Theorem konigthlem 9643

Description: Lemma for konigth 9644. (Contributed by Mario Carneiro, 22-Feb-2013.)

Hypotheses

Ref	Expression
konigth.1	⊢ 𝐴 ∈ V
konigth.2	⊢ 𝑆 = ∪ 𝑖 ∈ 𝐴 (𝑀‘𝑖)
konigth.3	⊢ 𝑃 = X𝑖 ∈ 𝐴 (𝑁‘𝑖)
konigth.4	⊢ 𝐷 = (𝑖 ∈ 𝐴 ↦ (𝑎 ∈ (𝑀‘𝑖) ↦ ((𝑓‘𝑎)‘𝑖)))
konigth.5	⊢ 𝐸 = (𝑖 ∈ 𝐴 ↦ (𝑒‘𝑖))

Assertion

Ref	Expression
konigthlem	⊢ (∀𝑖 ∈ 𝐴 (𝑀‘𝑖) ≺ (𝑁‘𝑖) → 𝑆 ≺ 𝑃)

Distinct variable groups:   𝐴,𝑎,𝑒,𝑓,𝑖   𝐷,𝑎,𝑒   𝐸,𝑎,𝑖   𝑀,𝑎,𝑓   𝑁,𝑎,𝑒,𝑓   𝑃,𝑎,𝑒,𝑓   𝑆,𝑎,𝑒,𝑓

Allowed substitution hints:   𝐷(𝑓,𝑖)   𝑃(𝑖)   𝑆(𝑖)   𝐸(𝑒,𝑓)   𝑀(𝑒,𝑖)   𝑁(𝑖)

Proof of Theorem konigthlem

Step	Hyp	Ref	Expression
1		fvex 6388	. . . . . . . . 9 ⊢ (𝑀‘𝑖) ∈ V
2		fvex 6388	. . . . . . . . . . 11 ⊢ ((𝑓‘𝑎)‘𝑖) ∈ V
3		eqid 2765	. . . . . . . . . . 11 ⊢ (𝑎 ∈ (𝑀‘𝑖) ↦ ((𝑓‘𝑎)‘𝑖)) = (𝑎 ∈ (𝑀‘𝑖) ↦ ((𝑓‘𝑎)‘𝑖))
4	2, 3	fnmpti 6200	. . . . . . . . . 10 ⊢ (𝑎 ∈ (𝑀‘𝑖) ↦ ((𝑓‘𝑎)‘𝑖)) Fn (𝑀‘𝑖)
5	1	mptex 6679	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ (𝑀‘𝑖) ↦ ((𝑓‘𝑎)‘𝑖)) ∈ V
6		konigth.4	. . . . . . . . . . . . 13 ⊢ 𝐷 = (𝑖 ∈ 𝐴 ↦ (𝑎 ∈ (𝑀‘𝑖) ↦ ((𝑓‘𝑎)‘𝑖)))
7	6	fvmpt2 6480	. . . . . . . . . . . 12 ⊢ ((𝑖 ∈ 𝐴 ∧ (𝑎 ∈ (𝑀‘𝑖) ↦ ((𝑓‘𝑎)‘𝑖)) ∈ V) → (𝐷‘𝑖) = (𝑎 ∈ (𝑀‘𝑖) ↦ ((𝑓‘𝑎)‘𝑖)))
8	5, 7	mpan2 682	. . . . . . . . . . 11 ⊢ (𝑖 ∈ 𝐴 → (𝐷‘𝑖) = (𝑎 ∈ (𝑀‘𝑖) ↦ ((𝑓‘𝑎)‘𝑖)))
9	8	fneq1d 6159	. . . . . . . . . 10 ⊢ (𝑖 ∈ 𝐴 → ((𝐷‘𝑖) Fn (𝑀‘𝑖) ↔ (𝑎 ∈ (𝑀‘𝑖) ↦ ((𝑓‘𝑎)‘𝑖)) Fn (𝑀‘𝑖)))
10	4, 9	mpbiri 249	. . . . . . . . 9 ⊢ (𝑖 ∈ 𝐴 → (𝐷‘𝑖) Fn (𝑀‘𝑖))
11		fnrndomg 9611	. . . . . . . . 9 ⊢ ((𝑀‘𝑖) ∈ V → ((𝐷‘𝑖) Fn (𝑀‘𝑖) → ran (𝐷‘𝑖) ≼ (𝑀‘𝑖)))
12	1, 10, 11	mpsyl 68	. . . . . . . 8 ⊢ (𝑖 ∈ 𝐴 → ran (𝐷‘𝑖) ≼ (𝑀‘𝑖))
13		domsdomtr 8302	. . . . . . . 8 ⊢ ((ran (𝐷‘𝑖) ≼ (𝑀‘𝑖) ∧ (𝑀‘𝑖) ≺ (𝑁‘𝑖)) → ran (𝐷‘𝑖) ≺ (𝑁‘𝑖))
14	12, 13	sylan 575	. . . . . . 7 ⊢ ((𝑖 ∈ 𝐴 ∧ (𝑀‘𝑖) ≺ (𝑁‘𝑖)) → ran (𝐷‘𝑖) ≺ (𝑁‘𝑖))
15		sdomdif 8315	. . . . . . 7 ⊢ (ran (𝐷‘𝑖) ≺ (𝑁‘𝑖) → ((𝑁‘𝑖) ∖ ran (𝐷‘𝑖)) ≠ ∅)
16	14, 15	syl 17	. . . . . 6 ⊢ ((𝑖 ∈ 𝐴 ∧ (𝑀‘𝑖) ≺ (𝑁‘𝑖)) → ((𝑁‘𝑖) ∖ ran (𝐷‘𝑖)) ≠ ∅)
17	16	ralimiaa 3098	. . . . 5 ⊢ (∀𝑖 ∈ 𝐴 (𝑀‘𝑖) ≺ (𝑁‘𝑖) → ∀𝑖 ∈ 𝐴 ((𝑁‘𝑖) ∖ ran (𝐷‘𝑖)) ≠ ∅)
18		konigth.1	. . . . . 6 ⊢ 𝐴 ∈ V
19		fvex 6388	. . . . . . 7 ⊢ (𝑁‘𝑖) ∈ V
20		difss 3899	. . . . . . 7 ⊢ ((𝑁‘𝑖) ∖ ran (𝐷‘𝑖)) ⊆ (𝑁‘𝑖)
21	19, 20	ssexi 4964	. . . . . 6 ⊢ ((𝑁‘𝑖) ∖ ran (𝐷‘𝑖)) ∈ V
22	18, 21	ac6c5 9557	. . . . 5 ⊢ (∀𝑖 ∈ 𝐴 ((𝑁‘𝑖) ∖ ran (𝐷‘𝑖)) ≠ ∅ → ∃𝑒∀𝑖 ∈ 𝐴 (𝑒‘𝑖) ∈ ((𝑁‘𝑖) ∖ ran (𝐷‘𝑖)))
23		equid 2109	. . . . . . 7 ⊢ 𝑓 = 𝑓
24		eldifi 3894	. . . . . . . . . . . . 13 ⊢ ((𝑒‘𝑖) ∈ ((𝑁‘𝑖) ∖ ran (𝐷‘𝑖)) → (𝑒‘𝑖) ∈ (𝑁‘𝑖))
25		fvex 6388	. . . . . . . . . . . . . . 15 ⊢ (𝑒‘𝑖) ∈ V
26		konigth.5	. . . . . . . . . . . . . . . 16 ⊢ 𝐸 = (𝑖 ∈ 𝐴 ↦ (𝑒‘𝑖))
27	26	fvmpt2 6480	. . . . . . . . . . . . . . 15 ⊢ ((𝑖 ∈ 𝐴 ∧ (𝑒‘𝑖) ∈ V) → (𝐸‘𝑖) = (𝑒‘𝑖))
28	25, 27	mpan2 682	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ 𝐴 → (𝐸‘𝑖) = (𝑒‘𝑖))
29	28	eleq1d 2829	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ 𝐴 → ((𝐸‘𝑖) ∈ (𝑁‘𝑖) ↔ (𝑒‘𝑖) ∈ (𝑁‘𝑖)))
30	24, 29	syl5ibr 237	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ 𝐴 → ((𝑒‘𝑖) ∈ ((𝑁‘𝑖) ∖ ran (𝐷‘𝑖)) → (𝐸‘𝑖) ∈ (𝑁‘𝑖)))
31	30	ralimia 3097	. . . . . . . . . . 11 ⊢ (∀𝑖 ∈ 𝐴 (𝑒‘𝑖) ∈ ((𝑁‘𝑖) ∖ ran (𝐷‘𝑖)) → ∀𝑖 ∈ 𝐴 (𝐸‘𝑖) ∈ (𝑁‘𝑖))
32	25, 26	fnmpti 6200	. . . . . . . . . . 11 ⊢ 𝐸 Fn 𝐴
33	31, 32	jctil 515	. . . . . . . . . 10 ⊢ (∀𝑖 ∈ 𝐴 (𝑒‘𝑖) ∈ ((𝑁‘𝑖) ∖ ran (𝐷‘𝑖)) → (𝐸 Fn 𝐴 ∧ ∀𝑖 ∈ 𝐴 (𝐸‘𝑖) ∈ (𝑁‘𝑖)))
34	18	mptex 6679	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ 𝐴 ↦ (𝑒‘𝑖)) ∈ V
35	26, 34	eqeltri 2840	. . . . . . . . . . 11 ⊢ 𝐸 ∈ V
36	35	elixp 8120	. . . . . . . . . 10 ⊢ (𝐸 ∈ X𝑖 ∈ 𝐴 (𝑁‘𝑖) ↔ (𝐸 Fn 𝐴 ∧ ∀𝑖 ∈ 𝐴 (𝐸‘𝑖) ∈ (𝑁‘𝑖)))
37	33, 36	sylibr 225	. . . . . . . . 9 ⊢ (∀𝑖 ∈ 𝐴 (𝑒‘𝑖) ∈ ((𝑁‘𝑖) ∖ ran (𝐷‘𝑖)) → 𝐸 ∈ X𝑖 ∈ 𝐴 (𝑁‘𝑖))
38		konigth.3	. . . . . . . . 9 ⊢ 𝑃 = X𝑖 ∈ 𝐴 (𝑁‘𝑖)
39	37, 38	syl6eleqr 2855	. . . . . . . 8 ⊢ (∀𝑖 ∈ 𝐴 (𝑒‘𝑖) ∈ ((𝑁‘𝑖) ∖ ran (𝐷‘𝑖)) → 𝐸 ∈ 𝑃)
40		foelrn 6568	. . . . . . . . . 10 ⊢ ((𝑓:𝑆–onto→𝑃 ∧ 𝐸 ∈ 𝑃) → ∃𝑎 ∈ 𝑆 𝐸 = (𝑓‘𝑎))
41	40	expcom 402	. . . . . . . . 9 ⊢ (𝐸 ∈ 𝑃 → (𝑓:𝑆–onto→𝑃 → ∃𝑎 ∈ 𝑆 𝐸 = (𝑓‘𝑎)))
42		konigth.2	. . . . . . . . . . . . . . 15 ⊢ 𝑆 = ∪ 𝑖 ∈ 𝐴 (𝑀‘𝑖)
43	42	eleq2i 2836	. . . . . . . . . . . . . 14 ⊢ (𝑎 ∈ 𝑆 ↔ 𝑎 ∈ ∪ 𝑖 ∈ 𝐴 (𝑀‘𝑖))
44		eliun 4680	. . . . . . . . . . . . . 14 ⊢ (𝑎 ∈ ∪ 𝑖 ∈ 𝐴 (𝑀‘𝑖) ↔ ∃𝑖 ∈ 𝐴 𝑎 ∈ (𝑀‘𝑖))
45	43, 44	bitri 266	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ 𝑆 ↔ ∃𝑖 ∈ 𝐴 𝑎 ∈ (𝑀‘𝑖))
46		nfra1 3088	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑖∀𝑖 ∈ 𝐴 (𝑒‘𝑖) ∈ ((𝑁‘𝑖) ∖ ran (𝐷‘𝑖))
47		nfv 2009	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑖 𝐸 = (𝑓‘𝑎)
48	46, 47	nfan 1998	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑖(∀𝑖 ∈ 𝐴 (𝑒‘𝑖) ∈ ((𝑁‘𝑖) ∖ ran (𝐷‘𝑖)) ∧ 𝐸 = (𝑓‘𝑎))
49		nfv 2009	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑖 ¬ 𝑓 = 𝑓
50	28	ad2antrl 719	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐸 = (𝑓‘𝑎) ∧ (𝑖 ∈ 𝐴 ∧ 𝑎 ∈ (𝑀‘𝑖))) → (𝐸‘𝑖) = (𝑒‘𝑖))
51		fveq1 6374	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐸 = (𝑓‘𝑎) → (𝐸‘𝑖) = ((𝑓‘𝑎)‘𝑖))
52	8	fveq1d 6377	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑖 ∈ 𝐴 → ((𝐷‘𝑖)‘𝑎) = ((𝑎 ∈ (𝑀‘𝑖) ↦ ((𝑓‘𝑎)‘𝑖))‘𝑎))
53	3	fvmpt2 6480	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑎 ∈ (𝑀‘𝑖) ∧ ((𝑓‘𝑎)‘𝑖) ∈ V) → ((𝑎 ∈ (𝑀‘𝑖) ↦ ((𝑓‘𝑎)‘𝑖))‘𝑎) = ((𝑓‘𝑎)‘𝑖))
54	2, 53	mpan2 682	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑎 ∈ (𝑀‘𝑖) → ((𝑎 ∈ (𝑀‘𝑖) ↦ ((𝑓‘𝑎)‘𝑖))‘𝑎) = ((𝑓‘𝑎)‘𝑖))
55	52, 54	sylan9eq 2819	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑖 ∈ 𝐴 ∧ 𝑎 ∈ (𝑀‘𝑖)) → ((𝐷‘𝑖)‘𝑎) = ((𝑓‘𝑎)‘𝑖))
56	55	eqcomd 2771	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑖 ∈ 𝐴 ∧ 𝑎 ∈ (𝑀‘𝑖)) → ((𝑓‘𝑎)‘𝑖) = ((𝐷‘𝑖)‘𝑎))
57	51, 56	sylan9eq 2819	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐸 = (𝑓‘𝑎) ∧ (𝑖 ∈ 𝐴 ∧ 𝑎 ∈ (𝑀‘𝑖))) → (𝐸‘𝑖) = ((𝐷‘𝑖)‘𝑎))
58	50, 57	eqtr3d 2801	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐸 = (𝑓‘𝑎) ∧ (𝑖 ∈ 𝐴 ∧ 𝑎 ∈ (𝑀‘𝑖))) → (𝑒‘𝑖) = ((𝐷‘𝑖)‘𝑎))
59		fnfvelrn 6546	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐷‘𝑖) Fn (𝑀‘𝑖) ∧ 𝑎 ∈ (𝑀‘𝑖)) → ((𝐷‘𝑖)‘𝑎) ∈ ran (𝐷‘𝑖))
60	10, 59	sylan 575	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑖 ∈ 𝐴 ∧ 𝑎 ∈ (𝑀‘𝑖)) → ((𝐷‘𝑖)‘𝑎) ∈ ran (𝐷‘𝑖))
61	60	adantl 473	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐸 = (𝑓‘𝑎) ∧ (𝑖 ∈ 𝐴 ∧ 𝑎 ∈ (𝑀‘𝑖))) → ((𝐷‘𝑖)‘𝑎) ∈ ran (𝐷‘𝑖))
62	58, 61	eqeltrd 2844	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐸 = (𝑓‘𝑎) ∧ (𝑖 ∈ 𝐴 ∧ 𝑎 ∈ (𝑀‘𝑖))) → (𝑒‘𝑖) ∈ ran (𝐷‘𝑖))
63	62	3adant1 1160	. . . . . . . . . . . . . . . . 17 ⊢ ((∀𝑖 ∈ 𝐴 (𝑒‘𝑖) ∈ ((𝑁‘𝑖) ∖ ran (𝐷‘𝑖)) ∧ 𝐸 = (𝑓‘𝑎) ∧ (𝑖 ∈ 𝐴 ∧ 𝑎 ∈ (𝑀‘𝑖))) → (𝑒‘𝑖) ∈ ran (𝐷‘𝑖))
64		simp1 1166	. . . . . . . . . . . . . . . . . 18 ⊢ ((∀𝑖 ∈ 𝐴 (𝑒‘𝑖) ∈ ((𝑁‘𝑖) ∖ ran (𝐷‘𝑖)) ∧ 𝐸 = (𝑓‘𝑎) ∧ (𝑖 ∈ 𝐴 ∧ 𝑎 ∈ (𝑀‘𝑖))) → ∀𝑖 ∈ 𝐴 (𝑒‘𝑖) ∈ ((𝑁‘𝑖) ∖ ran (𝐷‘𝑖)))
65		simp3l 1258	. . . . . . . . . . . . . . . . . 18 ⊢ ((∀𝑖 ∈ 𝐴 (𝑒‘𝑖) ∈ ((𝑁‘𝑖) ∖ ran (𝐷‘𝑖)) ∧ 𝐸 = (𝑓‘𝑎) ∧ (𝑖 ∈ 𝐴 ∧ 𝑎 ∈ (𝑀‘𝑖))) → 𝑖 ∈ 𝐴)
66		rsp 3076	. . . . . . . . . . . . . . . . . . 19 ⊢ (∀𝑖 ∈ 𝐴 (𝑒‘𝑖) ∈ ((𝑁‘𝑖) ∖ ran (𝐷‘𝑖)) → (𝑖 ∈ 𝐴 → (𝑒‘𝑖) ∈ ((𝑁‘𝑖) ∖ ran (𝐷‘𝑖))))
67		eldifn 3895	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑒‘𝑖) ∈ ((𝑁‘𝑖) ∖ ran (𝐷‘𝑖)) → ¬ (𝑒‘𝑖) ∈ ran (𝐷‘𝑖))
68	66, 67	syl6 35	. . . . . . . . . . . . . . . . . 18 ⊢ (∀𝑖 ∈ 𝐴 (𝑒‘𝑖) ∈ ((𝑁‘𝑖) ∖ ran (𝐷‘𝑖)) → (𝑖 ∈ 𝐴 → ¬ (𝑒‘𝑖) ∈ ran (𝐷‘𝑖)))
69	64, 65, 68	sylc 65	. . . . . . . . . . . . . . . . 17 ⊢ ((∀𝑖 ∈ 𝐴 (𝑒‘𝑖) ∈ ((𝑁‘𝑖) ∖ ran (𝐷‘𝑖)) ∧ 𝐸 = (𝑓‘𝑎) ∧ (𝑖 ∈ 𝐴 ∧ 𝑎 ∈ (𝑀‘𝑖))) → ¬ (𝑒‘𝑖) ∈ ran (𝐷‘𝑖))
70	63, 69	pm2.21dd 186	. . . . . . . . . . . . . . . 16 ⊢ ((∀𝑖 ∈ 𝐴 (𝑒‘𝑖) ∈ ((𝑁‘𝑖) ∖ ran (𝐷‘𝑖)) ∧ 𝐸 = (𝑓‘𝑎) ∧ (𝑖 ∈ 𝐴 ∧ 𝑎 ∈ (𝑀‘𝑖))) → ¬ 𝑓 = 𝑓)
71	70	3expia 1150	. . . . . . . . . . . . . . 15 ⊢ ((∀𝑖 ∈ 𝐴 (𝑒‘𝑖) ∈ ((𝑁‘𝑖) ∖ ran (𝐷‘𝑖)) ∧ 𝐸 = (𝑓‘𝑎)) → ((𝑖 ∈ 𝐴 ∧ 𝑎 ∈ (𝑀‘𝑖)) → ¬ 𝑓 = 𝑓))
72	71	expd 404	. . . . . . . . . . . . . 14 ⊢ ((∀𝑖 ∈ 𝐴 (𝑒‘𝑖) ∈ ((𝑁‘𝑖) ∖ ran (𝐷‘𝑖)) ∧ 𝐸 = (𝑓‘𝑎)) → (𝑖 ∈ 𝐴 → (𝑎 ∈ (𝑀‘𝑖) → ¬ 𝑓 = 𝑓)))
73	48, 49, 72	rexlimd 3173	. . . . . . . . . . . . 13 ⊢ ((∀𝑖 ∈ 𝐴 (𝑒‘𝑖) ∈ ((𝑁‘𝑖) ∖ ran (𝐷‘𝑖)) ∧ 𝐸 = (𝑓‘𝑎)) → (∃𝑖 ∈ 𝐴 𝑎 ∈ (𝑀‘𝑖) → ¬ 𝑓 = 𝑓))
74	45, 73	syl5bi 233	. . . . . . . . . . . 12 ⊢ ((∀𝑖 ∈ 𝐴 (𝑒‘𝑖) ∈ ((𝑁‘𝑖) ∖ ran (𝐷‘𝑖)) ∧ 𝐸 = (𝑓‘𝑎)) → (𝑎 ∈ 𝑆 → ¬ 𝑓 = 𝑓))
75	74	ex 401	. . . . . . . . . . 11 ⊢ (∀𝑖 ∈ 𝐴 (𝑒‘𝑖) ∈ ((𝑁‘𝑖) ∖ ran (𝐷‘𝑖)) → (𝐸 = (𝑓‘𝑎) → (𝑎 ∈ 𝑆 → ¬ 𝑓 = 𝑓)))
76	75	com23 86	. . . . . . . . . 10 ⊢ (∀𝑖 ∈ 𝐴 (𝑒‘𝑖) ∈ ((𝑁‘𝑖) ∖ ran (𝐷‘𝑖)) → (𝑎 ∈ 𝑆 → (𝐸 = (𝑓‘𝑎) → ¬ 𝑓 = 𝑓)))
77	76	rexlimdv 3177	. . . . . . . . 9 ⊢ (∀𝑖 ∈ 𝐴 (𝑒‘𝑖) ∈ ((𝑁‘𝑖) ∖ ran (𝐷‘𝑖)) → (∃𝑎 ∈ 𝑆 𝐸 = (𝑓‘𝑎) → ¬ 𝑓 = 𝑓))
78	41, 77	syl9r 78	. . . . . . . 8 ⊢ (∀𝑖 ∈ 𝐴 (𝑒‘𝑖) ∈ ((𝑁‘𝑖) ∖ ran (𝐷‘𝑖)) → (𝐸 ∈ 𝑃 → (𝑓:𝑆–onto→𝑃 → ¬ 𝑓 = 𝑓)))
79	39, 78	mpd 15	. . . . . . 7 ⊢ (∀𝑖 ∈ 𝐴 (𝑒‘𝑖) ∈ ((𝑁‘𝑖) ∖ ran (𝐷‘𝑖)) → (𝑓:𝑆–onto→𝑃 → ¬ 𝑓 = 𝑓))
80	23, 79	mt2i 134	. . . . . 6 ⊢ (∀𝑖 ∈ 𝐴 (𝑒‘𝑖) ∈ ((𝑁‘𝑖) ∖ ran (𝐷‘𝑖)) → ¬ 𝑓:𝑆–onto→𝑃)
81	80	exlimiv 2025	. . . . 5 ⊢ (∃𝑒∀𝑖 ∈ 𝐴 (𝑒‘𝑖) ∈ ((𝑁‘𝑖) ∖ ran (𝐷‘𝑖)) → ¬ 𝑓:𝑆–onto→𝑃)
82	17, 22, 81	3syl 18	. . . 4 ⊢ (∀𝑖 ∈ 𝐴 (𝑀‘𝑖) ≺ (𝑁‘𝑖) → ¬ 𝑓:𝑆–onto→𝑃)
83	82	nexdv 2031	. . 3 ⊢ (∀𝑖 ∈ 𝐴 (𝑀‘𝑖) ≺ (𝑁‘𝑖) → ¬ ∃𝑓 𝑓:𝑆–onto→𝑃)
84	1	0dom 8297	. . . . . . . 8 ⊢ ∅ ≼ (𝑀‘𝑖)
85		domsdomtr 8302	. . . . . . . 8 ⊢ ((∅ ≼ (𝑀‘𝑖) ∧ (𝑀‘𝑖) ≺ (𝑁‘𝑖)) → ∅ ≺ (𝑁‘𝑖))
86	84, 85	mpan 681	. . . . . . 7 ⊢ ((𝑀‘𝑖) ≺ (𝑁‘𝑖) → ∅ ≺ (𝑁‘𝑖))
87	19	0sdom 8298	. . . . . . 7 ⊢ (∅ ≺ (𝑁‘𝑖) ↔ (𝑁‘𝑖) ≠ ∅)
88	86, 87	sylib 209	. . . . . 6 ⊢ ((𝑀‘𝑖) ≺ (𝑁‘𝑖) → (𝑁‘𝑖) ≠ ∅)
89	88	ralimi 3099	. . . . 5 ⊢ (∀𝑖 ∈ 𝐴 (𝑀‘𝑖) ≺ (𝑁‘𝑖) → ∀𝑖 ∈ 𝐴 (𝑁‘𝑖) ≠ ∅)
90	38	neeq1i 3001	. . . . . 6 ⊢ (𝑃 ≠ ∅ ↔ X𝑖 ∈ 𝐴 (𝑁‘𝑖) ≠ ∅)
91	19	rgenw 3071	. . . . . . . . 9 ⊢ ∀𝑖 ∈ 𝐴 (𝑁‘𝑖) ∈ V
92		ixpexg 8137	. . . . . . . . 9 ⊢ (∀𝑖 ∈ 𝐴 (𝑁‘𝑖) ∈ V → X𝑖 ∈ 𝐴 (𝑁‘𝑖) ∈ V)
93	91, 92	ax-mp 5	. . . . . . . 8 ⊢ X𝑖 ∈ 𝐴 (𝑁‘𝑖) ∈ V
94	38, 93	eqeltri 2840	. . . . . . 7 ⊢ 𝑃 ∈ V
95	94	0sdom 8298	. . . . . 6 ⊢ (∅ ≺ 𝑃 ↔ 𝑃 ≠ ∅)
96	18, 19	ac9 9558	. . . . . 6 ⊢ (∀𝑖 ∈ 𝐴 (𝑁‘𝑖) ≠ ∅ ↔ X𝑖 ∈ 𝐴 (𝑁‘𝑖) ≠ ∅)
97	90, 95, 96	3bitr4i 294	. . . . 5 ⊢ (∅ ≺ 𝑃 ↔ ∀𝑖 ∈ 𝐴 (𝑁‘𝑖) ≠ ∅)
98	89, 97	sylibr 225	. . . 4 ⊢ (∀𝑖 ∈ 𝐴 (𝑀‘𝑖) ≺ (𝑁‘𝑖) → ∅ ≺ 𝑃)
99	18, 1	iunex 7345	. . . . . . 7 ⊢ ∪ 𝑖 ∈ 𝐴 (𝑀‘𝑖) ∈ V
100	42, 99	eqeltri 2840	. . . . . 6 ⊢ 𝑆 ∈ V
101		domtri 9631	. . . . . 6 ⊢ ((𝑃 ∈ V ∧ 𝑆 ∈ V) → (𝑃 ≼ 𝑆 ↔ ¬ 𝑆 ≺ 𝑃))
102	94, 100, 101	mp2an 683	. . . . 5 ⊢ (𝑃 ≼ 𝑆 ↔ ¬ 𝑆 ≺ 𝑃)
103	102	biimpri 219	. . . 4 ⊢ (¬ 𝑆 ≺ 𝑃 → 𝑃 ≼ 𝑆)
104		fodomr 8318	. . . 4 ⊢ ((∅ ≺ 𝑃 ∧ 𝑃 ≼ 𝑆) → ∃𝑓 𝑓:𝑆–onto→𝑃)
105	98, 103, 104	syl2an 589	. . 3 ⊢ ((∀𝑖 ∈ 𝐴 (𝑀‘𝑖) ≺ (𝑁‘𝑖) ∧ ¬ 𝑆 ≺ 𝑃) → ∃𝑓 𝑓:𝑆–onto→𝑃)
106	83, 105	mtand 850	. 2 ⊢ (∀𝑖 ∈ 𝐴 (𝑀‘𝑖) ≺ (𝑁‘𝑖) → ¬ ¬ 𝑆 ≺ 𝑃)
107	106	notnotrd 130	1 ⊢ (∀𝑖 ∈ 𝐴 (𝑀‘𝑖) ≺ (𝑁‘𝑖) → 𝑆 ≺ 𝑃)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 197   ∧ wa 384   ∧ w3a 1107   = wceq 1652  ∃wex 1874   ∈ wcel 2155   ≠ wne 2937  ∀wral 3055  ∃wrex 3056  Vcvv 3350   ∖ cdif 3729  ∅c0 4079  ∪ ciun 4676   class class class wbr 4809   ↦ cmpt 4888  ran crn 5278   Fn wfn 6063  –onto→wfo 6066  ‘cfv 6068  Xcixp 8113   ≼ cdom 8158   ≺ csdm 8159

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 2069  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 4930  ax-sep 4941  ax-nul 4949  ax-pow 5001  ax-pr 5062  ax-un 7147  ax-ac2 9538

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 2062  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 3597  df-csb 3692  df-dif 3735  df-un 3737  df-in 3739  df-ss 3746  df-pss 3748  df-nul 4080  df-if 4244  df-pw 4317  df-sn 4335  df-pr 4337  df-tp 4339  df-op 4341  df-uni 4595  df-int 4634  df-iun 4678  df-br 4810  df-opab 4872  df-mpt 4889  df-tr 4912  df-id 5185  df-eprel 5190  df-po 5198  df-so 5199  df-fr 5236  df-se 5237  df-we 5238  df-xp 5283  df-rel 5284  df-cnv 5285  df-co 5286  df-dm 5287  df-rn 5288  df-res 5289  df-ima 5290  df-pred 5865  df-ord 5911  df-on 5912  df-suc 5914  df-iota 6031  df-fun 6070  df-fn 6071  df-f 6072  df-f1 6073  df-fo 6074  df-f1o 6075  df-fv 6076  df-isom 6077  df-riota 6803  df-ov 6845  df-oprab 6846  df-mpt2 6847  df-1st 7366  df-2nd 7367  df-wrecs 7610  df-recs 7672  df-er 7947  df-map 8062  df-ixp 8114  df-en 8161  df-dom 8162  df-sdom 8163  df-card 9016  df-acn 9019  df-ac 9190

This theorem is referenced by:  konigth  9644