Theorem smogt 8368

Description: A strictly monotone ordinal function is greater than or equal to its argument. Exercise 1 in [TakeutiZaring] p. 50. (Contributed by Andrew Salmon, 23-Nov-2011.) (Revised by Mario Carneiro, 28-Feb-2013.)

Assertion

Ref	Expression
smogt	⊢ ((𝐹 Fn 𝐴 ∧ Smo 𝐹 ∧ 𝐶 ∈ 𝐴) → 𝐶 ⊆ (𝐹‘𝐶))

Proof of Theorem smogt

Dummy variables 𝑦 𝑥 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 23	. . . . . 6 ⊢ (𝑥 = 𝐶 → 𝑥 = 𝐶)
2		fveq2 6883	. . . . . 6 ⊢ (𝑥 = 𝐶 → (𝐹‘𝑥) = (𝐹‘𝐶))
3	1, 2	sseq12d 3964	. . . . 5 ⊢ (𝑥 = 𝐶 → (𝑥 ⊆ (𝐹‘𝑥) ↔ 𝐶 ⊆ (𝐹‘𝐶)))
4	3	imbi2d 343	. . . 4 ⊢ (𝑥 = 𝐶 → (((𝐹 Fn 𝐴 ∧ Smo 𝐹) → 𝑥 ⊆ (𝐹‘𝑥)) ↔ ((𝐹 Fn 𝐴 ∧ Smo 𝐹) → 𝐶 ⊆ (𝐹‘𝐶))))
5		smodm2 8356	. . . . . . . . . 10 ⊢ ((𝐹 Fn 𝐴 ∧ Smo 𝐹) → Ord 𝐴)
6	5	3adant3 1150	. . . . . . . . 9 ⊢ ((𝐹 Fn 𝐴 ∧ Smo 𝐹 ∧ 𝑥 ∈ 𝐴) → Ord 𝐴)
7		simp3 1156	. . . . . . . . 9 ⊢ ((𝐹 Fn 𝐴 ∧ Smo 𝐹 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ 𝐴)
8		ordelord 6383	. . . . . . . . 9 ⊢ ((Ord 𝐴 ∧ 𝑥 ∈ 𝐴) → Ord 𝑥)
9	6, 7, 8	syl2anc 596	. . . . . . . 8 ⊢ ((𝐹 Fn 𝐴 ∧ Smo 𝐹 ∧ 𝑥 ∈ 𝐴) → Ord 𝑥)
10		vex 3455	. . . . . . . . 9 ⊢ 𝑥 ∈ V
11	10	elon 6370	. . . . . . . 8 ⊢ (𝑥 ∈ On ↔ Ord 𝑥)
12	9, 11	sylibr 237	. . . . . . 7 ⊢ ((𝐹 Fn 𝐴 ∧ Smo 𝐹 ∧ 𝑥 ∈ 𝐴) → 𝑥 ∈ On)
13		eleq1w 2844	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝑥 ∈ 𝐴 ↔ 𝑦 ∈ 𝐴))
14	13	3anbi3d 1470	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → ((𝐹 Fn 𝐴 ∧ Smo 𝐹 ∧ 𝑥 ∈ 𝐴) ↔ (𝐹 Fn 𝐴 ∧ Smo 𝐹 ∧ 𝑦 ∈ 𝐴)))
15		id 23	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → 𝑥 = 𝑦)
16		fveq2 6883	. . . . . . . . . 10 ⊢ (𝑥 = 𝑦 → (𝐹‘𝑥) = (𝐹‘𝑦))
17	15, 16	sseq12d 3964	. . . . . . . . 9 ⊢ (𝑥 = 𝑦 → (𝑥 ⊆ (𝐹‘𝑥) ↔ 𝑦 ⊆ (𝐹‘𝑦)))
18	14, 17	imbi12d 347	. . . . . . . 8 ⊢ (𝑥 = 𝑦 → (((𝐹 Fn 𝐴 ∧ Smo 𝐹 ∧ 𝑥 ∈ 𝐴) → 𝑥 ⊆ (𝐹‘𝑥)) ↔ ((𝐹 Fn 𝐴 ∧ Smo 𝐹 ∧ 𝑦 ∈ 𝐴) → 𝑦 ⊆ (𝐹‘𝑦))))
19		simpl1 1210	. . . . . . . . . . . 12 ⊢ (((𝐹 Fn 𝐴 ∧ Smo 𝐹 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝑥) → 𝐹 Fn 𝐴)
20		simpl2 1211	. . . . . . . . . . . 12 ⊢ (((𝐹 Fn 𝐴 ∧ Smo 𝐹 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝑥) → Smo 𝐹)
21		ordtr1 6406	. . . . . . . . . . . . . . 15 ⊢ (Ord 𝐴 → ((𝑦 ∈ 𝑥 ∧ 𝑥 ∈ 𝐴) → 𝑦 ∈ 𝐴))
22	21	expcomd 422	. . . . . . . . . . . . . 14 ⊢ (Ord 𝐴 → (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐴)))
23	6, 7, 22	sylc 66	. . . . . . . . . . . . 13 ⊢ ((𝐹 Fn 𝐴 ∧ Smo 𝐹 ∧ 𝑥 ∈ 𝐴) → (𝑦 ∈ 𝑥 → 𝑦 ∈ 𝐴))
24	23	imp 412	. . . . . . . . . . . 12 ⊢ (((𝐹 Fn 𝐴 ∧ Smo 𝐹 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝑥) → 𝑦 ∈ 𝐴)
25		pm2.27 43	. . . . . . . . . . . 12 ⊢ ((𝐹 Fn 𝐴 ∧ Smo 𝐹 ∧ 𝑦 ∈ 𝐴) → (((𝐹 Fn 𝐴 ∧ Smo 𝐹 ∧ 𝑦 ∈ 𝐴) → 𝑦 ⊆ (𝐹‘𝑦)) → 𝑦 ⊆ (𝐹‘𝑦)))
26	19, 20, 24, 25	syl3anc 1398	. . . . . . . . . . 11 ⊢ (((𝐹 Fn 𝐴 ∧ Smo 𝐹 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝑥) → (((𝐹 Fn 𝐴 ∧ Smo 𝐹 ∧ 𝑦 ∈ 𝐴) → 𝑦 ⊆ (𝐹‘𝑦)) → 𝑦 ⊆ (𝐹‘𝑦)))
27	26	ralimdva 3175	. . . . . . . . . 10 ⊢ ((𝐹 Fn 𝐴 ∧ Smo 𝐹 ∧ 𝑥 ∈ 𝐴) → (∀𝑦 ∈ 𝑥 ((𝐹 Fn 𝐴 ∧ Smo 𝐹 ∧ 𝑦 ∈ 𝐴) → 𝑦 ⊆ (𝐹‘𝑦)) → ∀𝑦 ∈ 𝑥 𝑦 ⊆ (𝐹‘𝑦)))
28	5	3adant3 1150	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹 Fn 𝐴 ∧ Smo 𝐹 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥 ∧ 𝑦 ⊆ (𝐹‘𝑦))) → Ord 𝐴)
29		simp31 1228	. . . . . . . . . . . . . . . . . . 19 ⊢ ((𝐹 Fn 𝐴 ∧ Smo 𝐹 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥 ∧ 𝑦 ⊆ (𝐹‘𝑦))) → 𝑥 ∈ 𝐴)
30	28, 29, 8	syl2anc 596	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹 Fn 𝐴 ∧ Smo 𝐹 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥 ∧ 𝑦 ⊆ (𝐹‘𝑦))) → Ord 𝑥)
31		simp32 1229	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹 Fn 𝐴 ∧ Smo 𝐹 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥 ∧ 𝑦 ⊆ (𝐹‘𝑦))) → 𝑦 ∈ 𝑥)
32		ordelord 6383	. . . . . . . . . . . . . . . . . 18 ⊢ ((Ord 𝑥 ∧ 𝑦 ∈ 𝑥) → Ord 𝑦)
33	30, 31, 32	syl2anc 596	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹 Fn 𝐴 ∧ Smo 𝐹 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥 ∧ 𝑦 ⊆ (𝐹‘𝑦))) → Ord 𝑦)
34		smofvon2 8357	. . . . . . . . . . . . . . . . . . 19 ⊢ (Smo 𝐹 → (𝐹‘𝑥) ∈ On)
35	34	3ad2ant2 1152	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹 Fn 𝐴 ∧ Smo 𝐹 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥 ∧ 𝑦 ⊆ (𝐹‘𝑦))) → (𝐹‘𝑥) ∈ On)
36		eloni 6371	. . . . . . . . . . . . . . . . . 18 ⊢ ((𝐹‘𝑥) ∈ On → Ord (𝐹‘𝑥))
37	35, 36	syl 18	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹 Fn 𝐴 ∧ Smo 𝐹 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥 ∧ 𝑦 ⊆ (𝐹‘𝑦))) → Ord (𝐹‘𝑥))
38		simp33 1230	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹 Fn 𝐴 ∧ Smo 𝐹 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥 ∧ 𝑦 ⊆ (𝐹‘𝑦))) → 𝑦 ⊆ (𝐹‘𝑦))
39		smoel2 8364	. . . . . . . . . . . . . . . . . . 19 ⊢ (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥)) → (𝐹‘𝑦) ∈ (𝐹‘𝑥))
40	39	3adantr3 1190	. . . . . . . . . . . . . . . . . 18 ⊢ (((𝐹 Fn 𝐴 ∧ Smo 𝐹) ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥 ∧ 𝑦 ⊆ (𝐹‘𝑦))) → (𝐹‘𝑦) ∈ (𝐹‘𝑥))
41	40	3impa 1127	. . . . . . . . . . . . . . . . 17 ⊢ ((𝐹 Fn 𝐴 ∧ Smo 𝐹 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥 ∧ 𝑦 ⊆ (𝐹‘𝑦))) → (𝐹‘𝑦) ∈ (𝐹‘𝑥))
42		ordtr2 6407	. . . . . . . . . . . . . . . . . 18 ⊢ ((Ord 𝑦 ∧ Ord (𝐹‘𝑥)) → ((𝑦 ⊆ (𝐹‘𝑦) ∧ (𝐹‘𝑦) ∈ (𝐹‘𝑥)) → 𝑦 ∈ (𝐹‘𝑥)))
43	42	imp 412	. . . . . . . . . . . . . . . . 17 ⊢ (((Ord 𝑦 ∧ Ord (𝐹‘𝑥)) ∧ (𝑦 ⊆ (𝐹‘𝑦) ∧ (𝐹‘𝑦) ∈ (𝐹‘𝑥))) → 𝑦 ∈ (𝐹‘𝑥))
44	33, 37, 38, 41, 43	syl22anc 852	. . . . . . . . . . . . . . . 16 ⊢ ((𝐹 Fn 𝐴 ∧ Smo 𝐹 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥 ∧ 𝑦 ⊆ (𝐹‘𝑦))) → 𝑦 ∈ (𝐹‘𝑥))
45	44	3expia 1139	. . . . . . . . . . . . . . 15 ⊢ ((𝐹 Fn 𝐴 ∧ Smo 𝐹) → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝑥 ∧ 𝑦 ⊆ (𝐹‘𝑦)) → 𝑦 ∈ (𝐹‘𝑥)))
46	45	3expd 1372	. . . . . . . . . . . . . 14 ⊢ ((𝐹 Fn 𝐴 ∧ Smo 𝐹) → (𝑥 ∈ 𝐴 → (𝑦 ∈ 𝑥 → (𝑦 ⊆ (𝐹‘𝑦) → 𝑦 ∈ (𝐹‘𝑥)))))
47	46	3impia 1135	. . . . . . . . . . . . 13 ⊢ ((𝐹 Fn 𝐴 ∧ Smo 𝐹 ∧ 𝑥 ∈ 𝐴) → (𝑦 ∈ 𝑥 → (𝑦 ⊆ (𝐹‘𝑦) → 𝑦 ∈ (𝐹‘𝑥))))
48	47	imp 412	. . . . . . . . . . . 12 ⊢ (((𝐹 Fn 𝐴 ∧ Smo 𝐹 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝑥) → (𝑦 ⊆ (𝐹‘𝑦) → 𝑦 ∈ (𝐹‘𝑥)))
49	48	ralimdva 3175	. . . . . . . . . . 11 ⊢ ((𝐹 Fn 𝐴 ∧ Smo 𝐹 ∧ 𝑥 ∈ 𝐴) → (∀𝑦 ∈ 𝑥 𝑦 ⊆ (𝐹‘𝑦) → ∀𝑦 ∈ 𝑥 𝑦 ∈ (𝐹‘𝑥)))
50		dfss3 3920	. . . . . . . . . . 11 ⊢ (𝑥 ⊆ (𝐹‘𝑥) ↔ ∀𝑦 ∈ 𝑥 𝑦 ∈ (𝐹‘𝑥))
51	49, 50	imbitrrdi 255	. . . . . . . . . 10 ⊢ ((𝐹 Fn 𝐴 ∧ Smo 𝐹 ∧ 𝑥 ∈ 𝐴) → (∀𝑦 ∈ 𝑥 𝑦 ⊆ (𝐹‘𝑦) → 𝑥 ⊆ (𝐹‘𝑥)))
52	27, 51	syldc 49	. . . . . . . . 9 ⊢ (∀𝑦 ∈ 𝑥 ((𝐹 Fn 𝐴 ∧ Smo 𝐹 ∧ 𝑦 ∈ 𝐴) → 𝑦 ⊆ (𝐹‘𝑦)) → ((𝐹 Fn 𝐴 ∧ Smo 𝐹 ∧ 𝑥 ∈ 𝐴) → 𝑥 ⊆ (𝐹‘𝑥)))
53	52	a1i 11	. . . . . . . 8 ⊢ (𝑥 ∈ On → (∀𝑦 ∈ 𝑥 ((𝐹 Fn 𝐴 ∧ Smo 𝐹 ∧ 𝑦 ∈ 𝐴) → 𝑦 ⊆ (𝐹‘𝑦)) → ((𝐹 Fn 𝐴 ∧ Smo 𝐹 ∧ 𝑥 ∈ 𝐴) → 𝑥 ⊆ (𝐹‘𝑥))))
54	18, 53	tfis2 7866	. . . . . . 7 ⊢ (𝑥 ∈ On → ((𝐹 Fn 𝐴 ∧ Smo 𝐹 ∧ 𝑥 ∈ 𝐴) → 𝑥 ⊆ (𝐹‘𝑥)))
55	12, 54	mpcom 39	. . . . . 6 ⊢ ((𝐹 Fn 𝐴 ∧ Smo 𝐹 ∧ 𝑥 ∈ 𝐴) → 𝑥 ⊆ (𝐹‘𝑥))
56	55	3expia 1139	. . . . 5 ⊢ ((𝐹 Fn 𝐴 ∧ Smo 𝐹) → (𝑥 ∈ 𝐴 → 𝑥 ⊆ (𝐹‘𝑥)))
57	56	com12 33	. . . 4 ⊢ (𝑥 ∈ 𝐴 → ((𝐹 Fn 𝐴 ∧ Smo 𝐹) → 𝑥 ⊆ (𝐹‘𝑥)))
58	4, 57	vtoclga 3537	. . 3 ⊢ (𝐶 ∈ 𝐴 → ((𝐹 Fn 𝐴 ∧ Smo 𝐹) → 𝐶 ⊆ (𝐹‘𝐶)))
59	58	com12 33	. 2 ⊢ ((𝐹 Fn 𝐴 ∧ Smo 𝐹) → (𝐶 ∈ 𝐴 → 𝐶 ⊆ (𝐹‘𝐶)))
60	59	3impia 1135	1 ⊢ ((𝐹 Fn 𝐴 ∧ Smo 𝐹 ∧ 𝐶 ∈ 𝐴) → 𝐶 ⊆ (𝐹‘𝐶))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   ∧ w3a 1103   = wceq 1570   ∈ wcel 2145  ∀wral 3077   ⊆ wss 3899  Ord word 6360  Oncon0 6361   Fn wfn 6532  ‘cfv 6537  Smo wsmo 8346

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 2733  ax-sep 5249  ax-nul 5260  ax-pr 5391

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 2565  df-eu 2595  df-clab 2740  df-cleq 2753  df-clel 2836  df-nfc 2910  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  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-br 5104  df-opab 5168  df-tr 5213  df-id 5546  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-ord 6364  df-on 6365  df-iota 6493  df-fun 6539  df-fn 6540  df-f 6541  df-fv 6545  df-smo 8347

This theorem is used by:  smocdmdom  8369  oismo  9527