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Theorem cofsmo 9491
 Description: Any cofinal map implies the existence of a strictly monotone cofinal map with a domain no larger than the original. Proposition 11.7 of [TakeutiZaring] p. 101. (Contributed by Mario Carneiro, 20-Mar-2013.)
Hypotheses
Ref Expression
cofsmo.1 𝐶 = {𝑦𝐵 ∣ ∀𝑤𝑦 (𝑓𝑤) ∈ (𝑓𝑦)}
cofsmo.2 𝐾 = {𝑥𝐵𝑧 ⊆ (𝑓𝑥)}
cofsmo.3 𝑂 = OrdIso( E , 𝐶)
Assertion
Ref Expression
cofsmo ((Ord 𝐴𝐵 ∈ On) → (∃𝑓(𝑓:𝐵𝐴 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤)) → ∃𝑥 ∈ suc 𝐵𝑔(𝑔:𝑥𝐴 ∧ Smo 𝑔 ∧ ∀𝑧𝐴𝑣𝑥 𝑧 ⊆ (𝑔𝑣))))
Distinct variable groups:   𝑓,𝑔,𝑣,𝑤,𝑥,𝑧,𝐴   𝑦,𝑓,𝐵,𝑣,𝑤,𝑥,𝑧   𝑣,𝐶   𝑣,𝐾,𝑤,𝑦   𝑔,𝑂,𝑣,𝑥,𝑧
Allowed substitution hints:   𝐴(𝑦)   𝐵(𝑔)   𝐶(𝑥,𝑦,𝑧,𝑤,𝑓,𝑔)   𝐾(𝑥,𝑧,𝑓,𝑔)   𝑂(𝑦,𝑤,𝑓)

Proof of Theorem cofsmo
Dummy variables 𝑠 𝑡 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 cofsmo.1 . . . . . . . . . . . . 13 𝐶 = {𝑦𝐵 ∣ ∀𝑤𝑦 (𝑓𝑤) ∈ (𝑓𝑦)}
21ssrab3 3949 . . . . . . . . . . . 12 𝐶𝐵
3 ssexg 5084 . . . . . . . . . . . 12 ((𝐶𝐵𝐵 ∈ On) → 𝐶 ∈ V)
42, 3mpan 677 . . . . . . . . . . 11 (𝐵 ∈ On → 𝐶 ∈ V)
5 onss 7323 . . . . . . . . . . . . 13 (𝐵 ∈ On → 𝐵 ⊆ On)
62, 5syl5ss 3871 . . . . . . . . . . . 12 (𝐵 ∈ On → 𝐶 ⊆ On)
7 epweon 7315 . . . . . . . . . . . 12 E We On
8 wess 5395 . . . . . . . . . . . 12 (𝐶 ⊆ On → ( E We On → E We 𝐶))
96, 7, 8mpisyl 21 . . . . . . . . . . 11 (𝐵 ∈ On → E We 𝐶)
10 cofsmo.3 . . . . . . . . . . . 12 𝑂 = OrdIso( E , 𝐶)
1110oiiso 8798 . . . . . . . . . . 11 ((𝐶 ∈ V ∧ E We 𝐶) → 𝑂 Isom E , E (dom 𝑂, 𝐶))
124, 9, 11syl2anc 576 . . . . . . . . . 10 (𝐵 ∈ On → 𝑂 Isom E , E (dom 𝑂, 𝐶))
1312ad2antlr 714 . . . . . . . . 9 (((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) → 𝑂 Isom E , E (dom 𝑂, 𝐶))
14 isof1o 6901 . . . . . . . . 9 (𝑂 Isom E , E (dom 𝑂, 𝐶) → 𝑂:dom 𝑂1-1-onto𝐶)
15 f1ofo 6453 . . . . . . . . 9 (𝑂:dom 𝑂1-1-onto𝐶𝑂:dom 𝑂onto𝐶)
1613, 14, 153syl 18 . . . . . . . 8 (((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) → 𝑂:dom 𝑂onto𝐶)
17 fof 6421 . . . . . . . . 9 (𝑂:dom 𝑂onto𝐶𝑂:dom 𝑂𝐶)
18 fss 6359 . . . . . . . . 9 ((𝑂:dom 𝑂𝐶𝐶𝐵) → 𝑂:dom 𝑂𝐵)
1917, 2, 18sylancl 577 . . . . . . . 8 (𝑂:dom 𝑂onto𝐶𝑂:dom 𝑂𝐵)
2016, 19syl 17 . . . . . . 7 (((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) → 𝑂:dom 𝑂𝐵)
2110oion 8797 . . . . . . . . . 10 (𝐶 ∈ V → dom 𝑂 ∈ On)
224, 21syl 17 . . . . . . . . 9 (𝐵 ∈ On → dom 𝑂 ∈ On)
2322ad2antlr 714 . . . . . . . 8 (((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) → dom 𝑂 ∈ On)
24 simplr 756 . . . . . . . 8 (((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) → 𝐵 ∈ On)
25 eloni 6041 . . . . . . . . . . 11 (dom 𝑂 ∈ On → Ord dom 𝑂)
26 smoiso2 7812 . . . . . . . . . . 11 ((Ord dom 𝑂𝐶 ⊆ On) → ((𝑂:dom 𝑂onto𝐶 ∧ Smo 𝑂) ↔ 𝑂 Isom E , E (dom 𝑂, 𝐶)))
2725, 6, 26syl2an 586 . . . . . . . . . 10 ((dom 𝑂 ∈ On ∧ 𝐵 ∈ On) → ((𝑂:dom 𝑂onto𝐶 ∧ Smo 𝑂) ↔ 𝑂 Isom E , E (dom 𝑂, 𝐶)))
2827biimpar 470 . . . . . . . . 9 (((dom 𝑂 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑂 Isom E , E (dom 𝑂, 𝐶)) → (𝑂:dom 𝑂onto𝐶 ∧ Smo 𝑂))
2928simprd 488 . . . . . . . 8 (((dom 𝑂 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑂 Isom E , E (dom 𝑂, 𝐶)) → Smo 𝑂)
3023, 24, 13, 29syl21anc 825 . . . . . . 7 (((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) → Smo 𝑂)
31 eloni 6041 . . . . . . . 8 (𝐵 ∈ On → Ord 𝐵)
3231ad2antlr 714 . . . . . . 7 (((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) → Ord 𝐵)
33 smorndom 7811 . . . . . . 7 ((𝑂:dom 𝑂𝐵 ∧ Smo 𝑂 ∧ Ord 𝐵) → dom 𝑂𝐵)
3420, 30, 32, 33syl3anc 1351 . . . . . 6 (((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) → dom 𝑂𝐵)
35 onsssuc 6118 . . . . . . 7 ((dom 𝑂 ∈ On ∧ 𝐵 ∈ On) → (dom 𝑂𝐵 ↔ dom 𝑂 ∈ suc 𝐵))
3623, 24, 35syl2anc 576 . . . . . 6 (((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) → (dom 𝑂𝐵 ↔ dom 𝑂 ∈ suc 𝐵))
3734, 36mpbid 224 . . . . 5 (((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) → dom 𝑂 ∈ suc 𝐵)
3837adantrr 704 . . . 4 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤))) → dom 𝑂 ∈ suc 𝐵)
39 vex 3418 . . . . . 6 𝑓 ∈ V
4010oiexg 8796 . . . . . . . 8 (𝐶 ∈ V → 𝑂 ∈ V)
414, 40syl 17 . . . . . . 7 (𝐵 ∈ On → 𝑂 ∈ V)
4241ad2antlr 714 . . . . . 6 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤))) → 𝑂 ∈ V)
43 coexg 7451 . . . . . 6 ((𝑓 ∈ V ∧ 𝑂 ∈ V) → (𝑓𝑂) ∈ V)
4439, 42, 43sylancr 578 . . . . 5 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤))) → (𝑓𝑂) ∈ V)
45 simprl 758 . . . . . . 7 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤))) → 𝑓:𝐵𝐴)
4620adantrr 704 . . . . . . 7 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤))) → 𝑂:dom 𝑂𝐵)
47 fco 6363 . . . . . . 7 ((𝑓:𝐵𝐴𝑂:dom 𝑂𝐵) → (𝑓𝑂):dom 𝑂𝐴)
4845, 46, 47syl2anc 576 . . . . . 6 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤))) → (𝑓𝑂):dom 𝑂𝐴)
49 simpr 477 . . . . . . . . 9 (((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) → 𝑓:𝐵𝐴)
5049, 20, 47syl2anc 576 . . . . . . . 8 (((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) → (𝑓𝑂):dom 𝑂𝐴)
51 ordsson 7322 . . . . . . . . 9 (Ord 𝐴𝐴 ⊆ On)
5251ad2antrr 713 . . . . . . . 8 (((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) → 𝐴 ⊆ On)
5323, 25syl 17 . . . . . . . 8 (((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) → Ord dom 𝑂)
5416, 17syl 17 . . . . . . . . . . . 12 (((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) → 𝑂:dom 𝑂𝐶)
55 simpl 475 . . . . . . . . . . . 12 ((𝑠 ∈ dom 𝑂𝑡𝑠) → 𝑠 ∈ dom 𝑂)
56 ffvelrn 6676 . . . . . . . . . . . 12 ((𝑂:dom 𝑂𝐶𝑠 ∈ dom 𝑂) → (𝑂𝑠) ∈ 𝐶)
5754, 55, 56syl2an 586 . . . . . . . . . . 11 ((((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) ∧ (𝑠 ∈ dom 𝑂𝑡𝑠)) → (𝑂𝑠) ∈ 𝐶)
58 ffn 6346 . . . . . . . . . . . . . 14 (𝑂:dom 𝑂𝐶𝑂 Fn dom 𝑂)
5916, 17, 583syl 18 . . . . . . . . . . . . 13 (((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) → 𝑂 Fn dom 𝑂)
6059, 30jca 504 . . . . . . . . . . . 12 (((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) → (𝑂 Fn dom 𝑂 ∧ Smo 𝑂))
61 smoel2 7806 . . . . . . . . . . . 12 (((𝑂 Fn dom 𝑂 ∧ Smo 𝑂) ∧ (𝑠 ∈ dom 𝑂𝑡𝑠)) → (𝑂𝑡) ∈ (𝑂𝑠))
6260, 61sylan 572 . . . . . . . . . . 11 ((((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) ∧ (𝑠 ∈ dom 𝑂𝑡𝑠)) → (𝑂𝑡) ∈ (𝑂𝑠))
63 fveq2 6501 . . . . . . . . . . . . . . . 16 (𝑧 = (𝑂𝑠) → (𝑓𝑧) = (𝑓‘(𝑂𝑠)))
6463eleq2d 2851 . . . . . . . . . . . . . . 15 (𝑧 = (𝑂𝑠) → ((𝑓𝑥) ∈ (𝑓𝑧) ↔ (𝑓𝑥) ∈ (𝑓‘(𝑂𝑠))))
6564raleqbi1dv 3343 . . . . . . . . . . . . . 14 (𝑧 = (𝑂𝑠) → (∀𝑥𝑧 (𝑓𝑥) ∈ (𝑓𝑧) ↔ ∀𝑥 ∈ (𝑂𝑠)(𝑓𝑥) ∈ (𝑓‘(𝑂𝑠))))
66 fveq2 6501 . . . . . . . . . . . . . . . . . . 19 (𝑤 = 𝑥 → (𝑓𝑤) = (𝑓𝑥))
6766eleq1d 2850 . . . . . . . . . . . . . . . . . 18 (𝑤 = 𝑥 → ((𝑓𝑤) ∈ (𝑓𝑦) ↔ (𝑓𝑥) ∈ (𝑓𝑦)))
6867cbvralv 3383 . . . . . . . . . . . . . . . . 17 (∀𝑤𝑦 (𝑓𝑤) ∈ (𝑓𝑦) ↔ ∀𝑥𝑦 (𝑓𝑥) ∈ (𝑓𝑦))
69 fveq2 6501 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑧 → (𝑓𝑦) = (𝑓𝑧))
7069eleq2d 2851 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑧 → ((𝑓𝑥) ∈ (𝑓𝑦) ↔ (𝑓𝑥) ∈ (𝑓𝑧)))
7170raleqbi1dv 3343 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑧 → (∀𝑥𝑦 (𝑓𝑥) ∈ (𝑓𝑦) ↔ ∀𝑥𝑧 (𝑓𝑥) ∈ (𝑓𝑧)))
7268, 71syl5bb 275 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑧 → (∀𝑤𝑦 (𝑓𝑤) ∈ (𝑓𝑦) ↔ ∀𝑥𝑧 (𝑓𝑥) ∈ (𝑓𝑧)))
7372cbvrabv 3412 . . . . . . . . . . . . . . 15 {𝑦𝐵 ∣ ∀𝑤𝑦 (𝑓𝑤) ∈ (𝑓𝑦)} = {𝑧𝐵 ∣ ∀𝑥𝑧 (𝑓𝑥) ∈ (𝑓𝑧)}
741, 73eqtri 2802 . . . . . . . . . . . . . 14 𝐶 = {𝑧𝐵 ∣ ∀𝑥𝑧 (𝑓𝑥) ∈ (𝑓𝑧)}
7565, 74elrab2 3599 . . . . . . . . . . . . 13 ((𝑂𝑠) ∈ 𝐶 ↔ ((𝑂𝑠) ∈ 𝐵 ∧ ∀𝑥 ∈ (𝑂𝑠)(𝑓𝑥) ∈ (𝑓‘(𝑂𝑠))))
7675simprbi 489 . . . . . . . . . . . 12 ((𝑂𝑠) ∈ 𝐶 → ∀𝑥 ∈ (𝑂𝑠)(𝑓𝑥) ∈ (𝑓‘(𝑂𝑠)))
77 fveq2 6501 . . . . . . . . . . . . . 14 (𝑥 = (𝑂𝑡) → (𝑓𝑥) = (𝑓‘(𝑂𝑡)))
7877eleq1d 2850 . . . . . . . . . . . . 13 (𝑥 = (𝑂𝑡) → ((𝑓𝑥) ∈ (𝑓‘(𝑂𝑠)) ↔ (𝑓‘(𝑂𝑡)) ∈ (𝑓‘(𝑂𝑠))))
7978rspccv 3532 . . . . . . . . . . . 12 (∀𝑥 ∈ (𝑂𝑠)(𝑓𝑥) ∈ (𝑓‘(𝑂𝑠)) → ((𝑂𝑡) ∈ (𝑂𝑠) → (𝑓‘(𝑂𝑡)) ∈ (𝑓‘(𝑂𝑠))))
8076, 79syl 17 . . . . . . . . . . 11 ((𝑂𝑠) ∈ 𝐶 → ((𝑂𝑡) ∈ (𝑂𝑠) → (𝑓‘(𝑂𝑡)) ∈ (𝑓‘(𝑂𝑠))))
8157, 62, 80sylc 65 . . . . . . . . . 10 ((((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) ∧ (𝑠 ∈ dom 𝑂𝑡𝑠)) → (𝑓‘(𝑂𝑡)) ∈ (𝑓‘(𝑂𝑠)))
82 ordtr1 6074 . . . . . . . . . . . . . 14 (Ord dom 𝑂 → ((𝑡𝑠𝑠 ∈ dom 𝑂) → 𝑡 ∈ dom 𝑂))
8382ancomsd 458 . . . . . . . . . . . . 13 (Ord dom 𝑂 → ((𝑠 ∈ dom 𝑂𝑡𝑠) → 𝑡 ∈ dom 𝑂))
8423, 25, 833syl 18 . . . . . . . . . . . 12 (((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) → ((𝑠 ∈ dom 𝑂𝑡𝑠) → 𝑡 ∈ dom 𝑂))
8584imp 398 . . . . . . . . . . 11 ((((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) ∧ (𝑠 ∈ dom 𝑂𝑡𝑠)) → 𝑡 ∈ dom 𝑂)
86 fvco3 6590 . . . . . . . . . . 11 ((𝑂:dom 𝑂𝐵𝑡 ∈ dom 𝑂) → ((𝑓𝑂)‘𝑡) = (𝑓‘(𝑂𝑡)))
8720, 85, 86syl2an2r 672 . . . . . . . . . 10 ((((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) ∧ (𝑠 ∈ dom 𝑂𝑡𝑠)) → ((𝑓𝑂)‘𝑡) = (𝑓‘(𝑂𝑡)))
88 simprl 758 . . . . . . . . . . 11 ((((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) ∧ (𝑠 ∈ dom 𝑂𝑡𝑠)) → 𝑠 ∈ dom 𝑂)
89 fvco3 6590 . . . . . . . . . . 11 ((𝑂:dom 𝑂𝐵𝑠 ∈ dom 𝑂) → ((𝑓𝑂)‘𝑠) = (𝑓‘(𝑂𝑠)))
9020, 88, 89syl2an2r 672 . . . . . . . . . 10 ((((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) ∧ (𝑠 ∈ dom 𝑂𝑡𝑠)) → ((𝑓𝑂)‘𝑠) = (𝑓‘(𝑂𝑠)))
9181, 87, 903eltr4d 2881 . . . . . . . . 9 ((((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) ∧ (𝑠 ∈ dom 𝑂𝑡𝑠)) → ((𝑓𝑂)‘𝑡) ∈ ((𝑓𝑂)‘𝑠))
9291ralrimivva 3141 . . . . . . . 8 (((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) → ∀𝑠 ∈ dom 𝑂𝑡𝑠 ((𝑓𝑂)‘𝑡) ∈ ((𝑓𝑂)‘𝑠))
93 issmo2 7792 . . . . . . . . 9 ((𝑓𝑂):dom 𝑂𝐴 → ((𝐴 ⊆ On ∧ Ord dom 𝑂 ∧ ∀𝑠 ∈ dom 𝑂𝑡𝑠 ((𝑓𝑂)‘𝑡) ∈ ((𝑓𝑂)‘𝑠)) → Smo (𝑓𝑂)))
9493imp 398 . . . . . . . 8 (((𝑓𝑂):dom 𝑂𝐴 ∧ (𝐴 ⊆ On ∧ Ord dom 𝑂 ∧ ∀𝑠 ∈ dom 𝑂𝑡𝑠 ((𝑓𝑂)‘𝑡) ∈ ((𝑓𝑂)‘𝑠))) → Smo (𝑓𝑂))
9550, 52, 53, 92, 94syl13anc 1352 . . . . . . 7 (((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) → Smo (𝑓𝑂))
9695adantrr 704 . . . . . 6 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤))) → Smo (𝑓𝑂))
97 rabn0 4227 . . . . . . . . . . . . . . . . . 18 ({𝑤𝐵𝑧 ⊆ (𝑓𝑤)} ≠ ∅ ↔ ∃𝑤𝐵 𝑧 ⊆ (𝑓𝑤))
98 ssrab2 3948 . . . . . . . . . . . . . . . . . . . 20 {𝑤𝐵𝑧 ⊆ (𝑓𝑤)} ⊆ 𝐵
9998, 5syl5ss 3871 . . . . . . . . . . . . . . . . . . 19 (𝐵 ∈ On → {𝑤𝐵𝑧 ⊆ (𝑓𝑤)} ⊆ On)
100 cofsmo.2 . . . . . . . . . . . . . . . . . . . . 21 𝐾 = {𝑥𝐵𝑧 ⊆ (𝑓𝑥)}
101 fveq2 6501 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = 𝑤 → (𝑓𝑥) = (𝑓𝑤))
102101sseq2d 3891 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = 𝑤 → (𝑧 ⊆ (𝑓𝑥) ↔ 𝑧 ⊆ (𝑓𝑤)))
103102cbvrabv 3412 . . . . . . . . . . . . . . . . . . . . . 22 {𝑥𝐵𝑧 ⊆ (𝑓𝑥)} = {𝑤𝐵𝑧 ⊆ (𝑓𝑤)}
104103inteqi 4754 . . . . . . . . . . . . . . . . . . . . 21 {𝑥𝐵𝑧 ⊆ (𝑓𝑥)} = {𝑤𝐵𝑧 ⊆ (𝑓𝑤)}
105100, 104eqtri 2802 . . . . . . . . . . . . . . . . . . . 20 𝐾 = {𝑤𝐵𝑧 ⊆ (𝑓𝑤)}
106 onint 7328 . . . . . . . . . . . . . . . . . . . 20 (({𝑤𝐵𝑧 ⊆ (𝑓𝑤)} ⊆ On ∧ {𝑤𝐵𝑧 ⊆ (𝑓𝑤)} ≠ ∅) → {𝑤𝐵𝑧 ⊆ (𝑓𝑤)} ∈ {𝑤𝐵𝑧 ⊆ (𝑓𝑤)})
107105, 106syl5eqel 2870 . . . . . . . . . . . . . . . . . . 19 (({𝑤𝐵𝑧 ⊆ (𝑓𝑤)} ⊆ On ∧ {𝑤𝐵𝑧 ⊆ (𝑓𝑤)} ≠ ∅) → 𝐾 ∈ {𝑤𝐵𝑧 ⊆ (𝑓𝑤)})
10899, 107sylan 572 . . . . . . . . . . . . . . . . . 18 ((𝐵 ∈ On ∧ {𝑤𝐵𝑧 ⊆ (𝑓𝑤)} ≠ ∅) → 𝐾 ∈ {𝑤𝐵𝑧 ⊆ (𝑓𝑤)})
10997, 108sylan2br 585 . . . . . . . . . . . . . . . . 17 ((𝐵 ∈ On ∧ ∃𝑤𝐵 𝑧 ⊆ (𝑓𝑤)) → 𝐾 ∈ {𝑤𝐵𝑧 ⊆ (𝑓𝑤)})
110 fveq2 6501 . . . . . . . . . . . . . . . . . . 19 (𝑤 = 𝐾 → (𝑓𝑤) = (𝑓𝐾))
111110sseq2d 3891 . . . . . . . . . . . . . . . . . 18 (𝑤 = 𝐾 → (𝑧 ⊆ (𝑓𝑤) ↔ 𝑧 ⊆ (𝑓𝐾)))
112111elrab 3595 . . . . . . . . . . . . . . . . 17 (𝐾 ∈ {𝑤𝐵𝑧 ⊆ (𝑓𝑤)} ↔ (𝐾𝐵𝑧 ⊆ (𝑓𝐾)))
113109, 112sylib 210 . . . . . . . . . . . . . . . 16 ((𝐵 ∈ On ∧ ∃𝑤𝐵 𝑧 ⊆ (𝑓𝑤)) → (𝐾𝐵𝑧 ⊆ (𝑓𝐾)))
114113ex 405 . . . . . . . . . . . . . . 15 (𝐵 ∈ On → (∃𝑤𝐵 𝑧 ⊆ (𝑓𝑤) → (𝐾𝐵𝑧 ⊆ (𝑓𝐾))))
115114adantl 474 . . . . . . . . . . . . . 14 ((Ord 𝐴𝐵 ∈ On) → (∃𝑤𝐵 𝑧 ⊆ (𝑓𝑤) → (𝐾𝐵𝑧 ⊆ (𝑓𝐾))))
116 simpr2 1175 . . . . . . . . . . . . . . . . . 18 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴𝐾𝐵𝑧 ⊆ (𝑓𝐾))) → 𝐾𝐵)
117 simp3 1118 . . . . . . . . . . . . . . . . . . . . 21 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴𝐾𝐵𝑧 ⊆ (𝑓𝐾)) ∧ 𝑤𝐾) → 𝑤𝐾)
118105eleq2i 2857 . . . . . . . . . . . . . . . . . . . . . 22 (𝑤𝐾𝑤 {𝑤𝐵𝑧 ⊆ (𝑓𝑤)})
119 simp21 1186 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴𝐾𝐵𝑧 ⊆ (𝑓𝐾)) ∧ 𝑤𝐾) → 𝑓:𝐵𝐴)
120 simp1l 1177 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴𝐾𝐵𝑧 ⊆ (𝑓𝐾)) ∧ 𝑤𝐾) → Ord 𝐴)
121120, 51syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴𝐾𝐵𝑧 ⊆ (𝑓𝐾)) ∧ 𝑤𝐾) → 𝐴 ⊆ On)
122119, 121fssd 6360 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴𝐾𝐵𝑧 ⊆ (𝑓𝐾)) ∧ 𝑤𝐾) → 𝑓:𝐵⟶On)
123 simp22 1187 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴𝐾𝐵𝑧 ⊆ (𝑓𝐾)) ∧ 𝑤𝐾) → 𝐾𝐵)
124122, 123ffvelrnd 6679 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴𝐾𝐵𝑧 ⊆ (𝑓𝐾)) ∧ 𝑤𝐾) → (𝑓𝐾) ∈ On)
125 simp1r 1178 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴𝐾𝐵𝑧 ⊆ (𝑓𝐾)) ∧ 𝑤𝐾) → 𝐵 ∈ On)
126 ontr1 6077 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝐵 ∈ On → ((𝑤𝐾𝐾𝐵) → 𝑤𝐵))
1271263impib 1096 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝐵 ∈ On ∧ 𝑤𝐾𝐾𝐵) → 𝑤𝐵)
128125, 117, 123, 127syl3anc 1351 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴𝐾𝐵𝑧 ⊆ (𝑓𝐾)) ∧ 𝑤𝐾) → 𝑤𝐵)
129122, 128ffvelrnd 6679 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴𝐾𝐵𝑧 ⊆ (𝑓𝐾)) ∧ 𝑤𝐾) → (𝑓𝑤) ∈ On)
130 ontri1 6065 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑓𝐾) ∈ On ∧ (𝑓𝑤) ∈ On) → ((𝑓𝐾) ⊆ (𝑓𝑤) ↔ ¬ (𝑓𝑤) ∈ (𝑓𝐾)))
131124, 129, 130syl2anc 576 . . . . . . . . . . . . . . . . . . . . . . . 24 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴𝐾𝐵𝑧 ⊆ (𝑓𝐾)) ∧ 𝑤𝐾) → ((𝑓𝐾) ⊆ (𝑓𝑤) ↔ ¬ (𝑓𝑤) ∈ (𝑓𝐾)))
132 simp23 1188 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴𝐾𝐵𝑧 ⊆ (𝑓𝐾)) ∧ 𝑤𝐾) → 𝑧 ⊆ (𝑓𝐾))
133 simpl1 1171 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝐵 ∈ On ∧ 𝑤𝐾𝐾𝐵) ∧ (𝑧 ⊆ (𝑓𝐾) ∧ (𝑓𝐾) ⊆ (𝑓𝑤))) → 𝐵 ∈ On)
134133, 99syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝐵 ∈ On ∧ 𝑤𝐾𝐾𝐵) ∧ (𝑧 ⊆ (𝑓𝐾) ∧ (𝑓𝐾) ⊆ (𝑓𝑤))) → {𝑤𝐵𝑧 ⊆ (𝑓𝑤)} ⊆ On)
135 sstr 3868 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑧 ⊆ (𝑓𝐾) ∧ (𝑓𝐾) ⊆ (𝑓𝑤)) → 𝑧 ⊆ (𝑓𝑤))
136127, 135anim12i 603 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝐵 ∈ On ∧ 𝑤𝐾𝐾𝐵) ∧ (𝑧 ⊆ (𝑓𝐾) ∧ (𝑓𝐾) ⊆ (𝑓𝑤))) → (𝑤𝐵𝑧 ⊆ (𝑓𝑤)))
137 rabid 3317 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑤 ∈ {𝑤𝐵𝑧 ⊆ (𝑓𝑤)} ↔ (𝑤𝐵𝑧 ⊆ (𝑓𝑤)))
138136, 137sylibr 226 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝐵 ∈ On ∧ 𝑤𝐾𝐾𝐵) ∧ (𝑧 ⊆ (𝑓𝐾) ∧ (𝑓𝐾) ⊆ (𝑓𝑤))) → 𝑤 ∈ {𝑤𝐵𝑧 ⊆ (𝑓𝑤)})
139 onnmin 7336 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (({𝑤𝐵𝑧 ⊆ (𝑓𝑤)} ⊆ On ∧ 𝑤 ∈ {𝑤𝐵𝑧 ⊆ (𝑓𝑤)}) → ¬ 𝑤 {𝑤𝐵𝑧 ⊆ (𝑓𝑤)})
140134, 138, 139syl2anc 576 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝐵 ∈ On ∧ 𝑤𝐾𝐾𝐵) ∧ (𝑧 ⊆ (𝑓𝐾) ∧ (𝑓𝐾) ⊆ (𝑓𝑤))) → ¬ 𝑤 {𝑤𝐵𝑧 ⊆ (𝑓𝑤)})
141140expr 449 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝐵 ∈ On ∧ 𝑤𝐾𝐾𝐵) ∧ 𝑧 ⊆ (𝑓𝐾)) → ((𝑓𝐾) ⊆ (𝑓𝑤) → ¬ 𝑤 {𝑤𝐵𝑧 ⊆ (𝑓𝑤)}))
142125, 117, 123, 132, 141syl31anc 1353 . . . . . . . . . . . . . . . . . . . . . . . 24 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴𝐾𝐵𝑧 ⊆ (𝑓𝐾)) ∧ 𝑤𝐾) → ((𝑓𝐾) ⊆ (𝑓𝑤) → ¬ 𝑤 {𝑤𝐵𝑧 ⊆ (𝑓𝑤)}))
143131, 142sylbird 252 . . . . . . . . . . . . . . . . . . . . . . 23 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴𝐾𝐵𝑧 ⊆ (𝑓𝐾)) ∧ 𝑤𝐾) → (¬ (𝑓𝑤) ∈ (𝑓𝐾) → ¬ 𝑤 {𝑤𝐵𝑧 ⊆ (𝑓𝑤)}))
144143con4d 115 . . . . . . . . . . . . . . . . . . . . . 22 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴𝐾𝐵𝑧 ⊆ (𝑓𝐾)) ∧ 𝑤𝐾) → (𝑤 {𝑤𝐵𝑧 ⊆ (𝑓𝑤)} → (𝑓𝑤) ∈ (𝑓𝐾)))
145118, 144syl5bi 234 . . . . . . . . . . . . . . . . . . . . 21 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴𝐾𝐵𝑧 ⊆ (𝑓𝐾)) ∧ 𝑤𝐾) → (𝑤𝐾 → (𝑓𝑤) ∈ (𝑓𝐾)))
146117, 145mpd 15 . . . . . . . . . . . . . . . . . . . 20 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴𝐾𝐵𝑧 ⊆ (𝑓𝐾)) ∧ 𝑤𝐾) → (𝑓𝑤) ∈ (𝑓𝐾))
1471463expia 1101 . . . . . . . . . . . . . . . . . . 19 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴𝐾𝐵𝑧 ⊆ (𝑓𝐾))) → (𝑤𝐾 → (𝑓𝑤) ∈ (𝑓𝐾)))
148147ralrimiv 3131 . . . . . . . . . . . . . . . . . 18 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴𝐾𝐵𝑧 ⊆ (𝑓𝐾))) → ∀𝑤𝐾 (𝑓𝑤) ∈ (𝑓𝐾))
149 fveq2 6501 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 = 𝐾 → (𝑓𝑦) = (𝑓𝐾))
150149eleq2d 2851 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 𝐾 → ((𝑓𝑤) ∈ (𝑓𝑦) ↔ (𝑓𝑤) ∈ (𝑓𝐾)))
151150raleqbi1dv 3343 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝐾 → (∀𝑤𝑦 (𝑓𝑤) ∈ (𝑓𝑦) ↔ ∀𝑤𝐾 (𝑓𝑤) ∈ (𝑓𝐾)))
152151, 1elrab2 3599 . . . . . . . . . . . . . . . . . 18 (𝐾𝐶 ↔ (𝐾𝐵 ∧ ∀𝑤𝐾 (𝑓𝑤) ∈ (𝑓𝐾)))
153116, 148, 152sylanbrc 575 . . . . . . . . . . . . . . . . 17 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴𝐾𝐵𝑧 ⊆ (𝑓𝐾))) → 𝐾𝐶)
154153expcom 406 . . . . . . . . . . . . . . . 16 ((𝑓:𝐵𝐴𝐾𝐵𝑧 ⊆ (𝑓𝐾)) → ((Ord 𝐴𝐵 ∈ On) → 𝐾𝐶))
1551543expib 1102 . . . . . . . . . . . . . . 15 (𝑓:𝐵𝐴 → ((𝐾𝐵𝑧 ⊆ (𝑓𝐾)) → ((Ord 𝐴𝐵 ∈ On) → 𝐾𝐶)))
156155com13 88 . . . . . . . . . . . . . 14 ((Ord 𝐴𝐵 ∈ On) → ((𝐾𝐵𝑧 ⊆ (𝑓𝐾)) → (𝑓:𝐵𝐴𝐾𝐶)))
157115, 156syld 47 . . . . . . . . . . . . 13 ((Ord 𝐴𝐵 ∈ On) → (∃𝑤𝐵 𝑧 ⊆ (𝑓𝑤) → (𝑓:𝐵𝐴𝐾𝐶)))
158157com23 86 . . . . . . . . . . . 12 ((Ord 𝐴𝐵 ∈ On) → (𝑓:𝐵𝐴 → (∃𝑤𝐵 𝑧 ⊆ (𝑓𝑤) → 𝐾𝐶)))
159158imp31 410 . . . . . . . . . . 11 ((((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) ∧ ∃𝑤𝐵 𝑧 ⊆ (𝑓𝑤)) → 𝐾𝐶)
160 foelrn 6697 . . . . . . . . . . 11 ((𝑂:dom 𝑂onto𝐶𝐾𝐶) → ∃𝑣 ∈ dom 𝑂 𝐾 = (𝑂𝑣))
16116, 159, 160syl2an2r 672 . . . . . . . . . 10 ((((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) ∧ ∃𝑤𝐵 𝑧 ⊆ (𝑓𝑤)) → ∃𝑣 ∈ dom 𝑂 𝐾 = (𝑂𝑣))
162 eleq1 2853 . . . . . . . . . . . . . . . 16 (𝐾 = (𝑂𝑣) → (𝐾 ∈ {𝑤𝐵𝑧 ⊆ (𝑓𝑤)} ↔ (𝑂𝑣) ∈ {𝑤𝐵𝑧 ⊆ (𝑓𝑤)}))
163162biimpcd 241 . . . . . . . . . . . . . . 15 (𝐾 ∈ {𝑤𝐵𝑧 ⊆ (𝑓𝑤)} → (𝐾 = (𝑂𝑣) → (𝑂𝑣) ∈ {𝑤𝐵𝑧 ⊆ (𝑓𝑤)}))
164 fveq2 6501 . . . . . . . . . . . . . . . . . 18 (𝑥 = (𝑂𝑣) → (𝑓𝑥) = (𝑓‘(𝑂𝑣)))
165164sseq2d 3891 . . . . . . . . . . . . . . . . 17 (𝑥 = (𝑂𝑣) → (𝑧 ⊆ (𝑓𝑥) ↔ 𝑧 ⊆ (𝑓‘(𝑂𝑣))))
16666sseq2d 3891 . . . . . . . . . . . . . . . . . 18 (𝑤 = 𝑥 → (𝑧 ⊆ (𝑓𝑤) ↔ 𝑧 ⊆ (𝑓𝑥)))
167166cbvrabv 3412 . . . . . . . . . . . . . . . . 17 {𝑤𝐵𝑧 ⊆ (𝑓𝑤)} = {𝑥𝐵𝑧 ⊆ (𝑓𝑥)}
168165, 167elrab2 3599 . . . . . . . . . . . . . . . 16 ((𝑂𝑣) ∈ {𝑤𝐵𝑧 ⊆ (𝑓𝑤)} ↔ ((𝑂𝑣) ∈ 𝐵𝑧 ⊆ (𝑓‘(𝑂𝑣))))
169168simprbi 489 . . . . . . . . . . . . . . 15 ((𝑂𝑣) ∈ {𝑤𝐵𝑧 ⊆ (𝑓𝑤)} → 𝑧 ⊆ (𝑓‘(𝑂𝑣)))
170163, 169syl6 35 . . . . . . . . . . . . . 14 (𝐾 ∈ {𝑤𝐵𝑧 ⊆ (𝑓𝑤)} → (𝐾 = (𝑂𝑣) → 𝑧 ⊆ (𝑓‘(𝑂𝑣))))
171109, 170syl 17 . . . . . . . . . . . . 13 ((𝐵 ∈ On ∧ ∃𝑤𝐵 𝑧 ⊆ (𝑓𝑤)) → (𝐾 = (𝑂𝑣) → 𝑧 ⊆ (𝑓‘(𝑂𝑣))))
172171ad5ant24 748 . . . . . . . . . . . 12 (((((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) ∧ ∃𝑤𝐵 𝑧 ⊆ (𝑓𝑤)) ∧ 𝑣 ∈ dom 𝑂) → (𝐾 = (𝑂𝑣) → 𝑧 ⊆ (𝑓‘(𝑂𝑣))))
17320ad2antrr 713 . . . . . . . . . . . . . 14 (((((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) ∧ ∃𝑤𝐵 𝑧 ⊆ (𝑓𝑤)) ∧ 𝑣 ∈ dom 𝑂) → 𝑂:dom 𝑂𝐵)
174 fvco3 6590 . . . . . . . . . . . . . 14 ((𝑂:dom 𝑂𝐵𝑣 ∈ dom 𝑂) → ((𝑓𝑂)‘𝑣) = (𝑓‘(𝑂𝑣)))
175173, 174sylancom 579 . . . . . . . . . . . . 13 (((((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) ∧ ∃𝑤𝐵 𝑧 ⊆ (𝑓𝑤)) ∧ 𝑣 ∈ dom 𝑂) → ((𝑓𝑂)‘𝑣) = (𝑓‘(𝑂𝑣)))
176175sseq2d 3891 . . . . . . . . . . . 12 (((((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) ∧ ∃𝑤𝐵 𝑧 ⊆ (𝑓𝑤)) ∧ 𝑣 ∈ dom 𝑂) → (𝑧 ⊆ ((𝑓𝑂)‘𝑣) ↔ 𝑧 ⊆ (𝑓‘(𝑂𝑣))))
177172, 176sylibrd 251 . . . . . . . . . . 11 (((((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) ∧ ∃𝑤𝐵 𝑧 ⊆ (𝑓𝑤)) ∧ 𝑣 ∈ dom 𝑂) → (𝐾 = (𝑂𝑣) → 𝑧 ⊆ ((𝑓𝑂)‘𝑣)))
178177reximdva 3219 . . . . . . . . . 10 ((((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) ∧ ∃𝑤𝐵 𝑧 ⊆ (𝑓𝑤)) → (∃𝑣 ∈ dom 𝑂 𝐾 = (𝑂𝑣) → ∃𝑣 ∈ dom 𝑂 𝑧 ⊆ ((𝑓𝑂)‘𝑣)))
179161, 178mpd 15 . . . . . . . . 9 ((((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) ∧ ∃𝑤𝐵 𝑧 ⊆ (𝑓𝑤)) → ∃𝑣 ∈ dom 𝑂 𝑧 ⊆ ((𝑓𝑂)‘𝑣))
180179ex 405 . . . . . . . 8 (((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) → (∃𝑤𝐵 𝑧 ⊆ (𝑓𝑤) → ∃𝑣 ∈ dom 𝑂 𝑧 ⊆ ((𝑓𝑂)‘𝑣)))
181180ralimdv 3128 . . . . . . 7 (((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) → (∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤) → ∀𝑧𝐴𝑣 ∈ dom 𝑂 𝑧 ⊆ ((𝑓𝑂)‘𝑣)))
182181impr 447 . . . . . 6 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤))) → ∀𝑧𝐴𝑣 ∈ dom 𝑂 𝑧 ⊆ ((𝑓𝑂)‘𝑣))
18348, 96, 1823jca 1108 . . . . 5 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤))) → ((𝑓𝑂):dom 𝑂𝐴 ∧ Smo (𝑓𝑂) ∧ ∀𝑧𝐴𝑣 ∈ dom 𝑂 𝑧 ⊆ ((𝑓𝑂)‘𝑣)))
184 feq1 6327 . . . . . 6 (𝑔 = (𝑓𝑂) → (𝑔:dom 𝑂𝐴 ↔ (𝑓𝑂):dom 𝑂𝐴))
185 smoeq 7793 . . . . . 6 (𝑔 = (𝑓𝑂) → (Smo 𝑔 ↔ Smo (𝑓𝑂)))
186 fveq1 6500 . . . . . . . . 9 (𝑔 = (𝑓𝑂) → (𝑔𝑣) = ((𝑓𝑂)‘𝑣))
187186sseq2d 3891 . . . . . . . 8 (𝑔 = (𝑓𝑂) → (𝑧 ⊆ (𝑔𝑣) ↔ 𝑧 ⊆ ((𝑓𝑂)‘𝑣)))
188187rexbidv 3242 . . . . . . 7 (𝑔 = (𝑓𝑂) → (∃𝑣 ∈ dom 𝑂 𝑧 ⊆ (𝑔𝑣) ↔ ∃𝑣 ∈ dom 𝑂 𝑧 ⊆ ((𝑓𝑂)‘𝑣)))
189188ralbidv 3147 . . . . . 6 (𝑔 = (𝑓𝑂) → (∀𝑧𝐴𝑣 ∈ dom 𝑂 𝑧 ⊆ (𝑔𝑣) ↔ ∀𝑧𝐴𝑣 ∈ dom 𝑂 𝑧 ⊆ ((𝑓𝑂)‘𝑣)))
190184, 185, 1893anbi123d 1415 . . . . 5 (𝑔 = (𝑓𝑂) → ((𝑔:dom 𝑂𝐴 ∧ Smo 𝑔 ∧ ∀𝑧𝐴𝑣 ∈ dom 𝑂 𝑧 ⊆ (𝑔𝑣)) ↔ ((𝑓𝑂):dom 𝑂𝐴 ∧ Smo (𝑓𝑂) ∧ ∀𝑧𝐴𝑣 ∈ dom 𝑂 𝑧 ⊆ ((𝑓𝑂)‘𝑣))))
19144, 183, 190elabd 3583 . . . 4 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤))) → ∃𝑔(𝑔:dom 𝑂𝐴 ∧ Smo 𝑔 ∧ ∀𝑧𝐴𝑣 ∈ dom 𝑂 𝑧 ⊆ (𝑔𝑣)))
192 feq2 6328 . . . . . . 7 (𝑥 = dom 𝑂 → (𝑔:𝑥𝐴𝑔:dom 𝑂𝐴))
193 rexeq 3346 . . . . . . . 8 (𝑥 = dom 𝑂 → (∃𝑣𝑥 𝑧 ⊆ (𝑔𝑣) ↔ ∃𝑣 ∈ dom 𝑂 𝑧 ⊆ (𝑔𝑣)))
194193ralbidv 3147 . . . . . . 7 (𝑥 = dom 𝑂 → (∀𝑧𝐴𝑣𝑥 𝑧 ⊆ (𝑔𝑣) ↔ ∀𝑧𝐴𝑣 ∈ dom 𝑂 𝑧 ⊆ (𝑔𝑣)))
195192, 1943anbi13d 1417 . . . . . 6 (𝑥 = dom 𝑂 → ((𝑔:𝑥𝐴 ∧ Smo 𝑔 ∧ ∀𝑧𝐴𝑣𝑥 𝑧 ⊆ (𝑔𝑣)) ↔ (𝑔:dom 𝑂𝐴 ∧ Smo 𝑔 ∧ ∀𝑧𝐴𝑣 ∈ dom 𝑂 𝑧 ⊆ (𝑔𝑣))))
196195exbidv 1880 . . . . 5 (𝑥 = dom 𝑂 → (∃𝑔(𝑔:𝑥𝐴 ∧ Smo 𝑔 ∧ ∀𝑧𝐴𝑣𝑥 𝑧 ⊆ (𝑔𝑣)) ↔ ∃𝑔(𝑔:dom 𝑂𝐴 ∧ Smo 𝑔 ∧ ∀𝑧𝐴𝑣 ∈ dom 𝑂 𝑧 ⊆ (𝑔𝑣))))
197196rspcev 3535 . . . 4 ((dom 𝑂 ∈ suc 𝐵 ∧ ∃𝑔(𝑔:dom 𝑂𝐴 ∧ Smo 𝑔 ∧ ∀𝑧𝐴𝑣 ∈ dom 𝑂 𝑧 ⊆ (𝑔𝑣))) → ∃𝑥 ∈ suc 𝐵𝑔(𝑔:𝑥𝐴 ∧ Smo 𝑔 ∧ ∀𝑧𝐴𝑣𝑥 𝑧 ⊆ (𝑔𝑣)))
19838, 191, 197syl2anc 576 . . 3 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤))) → ∃𝑥 ∈ suc 𝐵𝑔(𝑔:𝑥𝐴 ∧ Smo 𝑔 ∧ ∀𝑧𝐴𝑣𝑥 𝑧 ⊆ (𝑔𝑣)))
199198ex 405 . 2 ((Ord 𝐴𝐵 ∈ On) → ((𝑓:𝐵𝐴 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤)) → ∃𝑥 ∈ suc 𝐵𝑔(𝑔:𝑥𝐴 ∧ Smo 𝑔 ∧ ∀𝑧𝐴𝑣𝑥 𝑧 ⊆ (𝑔𝑣))))
200199exlimdv 1892 1 ((Ord 𝐴𝐵 ∈ On) → (∃𝑓(𝑓:𝐵𝐴 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤)) → ∃𝑥 ∈ suc 𝐵𝑔(𝑔:𝑥𝐴 ∧ Smo 𝑔 ∧ ∀𝑧𝐴𝑣𝑥 𝑧 ⊆ (𝑔𝑣))))
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 198   ∧ wa 387   ∧ w3a 1068   = wceq 1507  ∃wex 1742   ∈ wcel 2050   ≠ wne 2967  ∀wral 3088  ∃wrex 3089  {crab 3092  Vcvv 3415   ⊆ wss 3831  ∅c0 4180  ∩ cint 4750   E cep 5317   We wwe 5366  dom cdm 5408   ∘ ccom 5412  Ord word 6030  Oncon0 6031  suc csuc 6033   Fn wfn 6185  ⟶wf 6186  –onto→wfo 6188  –1-1-onto→wf1o 6189  ‘cfv 6190   Isom wiso 6191  Smo wsmo 7788  OrdIsocoi 8770 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1758  ax-4 1772  ax-5 1869  ax-6 1928  ax-7 1965  ax-8 2052  ax-9 2059  ax-10 2079  ax-11 2093  ax-12 2106  ax-13 2301  ax-ext 2750  ax-rep 5050  ax-sep 5061  ax-nul 5068  ax-pow 5120  ax-pr 5187  ax-un 7281 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 834  df-3or 1069  df-3an 1070  df-tru 1510  df-ex 1743  df-nf 1747  df-sb 2016  df-mo 2547  df-eu 2583  df-clab 2759  df-cleq 2771  df-clel 2846  df-nfc 2918  df-ne 2968  df-ral 3093  df-rex 3094  df-reu 3095  df-rmo 3096  df-rab 3097  df-v 3417  df-sbc 3684  df-csb 3789  df-dif 3834  df-un 3836  df-in 3838  df-ss 3845  df-pss 3847  df-nul 4181  df-if 4352  df-pw 4425  df-sn 4443  df-pr 4445  df-tp 4447  df-op 4449  df-uni 4714  df-int 4751  df-iun 4795  df-br 4931  df-opab 4993  df-mpt 5010  df-tr 5032  df-id 5313  df-eprel 5318  df-po 5327  df-so 5328  df-fr 5367  df-se 5368  df-we 5369  df-xp 5414  df-rel 5415  df-cnv 5416  df-co 5417  df-dm 5418  df-rn 5419  df-res 5420  df-ima 5421  df-pred 5988  df-ord 6034  df-on 6035  df-lim 6036  df-suc 6037  df-iota 6154  df-fun 6192  df-fn 6193  df-f 6194  df-f1 6195  df-fo 6196  df-f1o 6197  df-fv 6198  df-isom 6199  df-riota 6939  df-wrecs 7752  df-smo 7789  df-recs 7814  df-oi 8771 This theorem is referenced by:  cfcof  9496
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