Theorem konigthlem 9269

Description: Lemma for konigth 9270. (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 6113	. . . . . . . . 9 ⊢ (𝑀‘𝑖) ∈ V
2		fvex 6113	. . . . . . . . . . 11 ⊢ ((𝑓‘𝑎)‘𝑖) ∈ V
3		eqid 2610	. . . . . . . . . . 11 ⊢ (𝑎 ∈ (𝑀‘𝑖) ↦ ((𝑓‘𝑎)‘𝑖)) = (𝑎 ∈ (𝑀‘𝑖) ↦ ((𝑓‘𝑎)‘𝑖))
4	2, 3	fnmpti 5935	. . . . . . . . . 10 ⊢ (𝑎 ∈ (𝑀‘𝑖) ↦ ((𝑓‘𝑎)‘𝑖)) Fn (𝑀‘𝑖)
5	1	mptex 6390	. . . . . . . . . . . 12 ⊢ (𝑎 ∈ (𝑀‘𝑖) ↦ ((𝑓‘𝑎)‘𝑖)) ∈ V
6		konigth.4	. . . . . . . . . . . . 13 ⊢ 𝐷 = (𝑖 ∈ 𝐴 ↦ (𝑎 ∈ (𝑀‘𝑖) ↦ ((𝑓‘𝑎)‘𝑖)))
7	6	fvmpt2 6200	. . . . . . . . . . . 12 ⊢ ((𝑖 ∈ 𝐴 ∧ (𝑎 ∈ (𝑀‘𝑖) ↦ ((𝑓‘𝑎)‘𝑖)) ∈ V) → (𝐷‘𝑖) = (𝑎 ∈ (𝑀‘𝑖) ↦ ((𝑓‘𝑎)‘𝑖)))
8	5, 7	mpan2 703	. . . . . . . . . . 11 ⊢ (𝑖 ∈ 𝐴 → (𝐷‘𝑖) = (𝑎 ∈ (𝑀‘𝑖) ↦ ((𝑓‘𝑎)‘𝑖)))
9	8	fneq1d 5895	. . . . . . . . . 10 ⊢ (𝑖 ∈ 𝐴 → ((𝐷‘𝑖) Fn (𝑀‘𝑖) ↔ (𝑎 ∈ (𝑀‘𝑖) ↦ ((𝑓‘𝑎)‘𝑖)) Fn (𝑀‘𝑖)))
10	4, 9	mpbiri 247	. . . . . . . . 9 ⊢ (𝑖 ∈ 𝐴 → (𝐷‘𝑖) Fn (𝑀‘𝑖))
11		fnrndomg 9239	. . . . . . . . 9 ⊢ ((𝑀‘𝑖) ∈ V → ((𝐷‘𝑖) Fn (𝑀‘𝑖) → ran (𝐷‘𝑖) ≼ (𝑀‘𝑖)))
12	1, 10, 11	mpsyl 66	. . . . . . . 8 ⊢ (𝑖 ∈ 𝐴 → ran (𝐷‘𝑖) ≼ (𝑀‘𝑖))
13		domsdomtr 7980	. . . . . . . 8 ⊢ ((ran (𝐷‘𝑖) ≼ (𝑀‘𝑖) ∧ (𝑀‘𝑖) ≺ (𝑁‘𝑖)) → ran (𝐷‘𝑖) ≺ (𝑁‘𝑖))
14	12, 13	sylan 487	. . . . . . 7 ⊢ ((𝑖 ∈ 𝐴 ∧ (𝑀‘𝑖) ≺ (𝑁‘𝑖)) → ran (𝐷‘𝑖) ≺ (𝑁‘𝑖))
15		sdomdif 7993	. . . . . . 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 2935	. . . . 5 ⊢ (∀𝑖 ∈ 𝐴 (𝑀‘𝑖) ≺ (𝑁‘𝑖) → ∀𝑖 ∈ 𝐴 ((𝑁‘𝑖) ∖ ran (𝐷‘𝑖)) ≠ ∅)
18		konigth.1	. . . . . 6 ⊢ 𝐴 ∈ V
19		fvex 6113	. . . . . . 7 ⊢ (𝑁‘𝑖) ∈ V
20		difss 3699	. . . . . . 7 ⊢ ((𝑁‘𝑖) ∖ ran (𝐷‘𝑖)) ⊆ (𝑁‘𝑖)
21	19, 20	ssexi 4731	. . . . . 6 ⊢ ((𝑁‘𝑖) ∖ ran (𝐷‘𝑖)) ∈ V
22	18, 21	ac6c5 9187	. . . . 5 ⊢ (∀𝑖 ∈ 𝐴 ((𝑁‘𝑖) ∖ ran (𝐷‘𝑖)) ≠ ∅ → ∃𝑒∀𝑖 ∈ 𝐴 (𝑒‘𝑖) ∈ ((𝑁‘𝑖) ∖ ran (𝐷‘𝑖)))
23		equid 1926	. . . . . . 7 ⊢ 𝑓 = 𝑓
24		eldifi 3694	. . . . . . . . . . . . 13 ⊢ ((𝑒‘𝑖) ∈ ((𝑁‘𝑖) ∖ ran (𝐷‘𝑖)) → (𝑒‘𝑖) ∈ (𝑁‘𝑖))
25		fvex 6113	. . . . . . . . . . . . . . 15 ⊢ (𝑒‘𝑖) ∈ V
26		konigth.5	. . . . . . . . . . . . . . . 16 ⊢ 𝐸 = (𝑖 ∈ 𝐴 ↦ (𝑒‘𝑖))
27	26	fvmpt2 6200	. . . . . . . . . . . . . . 15 ⊢ ((𝑖 ∈ 𝐴 ∧ (𝑒‘𝑖) ∈ V) → (𝐸‘𝑖) = (𝑒‘𝑖))
28	25, 27	mpan2 703	. . . . . . . . . . . . . 14 ⊢ (𝑖 ∈ 𝐴 → (𝐸‘𝑖) = (𝑒‘𝑖))
29	28	eleq1d 2672	. . . . . . . . . . . . 13 ⊢ (𝑖 ∈ 𝐴 → ((𝐸‘𝑖) ∈ (𝑁‘𝑖) ↔ (𝑒‘𝑖) ∈ (𝑁‘𝑖)))
30	24, 29	syl5ibr 235	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ 𝐴 → ((𝑒‘𝑖) ∈ ((𝑁‘𝑖) ∖ ran (𝐷‘𝑖)) → (𝐸‘𝑖) ∈ (𝑁‘𝑖)))
31	30	ralimia 2934	. . . . . . . . . . 11 ⊢ (∀𝑖 ∈ 𝐴 (𝑒‘𝑖) ∈ ((𝑁‘𝑖) ∖ ran (𝐷‘𝑖)) → ∀𝑖 ∈ 𝐴 (𝐸‘𝑖) ∈ (𝑁‘𝑖))
32	25, 26	fnmpti 5935	. . . . . . . . . . 11 ⊢ 𝐸 Fn 𝐴
33	31, 32	jctil 558	. . . . . . . . . 10 ⊢ (∀𝑖 ∈ 𝐴 (𝑒‘𝑖) ∈ ((𝑁‘𝑖) ∖ ran (𝐷‘𝑖)) → (𝐸 Fn 𝐴 ∧ ∀𝑖 ∈ 𝐴 (𝐸‘𝑖) ∈ (𝑁‘𝑖)))
34	18	mptex 6390	. . . . . . . . . . . 12 ⊢ (𝑖 ∈ 𝐴 ↦ (𝑒‘𝑖)) ∈ V
35	26, 34	eqeltri 2684	. . . . . . . . . . 11 ⊢ 𝐸 ∈ V
36	35	elixp 7801	. . . . . . . . . 10 ⊢ (𝐸 ∈ X𝑖 ∈ 𝐴 (𝑁‘𝑖) ↔ (𝐸 Fn 𝐴 ∧ ∀𝑖 ∈ 𝐴 (𝐸‘𝑖) ∈ (𝑁‘𝑖)))
37	33, 36	sylibr 223	. . . . . . . . 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 2699	. . . . . . . 8 ⊢ (∀𝑖 ∈ 𝐴 (𝑒‘𝑖) ∈ ((𝑁‘𝑖) ∖ ran (𝐷‘𝑖)) → 𝐸 ∈ 𝑃)
40		foelrn 6286	. . . . . . . . . 10 ⊢ ((𝑓:𝑆–onto→𝑃 ∧ 𝐸 ∈ 𝑃) → ∃𝑎 ∈ 𝑆 𝐸 = (𝑓‘𝑎))
41	40	expcom 450	. . . . . . . . 9 ⊢ (𝐸 ∈ 𝑃 → (𝑓:𝑆–onto→𝑃 → ∃𝑎 ∈ 𝑆 𝐸 = (𝑓‘𝑎)))
42		konigth.2	. . . . . . . . . . . . . . 15 ⊢ 𝑆 = ∪ 𝑖 ∈ 𝐴 (𝑀‘𝑖)
43	42	eleq2i 2680	. . . . . . . . . . . . . 14 ⊢ (𝑎 ∈ 𝑆 ↔ 𝑎 ∈ ∪ 𝑖 ∈ 𝐴 (𝑀‘𝑖))
44		eliun 4460	. . . . . . . . . . . . . 14 ⊢ (𝑎 ∈ ∪ 𝑖 ∈ 𝐴 (𝑀‘𝑖) ↔ ∃𝑖 ∈ 𝐴 𝑎 ∈ (𝑀‘𝑖))
45	43, 44	bitri 263	. . . . . . . . . . . . 13 ⊢ (𝑎 ∈ 𝑆 ↔ ∃𝑖 ∈ 𝐴 𝑎 ∈ (𝑀‘𝑖))
46		nfra1 2925	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑖∀𝑖 ∈ 𝐴 (𝑒‘𝑖) ∈ ((𝑁‘𝑖) ∖ ran (𝐷‘𝑖))
47		nfv 1830	. . . . . . . . . . . . . . 15 ⊢ Ⅎ𝑖 𝐸 = (𝑓‘𝑎)
48	46, 47	nfan 1816	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑖(∀𝑖 ∈ 𝐴 (𝑒‘𝑖) ∈ ((𝑁‘𝑖) ∖ ran (𝐷‘𝑖)) ∧ 𝐸 = (𝑓‘𝑎))
49		nfv 1830	. . . . . . . . . . . . . 14 ⊢ Ⅎ𝑖 ¬ 𝑓 = 𝑓
50	28	ad2antrl 760	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐸 = (𝑓‘𝑎) ∧ (𝑖 ∈ 𝐴 ∧ 𝑎 ∈ (𝑀‘𝑖))) → (𝐸‘𝑖) = (𝑒‘𝑖))
51		fveq1 6102	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (𝐸 = (𝑓‘𝑎) → (𝐸‘𝑖) = ((𝑓‘𝑎)‘𝑖))
52	8	fveq1d 6105	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑖 ∈ 𝐴 → ((𝐷‘𝑖)‘𝑎) = ((𝑎 ∈ (𝑀‘𝑖) ↦ ((𝑓‘𝑎)‘𝑖))‘𝑎))
53	3	fvmpt2 6200	. . . . . . . . . . . . . . . . . . . . . . . 24 ⊢ ((𝑎 ∈ (𝑀‘𝑖) ∧ ((𝑓‘𝑎)‘𝑖) ∈ V) → ((𝑎 ∈ (𝑀‘𝑖) ↦ ((𝑓‘𝑎)‘𝑖))‘𝑎) = ((𝑓‘𝑎)‘𝑖))
54	2, 53	mpan2 703	. . . . . . . . . . . . . . . . . . . . . . 23 ⊢ (𝑎 ∈ (𝑀‘𝑖) → ((𝑎 ∈ (𝑀‘𝑖) ↦ ((𝑓‘𝑎)‘𝑖))‘𝑎) = ((𝑓‘𝑎)‘𝑖))
55	52, 54	sylan9eq 2664	. . . . . . . . . . . . . . . . . . . . . 22 ⊢ ((𝑖 ∈ 𝐴 ∧ 𝑎 ∈ (𝑀‘𝑖)) → ((𝐷‘𝑖)‘𝑎) = ((𝑓‘𝑎)‘𝑖))
56	55	eqcomd 2616	. . . . . . . . . . . . . . . . . . . . 21 ⊢ ((𝑖 ∈ 𝐴 ∧ 𝑎 ∈ (𝑀‘𝑖)) → ((𝑓‘𝑎)‘𝑖) = ((𝐷‘𝑖)‘𝑎))
57	51, 56	sylan9eq 2664	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝐸 = (𝑓‘𝑎) ∧ (𝑖 ∈ 𝐴 ∧ 𝑎 ∈ (𝑀‘𝑖))) → (𝐸‘𝑖) = ((𝐷‘𝑖)‘𝑎))
58	50, 57	eqtr3d 2646	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐸 = (𝑓‘𝑎) ∧ (𝑖 ∈ 𝐴 ∧ 𝑎 ∈ (𝑀‘𝑖))) → (𝑒‘𝑖) = ((𝐷‘𝑖)‘𝑎))
59		fnfvelrn 6264	. . . . . . . . . . . . . . . . . . . . 21 ⊢ (((𝐷‘𝑖) Fn (𝑀‘𝑖) ∧ 𝑎 ∈ (𝑀‘𝑖)) → ((𝐷‘𝑖)‘𝑎) ∈ ran (𝐷‘𝑖))
60	10, 59	sylan 487	. . . . . . . . . . . . . . . . . . . 20 ⊢ ((𝑖 ∈ 𝐴 ∧ 𝑎 ∈ (𝑀‘𝑖)) → ((𝐷‘𝑖)‘𝑎) ∈ ran (𝐷‘𝑖))
61	60	adantl 481	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐸 = (𝑓‘𝑎) ∧ (𝑖 ∈ 𝐴 ∧ 𝑎 ∈ (𝑀‘𝑖))) → ((𝐷‘𝑖)‘𝑎) ∈ ran (𝐷‘𝑖))
62	58, 61	eqeltrd 2688	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐸 = (𝑓‘𝑎) ∧ (𝑖 ∈ 𝐴 ∧ 𝑎 ∈ (𝑀‘𝑖))) → (𝑒‘𝑖) ∈ ran (𝐷‘𝑖))
63	62	3adant1 1072	. . . . . . . . . . . . . . . . 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 1054	. . . . . . . . . . . . . . . . . 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 1082	. . . . . . . . . . . . . . . . . 18 ⊢ ((∀𝑖 ∈ 𝐴 (𝑒‘𝑖) ∈ ((𝑁‘𝑖) ∖ ran (𝐷‘𝑖)) ∧ 𝐸 = (𝑓‘𝑎) ∧ (𝑖 ∈ 𝐴 ∧ 𝑎 ∈ (𝑀‘𝑖))) → 𝑖 ∈ 𝐴)
66		rsp 2913	. . . . . . . . . . . . . . . . . . 19 ⊢ (∀𝑖 ∈ 𝐴 (𝑒‘𝑖) ∈ ((𝑁‘𝑖) ∖ ran (𝐷‘𝑖)) → (𝑖 ∈ 𝐴 → (𝑒‘𝑖) ∈ ((𝑁‘𝑖) ∖ ran (𝐷‘𝑖))))
67		eldifn 3695	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝑒‘𝑖) ∈ ((𝑁‘𝑖) ∖ ran (𝐷‘𝑖)) → ¬ (𝑒‘𝑖) ∈ ran (𝐷‘𝑖))
68	66, 67	syl6 34	. . . . . . . . . . . . . . . . . 18 ⊢ (∀𝑖 ∈ 𝐴 (𝑒‘𝑖) ∈ ((𝑁‘𝑖) ∖ ran (𝐷‘𝑖)) → (𝑖 ∈ 𝐴 → ¬ (𝑒‘𝑖) ∈ ran (𝐷‘𝑖)))
69	64, 65, 68	sylc 63	. . . . . . . . . . . . . . . . 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 185	. . . . . . . . . . . . . . . 16 ⊢ ((∀𝑖 ∈ 𝐴 (𝑒‘𝑖) ∈ ((𝑁‘𝑖) ∖ ran (𝐷‘𝑖)) ∧ 𝐸 = (𝑓‘𝑎) ∧ (𝑖 ∈ 𝐴 ∧ 𝑎 ∈ (𝑀‘𝑖))) → ¬ 𝑓 = 𝑓)
71	70	3expia 1259	. . . . . . . . . . . . . . 15 ⊢ ((∀𝑖 ∈ 𝐴 (𝑒‘𝑖) ∈ ((𝑁‘𝑖) ∖ ran (𝐷‘𝑖)) ∧ 𝐸 = (𝑓‘𝑎)) → ((𝑖 ∈ 𝐴 ∧ 𝑎 ∈ (𝑀‘𝑖)) → ¬ 𝑓 = 𝑓))
72	71	expd 451	. . . . . . . . . . . . . 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 3008	. . . . . . . . . . . . 13 ⊢ ((∀𝑖 ∈ 𝐴 (𝑒‘𝑖) ∈ ((𝑁‘𝑖) ∖ ran (𝐷‘𝑖)) ∧ 𝐸 = (𝑓‘𝑎)) → (∃𝑖 ∈ 𝐴 𝑎 ∈ (𝑀‘𝑖) → ¬ 𝑓 = 𝑓))
74	45, 73	syl5bi 231	. . . . . . . . . . . 12 ⊢ ((∀𝑖 ∈ 𝐴 (𝑒‘𝑖) ∈ ((𝑁‘𝑖) ∖ ran (𝐷‘𝑖)) ∧ 𝐸 = (𝑓‘𝑎)) → (𝑎 ∈ 𝑆 → ¬ 𝑓 = 𝑓))
75	74	ex 449	. . . . . . . . . . 11 ⊢ (∀𝑖 ∈ 𝐴 (𝑒‘𝑖) ∈ ((𝑁‘𝑖) ∖ ran (𝐷‘𝑖)) → (𝐸 = (𝑓‘𝑎) → (𝑎 ∈ 𝑆 → ¬ 𝑓 = 𝑓)))
76	75	com23 84	. . . . . . . . . 10 ⊢ (∀𝑖 ∈ 𝐴 (𝑒‘𝑖) ∈ ((𝑁‘𝑖) ∖ ran (𝐷‘𝑖)) → (𝑎 ∈ 𝑆 → (𝐸 = (𝑓‘𝑎) → ¬ 𝑓 = 𝑓)))
77	76	rexlimdv 3012	. . . . . . . . 9 ⊢ (∀𝑖 ∈ 𝐴 (𝑒‘𝑖) ∈ ((𝑁‘𝑖) ∖ ran (𝐷‘𝑖)) → (∃𝑎 ∈ 𝑆 𝐸 = (𝑓‘𝑎) → ¬ 𝑓 = 𝑓))
78	41, 77	syl9r 76	. . . . . . . 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 131	. . . . . 6 ⊢ (∀𝑖 ∈ 𝐴 (𝑒‘𝑖) ∈ ((𝑁‘𝑖) ∖ ran (𝐷‘𝑖)) → ¬ 𝑓:𝑆–onto→𝑃)
81	80	exlimiv 1845	. . . . 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 1851	. . 3 ⊢ (∀𝑖 ∈ 𝐴 (𝑀‘𝑖) ≺ (𝑁‘𝑖) → ¬ ∃𝑓 𝑓:𝑆–onto→𝑃)
84	1	0dom 7975	. . . . . . . 8 ⊢ ∅ ≼ (𝑀‘𝑖)
85		domsdomtr 7980	. . . . . . . 8 ⊢ ((∅ ≼ (𝑀‘𝑖) ∧ (𝑀‘𝑖) ≺ (𝑁‘𝑖)) → ∅ ≺ (𝑁‘𝑖))
86	84, 85	mpan 702	. . . . . . 7 ⊢ ((𝑀‘𝑖) ≺ (𝑁‘𝑖) → ∅ ≺ (𝑁‘𝑖))
87	19	0sdom 7976	. . . . . . 7 ⊢ (∅ ≺ (𝑁‘𝑖) ↔ (𝑁‘𝑖) ≠ ∅)
88	86, 87	sylib 207	. . . . . 6 ⊢ ((𝑀‘𝑖) ≺ (𝑁‘𝑖) → (𝑁‘𝑖) ≠ ∅)
89	88	ralimi 2936	. . . . 5 ⊢ (∀𝑖 ∈ 𝐴 (𝑀‘𝑖) ≺ (𝑁‘𝑖) → ∀𝑖 ∈ 𝐴 (𝑁‘𝑖) ≠ ∅)
90	38	neeq1i 2846	. . . . . 6 ⊢ (𝑃 ≠ ∅ ↔ X𝑖 ∈ 𝐴 (𝑁‘𝑖) ≠ ∅)
91	19	rgenw 2908	. . . . . . . . 9 ⊢ ∀𝑖 ∈ 𝐴 (𝑁‘𝑖) ∈ V
92		ixpexg 7818	. . . . . . . . 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 2684	. . . . . . 7 ⊢ 𝑃 ∈ V
95	94	0sdom 7976	. . . . . 6 ⊢ (∅ ≺ 𝑃 ↔ 𝑃 ≠ ∅)
96	18, 19	ac9 9188	. . . . . 6 ⊢ (∀𝑖 ∈ 𝐴 (𝑁‘𝑖) ≠ ∅ ↔ X𝑖 ∈ 𝐴 (𝑁‘𝑖) ≠ ∅)
97	90, 95, 96	3bitr4i 291	. . . . 5 ⊢ (∅ ≺ 𝑃 ↔ ∀𝑖 ∈ 𝐴 (𝑁‘𝑖) ≠ ∅)
98	89, 97	sylibr 223	. . . 4 ⊢ (∀𝑖 ∈ 𝐴 (𝑀‘𝑖) ≺ (𝑁‘𝑖) → ∅ ≺ 𝑃)
99	18, 1	iunex 7039	. . . . . . 7 ⊢ ∪ 𝑖 ∈ 𝐴 (𝑀‘𝑖) ∈ V
100	42, 99	eqeltri 2684	. . . . . 6 ⊢ 𝑆 ∈ V
101		domtri 9257	. . . . . 6 ⊢ ((𝑃 ∈ V ∧ 𝑆 ∈ V) → (𝑃 ≼ 𝑆 ↔ ¬ 𝑆 ≺ 𝑃))
102	94, 100, 101	mp2an 704	. . . . 5 ⊢ (𝑃 ≼ 𝑆 ↔ ¬ 𝑆 ≺ 𝑃)
103	102	biimpri 217	. . . 4 ⊢ (¬ 𝑆 ≺ 𝑃 → 𝑃 ≼ 𝑆)
104		fodomr 7996	. . . 4 ⊢ ((∅ ≺ 𝑃 ∧ 𝑃 ≼ 𝑆) → ∃𝑓 𝑓:𝑆–onto→𝑃)
105	98, 103, 104	syl2an 493	. . 3 ⊢ ((∀𝑖 ∈ 𝐴 (𝑀‘𝑖) ≺ (𝑁‘𝑖) ∧ ¬ 𝑆 ≺ 𝑃) → ∃𝑓 𝑓:𝑆–onto→𝑃)
106	83, 105	mtand 689	. 2 ⊢ (∀𝑖 ∈ 𝐴 (𝑀‘𝑖) ≺ (𝑁‘𝑖) → ¬ ¬ 𝑆 ≺ 𝑃)
107	106	notnotrd 127	1 ⊢ (∀𝑖 ∈ 𝐴 (𝑀‘𝑖) ≺ (𝑁‘𝑖) → 𝑆 ≺ 𝑃)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 195   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977   ≠ wne 2780  ∀wral 2896  ∃wrex 2897  Vcvv 3173   ∖ cdif 3537  ∅c0 3874  ∪ ciun 4455   class class class wbr 4583   ↦ cmpt 4643  ran crn 5039   Fn wfn 5799  –onto→wfo 5802  ‘cfv 5804  Xcixp 7794   ≼ cdom 7839   ≺ csdm 7840

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-ac2 9168

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rmo 2904  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-int 4411  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-ord 5643  df-on 5644  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-isom 5813  df-riota 6511  df-ov 6552  df-oprab 6553  df-mpt2 6554  df-1st 7059  df-2nd 7060  df-wrecs 7294  df-recs 7355  df-er 7629  df-map 7746  df-ixp 7795  df-en 7842  df-dom 7843  df-sdom 7844  df-card 8648  df-acn 8651  df-ac 8822

This theorem is referenced by:  konigth  9270