Theorem cfcoflem 9094

Description: Lemma for cfcof 9096, 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 9080	. . 3 ⊢ (𝐵 ∈ On → ∃𝑔(𝑔:(cf‘𝐵)–1-1→𝐵 ∧ ∀𝑦 ∈ 𝐵 ∃𝑧 ∈ (cf‘𝐵)𝑦 ⊆ (𝑔‘𝑧)))
2		f1f 6101	. . . . . 6 ⊢ (𝑔:(cf‘𝐵)–1-1→𝐵 → 𝑔:(cf‘𝐵)⟶𝐵)
3		fco 6058	. . . . . . . . . . . . . 14 ⊢ ((𝑓:𝐵⟶𝐴 ∧ 𝑔:(cf‘𝐵)⟶𝐵) → (𝑓 ∘ 𝑔):(cf‘𝐵)⟶𝐴)
4	3	adantlr 751	. . . . . . . . . . . . 13 ⊢ (((𝑓:𝐵⟶𝐴 ∧ Smo 𝑓) ∧ 𝑔:(cf‘𝐵)⟶𝐵) → (𝑓 ∘ 𝑔):(cf‘𝐵)⟶𝐴)
5	4	adantr 481	. . . . . . . . . . . 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 3072	. . . . . . . . . . . . . . . 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 6357	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 ⊢ ((𝑔:(cf‘𝐵)⟶𝐵 ∧ 𝑧 ∈ (cf‘𝐵)) → (𝑔‘𝑧) ∈ 𝐵)
8		ffn 6045	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ (𝑓:𝐵⟶𝐴 → 𝑓 Fn 𝐵)
9		smoword 7463	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 ⊢ (((𝑓 Fn 𝐵 ∧ Smo 𝑓) ∧ (𝑦 ∈ 𝐵 ∧ (𝑔‘𝑧) ∈ 𝐵)) → (𝑦 ⊆ (𝑔‘𝑧) ↔ (𝑓‘𝑦) ⊆ (𝑓‘(𝑔‘𝑧))))
10	9	biimpd 219	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 ⊢ (((𝑓 Fn 𝐵 ∧ Smo 𝑓) ∧ (𝑦 ∈ 𝐵 ∧ (𝑔‘𝑧) ∈ 𝐵)) → (𝑦 ⊆ (𝑔‘𝑧) → (𝑓‘𝑦) ⊆ (𝑓‘(𝑔‘𝑧))))
11	10	exp32 631	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 ⊢ ((𝑓 Fn 𝐵 ∧ Smo 𝑓) → (𝑦 ∈ 𝐵 → ((𝑔‘𝑧) ∈ 𝐵 → (𝑦 ⊆ (𝑔‘𝑧) → (𝑓‘𝑦) ⊆ (𝑓‘(𝑔‘𝑧))))))
12	8, 11	sylan 488	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 453	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1282	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 3610	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 6275	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 ⊢ ((𝑔:(cf‘𝐵)⟶𝐵 ∧ 𝑧 ∈ (cf‘𝐵)) → ((𝑓 ∘ 𝑔)‘𝑧) = (𝑓‘(𝑔‘𝑧)))
20	19	sseq2d 3633	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 ⊢ ((𝑔:(cf‘𝐵)⟶𝐵 ∧ 𝑧 ∈ (cf‘𝐵)) → (𝑥 ⊆ ((𝑓 ∘ 𝑔)‘𝑧) ↔ 𝑥 ⊆ (𝑓‘(𝑔‘𝑧))))
21	20	adantll 750	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 31 ⊢ ((((𝑓:𝐵⟶𝐴 ∧ Smo 𝑓) ∧ 𝑔:(cf‘𝐵)⟶𝐵) ∧ 𝑧 ∈ (cf‘𝐵)) → (𝑥 ⊆ ((𝑓 ∘ 𝑔)‘𝑧) ↔ 𝑥 ⊆ (𝑓‘(𝑔‘𝑧))))
22	21	3ad2antr1 1226	. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 249	. . . . . . . . . . . . . . . . . . . . . . . . . . . . 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 451	. . . . . . . . . . . . . . . . . . . . . . . . . . . 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 1267	. . . . . . . . . . . . . . . . . . . . . . . . . . 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 445	. . . . . . . . . . . . . . . . . . . . . . . . 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 454	. . . . . . . . . . . . . . . . . . . . . . . 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 448	. . . . . . . . . . . . . . . . . . . . . . 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 3017	. . . . . . . . . . . . . . . . . . . . . 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 630	. . . . . . . . . . . . . . . . . . . . 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 447	. . . . . . . . . . . . . . . . . 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 3030	. . . . . . . . . . . . . . . 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 453	. . . . . . . . . . . . . 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 2963	. . . . . . . . . . . . 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 649	. . . . . . . . . . . 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 3203	. . . . . . . . . . . . . 14 ⊢ 𝑓 ∈ V
42		vex 3203	. . . . . . . . . . . . . 14 ⊢ 𝑔 ∈ V
43	41, 42	coex 7118	. . . . . . . . . . . . 13 ⊢ (𝑓 ∘ 𝑔) ∈ V
44		feq1 6026	. . . . . . . . . . . . . 14 ⊢ (ℎ = (𝑓 ∘ 𝑔) → (ℎ:(cf‘𝐵)⟶𝐴 ↔ (𝑓 ∘ 𝑔):(cf‘𝐵)⟶𝐴))
45		fveq1 6190	. . . . . . . . . . . . . . . . 17 ⊢ (ℎ = (𝑓 ∘ 𝑔) → (ℎ‘𝑧) = ((𝑓 ∘ 𝑔)‘𝑧))
46	45	sseq2d 3633	. . . . . . . . . . . . . . . 16 ⊢ (ℎ = (𝑓 ∘ 𝑔) → (𝑥 ⊆ (ℎ‘𝑧) ↔ 𝑥 ⊆ ((𝑓 ∘ 𝑔)‘𝑧)))
47	46	rexbidv 3052	. . . . . . . . . . . . . . 15 ⊢ (ℎ = (𝑓 ∘ 𝑔) → (∃𝑧 ∈ (cf‘𝐵)𝑥 ⊆ (ℎ‘𝑧) ↔ ∃𝑧 ∈ (cf‘𝐵)𝑥 ⊆ ((𝑓 ∘ 𝑔)‘𝑧)))
48	47	ralbidv 2986	. . . . . . . . . . . . . 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 747	. . . . . . . . . . . . 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 3300	. . . . . . . . . . . 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 693	. . . . . . . . . . 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 640	. . . . . . . . . 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 1261	. . . . . . . 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 1858	. . . . . . 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 445	. . . 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 1858	. . 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 9077	. . 3 ⊢ (cf‘𝐵) ∈ On
62		cfflb 9081	. . 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 707	. 2 ⊢ (𝐴 ∈ On → (∃ℎ(ℎ:(cf‘𝐵)⟶𝐴 ∧ ∀𝑥 ∈ 𝐴 ∃𝑧 ∈ (cf‘𝐵)𝑥 ⊆ (ℎ‘𝑧)) → (cf‘𝐴) ⊆ (cf‘𝐵)))
64	60, 63	sylan9r 690	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 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483  ∃wex 1704   ∈ wcel 1990  ∀wral 2912  ∃wrex 2913   ⊆ wss 3574   ∘ ccom 5118  Oncon0 5723   Fn wfn 5883  ⟶wf 5884  –1-1→wf1 5885  ‘cfv 5888  Smo wsmo 7442  cfccf 8763

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rmo 2920  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-int 4476  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-ord 5726  df-on 5727  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-isom 5897  df-riota 6611  df-ov 6653  df-oprab 6654  df-mpt2 6655  df-1st 7168  df-2nd 7169  df-wrecs 7407  df-smo 7443  df-recs 7468  df-er 7742  df-map 7859  df-en 7956  df-dom 7957  df-sdom 7958  df-card 8765  df-cf 8767  df-acn 8768

This theorem is referenced by:  cfcof  9096  cfidm  9097