 ∘ 𝑔)‘𝑧) ↔ 𝑥 ⊆ (𝑓‘(𝑔‘𝑧))))
21	20	3ad2antr1 1207	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 419	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 1139	. . . . . . . . . . . . . . . . . . . . . . . . . 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 94	. . . . . . . . . . . . . . . . . . . . . . . . 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 412	. . . . . . . . . . . . . . . . . . . . . . . 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 422	. . . . . . . . . . . . . . . . . . . . . . 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 423	. . . . . . . . . . . . . . . . . . . . . 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 3175	. . . . . . . . . . . . . . . . . . . . 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 425	. . . . . . . . . . . . . . . . . . . 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 92	. . . . . . . . . . . . . . . . . . 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 417	. . . . . . . . . . . . . . . . . 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 87	. . . . . . . . . . . . . . . . 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 3161	. . . . . . . . . . . . . . . 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 35	. . . . . . . . . . . . . . 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 458	. . . . . . . . . . . . . 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 3176	. . . . . . . . . . . . 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 460	. . . . . . . . . . . 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 3454	. . . . . . . . . . . . . 14 ⊢ 𝑓 ∈ V
40		vex 3454	. . . . . . . . . . . . . 14 ⊢ 𝑔 ∈ V
41	39, 40	coex 7928	. . . . . . . . . . . . 13 ⊢ (𝑓 ∘ 𝑔) ∈ V
42		feq1 6681	. . . . . . . . . . . . . 14 ⊢ (ℎ = (𝑓 ∘ 𝑔) → (ℎ:(cf‘𝐵)⟶𝐴 ↔ (𝑓 ∘ 𝑔):(cf‘𝐵)⟶𝐴))
43		fveq1 6878	. . . . . . . . . . . . . . . . 17 ⊢ (ℎ = (𝑓 ∘ 𝑔) → (ℎ‘𝑧) = ((𝑓 ∘ 𝑔)‘𝑧))
44	43	sseq2d 3963	. . . . . . . . . . . . . . . 16 ⊢ (ℎ = (𝑓 ∘ 𝑔) → (𝑥 ⊆ (ℎ‘𝑧) ↔ 𝑥 ⊆ ((𝑓 ∘ 𝑔)‘𝑧)))
45	44	rexbidv 3186	. . . . . . . . . . . . . . 15 ⊢ (ℎ = (𝑓 ∘ 𝑔) → (∃𝑧 ∈ (cf‘𝐵)𝑥 ⊆ (ℎ‘𝑧) ↔ ∃𝑧 ∈ (cf‘𝐵)𝑥 ⊆ ((𝑓 ∘ 𝑔)‘𝑧)))
46	45	ralbidv 3185	. . . . . . . . . . . . . 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 644	. . . . . . . . . . . . 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 3560	. . . . . . . . . . . 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 698	. . . . . . . . . . 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 442	. . . . . . . . . 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 96	. . . . . . . . 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 1135	. . . . . . . 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 1963	. . . . . . 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 89	. . . . . 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 18	. . . . 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 412	. . . 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 1963	. . 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 18	. 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 10259	. . 3 ⊢ (cf‘𝐵) ∈ On
60		cfflb 10264	. . 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 704	. 2 ⊢ (𝐴 ∈ On → (∃ℎ(ℎ:(cf‘𝐵)⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃𝑧 ∈ (cf‘𝐵)𝑥 ⊆ (ℎ‘𝑧)) → (cf‘𝐴) ⊆ (cf‘𝐵)))
62	58, 61	sylan9r 518	1 ⊢ ((𝐴 ∈ On ∧ 𝐵 ∈ On) → (∃𝑓(𝑓:𝐵⟶𝐴 ∧ Smo 𝑓 ∧ ∀𝑥 ∈ 𝐴 ∃𝑦 ∈ 𝐵 𝑥 ⊆ (𝑓‘𝑦)) → (cf‘𝐴) ⊆ (cf‘𝐵)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2145  ∀wral 3076  ∃wrex 3086   ⊆ wss 3899   ∘ ccom 5659  Oncon0 6357   Fn wfn 6528  ⟶wf 6529  –1-1→wf1 6530  ‘cfv 6533  Smo wsmo 8335  cfccf 9945

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-rep 5232  ax-sep 5251  ax-nul 5263  ax-pow 5330  ax-pr 5398  ax-un 7737

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3740  df-csb 3848  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5104  df-opab 5168  df-mpt 5187  df-tr 5213  df-id 5550  df-eprel 5555  df-po 5563  df-so 5564  df-fr 5608  df-se 5609  df-we 5610  df-xp 5661  df-rel 5662  df-cnv 5663  df-co 5664  df-dm 5665  df-rn 5666  df-res 5667  df-ima 5668  df-pred 6299  df-ord 6360  df-on 6361  df-suc 6363  df-iota 6489  df-fun 6535  df-fn 6536  df-f 6537  df-f1 6538  df-fo 6539  df-f1o 6540  df-fv 6541  df-isom 6542  df-riota 7371  df-ov 7417  df-oprab 7418  df-mpo 7419  df-1st 7987  df-2nd 7988  df-frecs 8281  df-wrecs 8312  df-smo 8336  df-recs 8361  df-er 8699  df-map 8831  df-en 8956  df-dom 8957  df-sdom 8958  df-card 9947  df-cf 9949  df-acn 9950

This theorem is used by:  cfcof  10279  cfidm  10280