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Theorem cofsmo 10310
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 4081 . . . . . . . . . . . 12 𝐶𝐵
3 ssexg 5322 . . . . . . . . . . . 12 ((𝐶𝐵𝐵 ∈ On) → 𝐶 ∈ V)
42, 3mpan 690 . . . . . . . . . . 11 (𝐵 ∈ On → 𝐶 ∈ V)
5 onss 7806 . . . . . . . . . . . . 13 (𝐵 ∈ On → 𝐵 ⊆ On)
62, 5sstrid 3994 . . . . . . . . . . . 12 (𝐵 ∈ On → 𝐶 ⊆ On)
7 epweon 7796 . . . . . . . . . . . 12 E We On
8 wess 5670 . . . . . . . . . . . 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 9578 . . . . . . . . . . 11 ((𝐶 ∈ V ∧ E We 𝐶) → 𝑂 Isom E , E (dom 𝑂, 𝐶))
124, 9, 11syl2anc 584 . . . . . . . . . 10 (𝐵 ∈ On → 𝑂 Isom E , E (dom 𝑂, 𝐶))
1312ad2antlr 727 . . . . . . . . 9 (((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) → 𝑂 Isom E , E (dom 𝑂, 𝐶))
14 isof1o 7344 . . . . . . . . 9 (𝑂 Isom E , E (dom 𝑂, 𝐶) → 𝑂:dom 𝑂1-1-onto𝐶)
15 f1ofo 6854 . . . . . . . . 9 (𝑂:dom 𝑂1-1-onto𝐶𝑂:dom 𝑂onto𝐶)
1613, 14, 153syl 18 . . . . . . . 8 (((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) → 𝑂:dom 𝑂onto𝐶)
17 fof 6819 . . . . . . . . 9 (𝑂:dom 𝑂onto𝐶𝑂:dom 𝑂𝐶)
18 fss 6751 . . . . . . . . 9 ((𝑂:dom 𝑂𝐶𝐶𝐵) → 𝑂:dom 𝑂𝐵)
1917, 2, 18sylancl 586 . . . . . . . 8 (𝑂:dom 𝑂onto𝐶𝑂:dom 𝑂𝐵)
2016, 19syl 17 . . . . . . 7 (((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) → 𝑂:dom 𝑂𝐵)
2110oion 9577 . . . . . . . . . 10 (𝐶 ∈ V → dom 𝑂 ∈ On)
224, 21syl 17 . . . . . . . . 9 (𝐵 ∈ On → dom 𝑂 ∈ On)
2322ad2antlr 727 . . . . . . . 8 (((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) → dom 𝑂 ∈ On)
24 simplr 768 . . . . . . . 8 (((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) → 𝐵 ∈ On)
25 eloni 6393 . . . . . . . . . . 11 (dom 𝑂 ∈ On → Ord dom 𝑂)
26 smoiso2 8410 . . . . . . . . . . 11 ((Ord dom 𝑂𝐶 ⊆ On) → ((𝑂:dom 𝑂onto𝐶 ∧ Smo 𝑂) ↔ 𝑂 Isom E , E (dom 𝑂, 𝐶)))
2725, 6, 26syl2an 596 . . . . . . . . . 10 ((dom 𝑂 ∈ On ∧ 𝐵 ∈ On) → ((𝑂:dom 𝑂onto𝐶 ∧ Smo 𝑂) ↔ 𝑂 Isom E , E (dom 𝑂, 𝐶)))
2827biimpar 477 . . . . . . . . 9 (((dom 𝑂 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑂 Isom E , E (dom 𝑂, 𝐶)) → (𝑂:dom 𝑂onto𝐶 ∧ Smo 𝑂))
2928simprd 495 . . . . . . . 8 (((dom 𝑂 ∈ On ∧ 𝐵 ∈ On) ∧ 𝑂 Isom E , E (dom 𝑂, 𝐶)) → Smo 𝑂)
3023, 24, 13, 29syl21anc 837 . . . . . . 7 (((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) → Smo 𝑂)
31 eloni 6393 . . . . . . . 8 (𝐵 ∈ On → Ord 𝐵)
3231ad2antlr 727 . . . . . . 7 (((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) → Ord 𝐵)
33 smocdmdom 8409 . . . . . . 7 ((𝑂:dom 𝑂𝐵 ∧ Smo 𝑂 ∧ Ord 𝐵) → dom 𝑂𝐵)
3420, 30, 32, 33syl3anc 1372 . . . . . 6 (((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) → dom 𝑂𝐵)
35 onsssuc 6473 . . . . . . 7 ((dom 𝑂 ∈ On ∧ 𝐵 ∈ On) → (dom 𝑂𝐵 ↔ dom 𝑂 ∈ suc 𝐵))
3623, 24, 35syl2anc 584 . . . . . 6 (((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) → (dom 𝑂𝐵 ↔ dom 𝑂 ∈ suc 𝐵))
3734, 36mpbid 232 . . . . 5 (((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) → dom 𝑂 ∈ suc 𝐵)
3837adantrr 717 . . . 4 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤))) → dom 𝑂 ∈ suc 𝐵)
39 vex 3483 . . . . . 6 𝑓 ∈ V
4010oiexg 9576 . . . . . . . 8 (𝐶 ∈ V → 𝑂 ∈ V)
414, 40syl 17 . . . . . . 7 (𝐵 ∈ On → 𝑂 ∈ V)
4241ad2antlr 727 . . . . . 6 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤))) → 𝑂 ∈ V)
43 coexg 7952 . . . . . 6 ((𝑓 ∈ V ∧ 𝑂 ∈ V) → (𝑓𝑂) ∈ V)
4439, 42, 43sylancr 587 . . . . 5 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤))) → (𝑓𝑂) ∈ V)
45 simprl 770 . . . . . . 7 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤))) → 𝑓:𝐵𝐴)
4620adantrr 717 . . . . . . 7 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤))) → 𝑂:dom 𝑂𝐵)
4745, 46fcod 6760 . . . . . 6 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤))) → (𝑓𝑂):dom 𝑂𝐴)
48 simpr 484 . . . . . . . . 9 (((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) → 𝑓:𝐵𝐴)
4948, 20fcod 6760 . . . . . . . 8 (((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) → (𝑓𝑂):dom 𝑂𝐴)
50 ordsson 7804 . . . . . . . . 9 (Ord 𝐴𝐴 ⊆ On)
5150ad2antrr 726 . . . . . . . 8 (((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) → 𝐴 ⊆ On)
5223, 25syl 17 . . . . . . . 8 (((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) → Ord dom 𝑂)
5316, 17syl 17 . . . . . . . . . . . 12 (((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) → 𝑂:dom 𝑂𝐶)
54 simpl 482 . . . . . . . . . . . 12 ((𝑠 ∈ dom 𝑂𝑡𝑠) → 𝑠 ∈ dom 𝑂)
55 ffvelcdm 7100 . . . . . . . . . . . 12 ((𝑂:dom 𝑂𝐶𝑠 ∈ dom 𝑂) → (𝑂𝑠) ∈ 𝐶)
5653, 54, 55syl2an 596 . . . . . . . . . . 11 ((((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) ∧ (𝑠 ∈ dom 𝑂𝑡𝑠)) → (𝑂𝑠) ∈ 𝐶)
57 ffn 6735 . . . . . . . . . . . . . 14 (𝑂:dom 𝑂𝐶𝑂 Fn dom 𝑂)
5816, 17, 573syl 18 . . . . . . . . . . . . 13 (((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) → 𝑂 Fn dom 𝑂)
5958, 30jca 511 . . . . . . . . . . . 12 (((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) → (𝑂 Fn dom 𝑂 ∧ Smo 𝑂))
60 smoel2 8404 . . . . . . . . . . . 12 (((𝑂 Fn dom 𝑂 ∧ Smo 𝑂) ∧ (𝑠 ∈ dom 𝑂𝑡𝑠)) → (𝑂𝑡) ∈ (𝑂𝑠))
6159, 60sylan 580 . . . . . . . . . . 11 ((((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) ∧ (𝑠 ∈ dom 𝑂𝑡𝑠)) → (𝑂𝑡) ∈ (𝑂𝑠))
62 fveq2 6905 . . . . . . . . . . . . . . . 16 (𝑧 = (𝑂𝑠) → (𝑓𝑧) = (𝑓‘(𝑂𝑠)))
6362eleq2d 2826 . . . . . . . . . . . . . . 15 (𝑧 = (𝑂𝑠) → ((𝑓𝑥) ∈ (𝑓𝑧) ↔ (𝑓𝑥) ∈ (𝑓‘(𝑂𝑠))))
6463raleqbi1dv 3337 . . . . . . . . . . . . . 14 (𝑧 = (𝑂𝑠) → (∀𝑥𝑧 (𝑓𝑥) ∈ (𝑓𝑧) ↔ ∀𝑥 ∈ (𝑂𝑠)(𝑓𝑥) ∈ (𝑓‘(𝑂𝑠))))
65 fveq2 6905 . . . . . . . . . . . . . . . . . . 19 (𝑤 = 𝑥 → (𝑓𝑤) = (𝑓𝑥))
6665eleq1d 2825 . . . . . . . . . . . . . . . . . 18 (𝑤 = 𝑥 → ((𝑓𝑤) ∈ (𝑓𝑦) ↔ (𝑓𝑥) ∈ (𝑓𝑦)))
6766cbvralvw 3236 . . . . . . . . . . . . . . . . 17 (∀𝑤𝑦 (𝑓𝑤) ∈ (𝑓𝑦) ↔ ∀𝑥𝑦 (𝑓𝑥) ∈ (𝑓𝑦))
68 fveq2 6905 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝑧 → (𝑓𝑦) = (𝑓𝑧))
6968eleq2d 2826 . . . . . . . . . . . . . . . . . 18 (𝑦 = 𝑧 → ((𝑓𝑥) ∈ (𝑓𝑦) ↔ (𝑓𝑥) ∈ (𝑓𝑧)))
7069raleqbi1dv 3337 . . . . . . . . . . . . . . . . 17 (𝑦 = 𝑧 → (∀𝑥𝑦 (𝑓𝑥) ∈ (𝑓𝑦) ↔ ∀𝑥𝑧 (𝑓𝑥) ∈ (𝑓𝑧)))
7167, 70bitrid 283 . . . . . . . . . . . . . . . 16 (𝑦 = 𝑧 → (∀𝑤𝑦 (𝑓𝑤) ∈ (𝑓𝑦) ↔ ∀𝑥𝑧 (𝑓𝑥) ∈ (𝑓𝑧)))
7271cbvrabv 3446 . . . . . . . . . . . . . . 15 {𝑦𝐵 ∣ ∀𝑤𝑦 (𝑓𝑤) ∈ (𝑓𝑦)} = {𝑧𝐵 ∣ ∀𝑥𝑧 (𝑓𝑥) ∈ (𝑓𝑧)}
731, 72eqtri 2764 . . . . . . . . . . . . . 14 𝐶 = {𝑧𝐵 ∣ ∀𝑥𝑧 (𝑓𝑥) ∈ (𝑓𝑧)}
7464, 73elrab2 3694 . . . . . . . . . . . . 13 ((𝑂𝑠) ∈ 𝐶 ↔ ((𝑂𝑠) ∈ 𝐵 ∧ ∀𝑥 ∈ (𝑂𝑠)(𝑓𝑥) ∈ (𝑓‘(𝑂𝑠))))
7574simprbi 496 . . . . . . . . . . . 12 ((𝑂𝑠) ∈ 𝐶 → ∀𝑥 ∈ (𝑂𝑠)(𝑓𝑥) ∈ (𝑓‘(𝑂𝑠)))
76 fveq2 6905 . . . . . . . . . . . . . 14 (𝑥 = (𝑂𝑡) → (𝑓𝑥) = (𝑓‘(𝑂𝑡)))
7776eleq1d 2825 . . . . . . . . . . . . 13 (𝑥 = (𝑂𝑡) → ((𝑓𝑥) ∈ (𝑓‘(𝑂𝑠)) ↔ (𝑓‘(𝑂𝑡)) ∈ (𝑓‘(𝑂𝑠))))
7877rspccv 3618 . . . . . . . . . . . 12 (∀𝑥 ∈ (𝑂𝑠)(𝑓𝑥) ∈ (𝑓‘(𝑂𝑠)) → ((𝑂𝑡) ∈ (𝑂𝑠) → (𝑓‘(𝑂𝑡)) ∈ (𝑓‘(𝑂𝑠))))
7975, 78syl 17 . . . . . . . . . . 11 ((𝑂𝑠) ∈ 𝐶 → ((𝑂𝑡) ∈ (𝑂𝑠) → (𝑓‘(𝑂𝑡)) ∈ (𝑓‘(𝑂𝑠))))
8056, 61, 79sylc 65 . . . . . . . . . 10 ((((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) ∧ (𝑠 ∈ dom 𝑂𝑡𝑠)) → (𝑓‘(𝑂𝑡)) ∈ (𝑓‘(𝑂𝑠)))
81 ordtr1 6426 . . . . . . . . . . . . . 14 (Ord dom 𝑂 → ((𝑡𝑠𝑠 ∈ dom 𝑂) → 𝑡 ∈ dom 𝑂))
8281ancomsd 465 . . . . . . . . . . . . 13 (Ord dom 𝑂 → ((𝑠 ∈ dom 𝑂𝑡𝑠) → 𝑡 ∈ dom 𝑂))
8323, 25, 823syl 18 . . . . . . . . . . . 12 (((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) → ((𝑠 ∈ dom 𝑂𝑡𝑠) → 𝑡 ∈ dom 𝑂))
8483imp 406 . . . . . . . . . . 11 ((((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) ∧ (𝑠 ∈ dom 𝑂𝑡𝑠)) → 𝑡 ∈ dom 𝑂)
85 fvco3 7007 . . . . . . . . . . 11 ((𝑂:dom 𝑂𝐵𝑡 ∈ dom 𝑂) → ((𝑓𝑂)‘𝑡) = (𝑓‘(𝑂𝑡)))
8620, 84, 85syl2an2r 685 . . . . . . . . . 10 ((((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) ∧ (𝑠 ∈ dom 𝑂𝑡𝑠)) → ((𝑓𝑂)‘𝑡) = (𝑓‘(𝑂𝑡)))
87 simprl 770 . . . . . . . . . . 11 ((((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) ∧ (𝑠 ∈ dom 𝑂𝑡𝑠)) → 𝑠 ∈ dom 𝑂)
88 fvco3 7007 . . . . . . . . . . 11 ((𝑂:dom 𝑂𝐵𝑠 ∈ dom 𝑂) → ((𝑓𝑂)‘𝑠) = (𝑓‘(𝑂𝑠)))
8920, 87, 88syl2an2r 685 . . . . . . . . . 10 ((((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) ∧ (𝑠 ∈ dom 𝑂𝑡𝑠)) → ((𝑓𝑂)‘𝑠) = (𝑓‘(𝑂𝑠)))
9080, 86, 893eltr4d 2855 . . . . . . . . 9 ((((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) ∧ (𝑠 ∈ dom 𝑂𝑡𝑠)) → ((𝑓𝑂)‘𝑡) ∈ ((𝑓𝑂)‘𝑠))
9190ralrimivva 3201 . . . . . . . 8 (((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) → ∀𝑠 ∈ dom 𝑂𝑡𝑠 ((𝑓𝑂)‘𝑡) ∈ ((𝑓𝑂)‘𝑠))
92 issmo2 8390 . . . . . . . . 9 ((𝑓𝑂):dom 𝑂𝐴 → ((𝐴 ⊆ On ∧ Ord dom 𝑂 ∧ ∀𝑠 ∈ dom 𝑂𝑡𝑠 ((𝑓𝑂)‘𝑡) ∈ ((𝑓𝑂)‘𝑠)) → Smo (𝑓𝑂)))
9392imp 406 . . . . . . . 8 (((𝑓𝑂):dom 𝑂𝐴 ∧ (𝐴 ⊆ On ∧ Ord dom 𝑂 ∧ ∀𝑠 ∈ dom 𝑂𝑡𝑠 ((𝑓𝑂)‘𝑡) ∈ ((𝑓𝑂)‘𝑠))) → Smo (𝑓𝑂))
9449, 51, 52, 91, 93syl13anc 1373 . . . . . . 7 (((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) → Smo (𝑓𝑂))
9594adantrr 717 . . . . . 6 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤))) → Smo (𝑓𝑂))
96 rabn0 4388 . . . . . . . . . . . . . . . . . 18 ({𝑤𝐵𝑧 ⊆ (𝑓𝑤)} ≠ ∅ ↔ ∃𝑤𝐵 𝑧 ⊆ (𝑓𝑤))
97 ssrab2 4079 . . . . . . . . . . . . . . . . . . . 20 {𝑤𝐵𝑧 ⊆ (𝑓𝑤)} ⊆ 𝐵
9897, 5sstrid 3994 . . . . . . . . . . . . . . . . . . 19 (𝐵 ∈ On → {𝑤𝐵𝑧 ⊆ (𝑓𝑤)} ⊆ On)
99 cofsmo.2 . . . . . . . . . . . . . . . . . . . . 21 𝐾 = {𝑥𝐵𝑧 ⊆ (𝑓𝑥)}
100 fveq2 6905 . . . . . . . . . . . . . . . . . . . . . . . 24 (𝑥 = 𝑤 → (𝑓𝑥) = (𝑓𝑤))
101100sseq2d 4015 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 = 𝑤 → (𝑧 ⊆ (𝑓𝑥) ↔ 𝑧 ⊆ (𝑓𝑤)))
102101cbvrabv 3446 . . . . . . . . . . . . . . . . . . . . . 22 {𝑥𝐵𝑧 ⊆ (𝑓𝑥)} = {𝑤𝐵𝑧 ⊆ (𝑓𝑤)}
103102inteqi 4949 . . . . . . . . . . . . . . . . . . . . 21 {𝑥𝐵𝑧 ⊆ (𝑓𝑥)} = {𝑤𝐵𝑧 ⊆ (𝑓𝑤)}
10499, 103eqtri 2764 . . . . . . . . . . . . . . . . . . . 20 𝐾 = {𝑤𝐵𝑧 ⊆ (𝑓𝑤)}
105 onint 7811 . . . . . . . . . . . . . . . . . . . 20 (({𝑤𝐵𝑧 ⊆ (𝑓𝑤)} ⊆ On ∧ {𝑤𝐵𝑧 ⊆ (𝑓𝑤)} ≠ ∅) → {𝑤𝐵𝑧 ⊆ (𝑓𝑤)} ∈ {𝑤𝐵𝑧 ⊆ (𝑓𝑤)})
106104, 105eqeltrid 2844 . . . . . . . . . . . . . . . . . . 19 (({𝑤𝐵𝑧 ⊆ (𝑓𝑤)} ⊆ On ∧ {𝑤𝐵𝑧 ⊆ (𝑓𝑤)} ≠ ∅) → 𝐾 ∈ {𝑤𝐵𝑧 ⊆ (𝑓𝑤)})
10798, 106sylan 580 . . . . . . . . . . . . . . . . . 18 ((𝐵 ∈ On ∧ {𝑤𝐵𝑧 ⊆ (𝑓𝑤)} ≠ ∅) → 𝐾 ∈ {𝑤𝐵𝑧 ⊆ (𝑓𝑤)})
10896, 107sylan2br 595 . . . . . . . . . . . . . . . . 17 ((𝐵 ∈ On ∧ ∃𝑤𝐵 𝑧 ⊆ (𝑓𝑤)) → 𝐾 ∈ {𝑤𝐵𝑧 ⊆ (𝑓𝑤)})
109 fveq2 6905 . . . . . . . . . . . . . . . . . . 19 (𝑤 = 𝐾 → (𝑓𝑤) = (𝑓𝐾))
110109sseq2d 4015 . . . . . . . . . . . . . . . . . 18 (𝑤 = 𝐾 → (𝑧 ⊆ (𝑓𝑤) ↔ 𝑧 ⊆ (𝑓𝐾)))
111110elrab 3691 . . . . . . . . . . . . . . . . 17 (𝐾 ∈ {𝑤𝐵𝑧 ⊆ (𝑓𝑤)} ↔ (𝐾𝐵𝑧 ⊆ (𝑓𝐾)))
112108, 111sylib 218 . . . . . . . . . . . . . . . 16 ((𝐵 ∈ On ∧ ∃𝑤𝐵 𝑧 ⊆ (𝑓𝑤)) → (𝐾𝐵𝑧 ⊆ (𝑓𝐾)))
113112ex 412 . . . . . . . . . . . . . . 15 (𝐵 ∈ On → (∃𝑤𝐵 𝑧 ⊆ (𝑓𝑤) → (𝐾𝐵𝑧 ⊆ (𝑓𝐾))))
114113adantl 481 . . . . . . . . . . . . . 14 ((Ord 𝐴𝐵 ∈ On) → (∃𝑤𝐵 𝑧 ⊆ (𝑓𝑤) → (𝐾𝐵𝑧 ⊆ (𝑓𝐾))))
115 simpr2 1195 . . . . . . . . . . . . . . . . . 18 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴𝐾𝐵𝑧 ⊆ (𝑓𝐾))) → 𝐾𝐵)
116 simp3 1138 . . . . . . . . . . . . . . . . . . . . 21 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴𝐾𝐵𝑧 ⊆ (𝑓𝐾)) ∧ 𝑤𝐾) → 𝑤𝐾)
117104eleq2i 2832 . . . . . . . . . . . . . . . . . . . . . 22 (𝑤𝐾𝑤 {𝑤𝐵𝑧 ⊆ (𝑓𝑤)})
118 simp21 1206 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴𝐾𝐵𝑧 ⊆ (𝑓𝐾)) ∧ 𝑤𝐾) → 𝑓:𝐵𝐴)
119 simp1l 1197 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴𝐾𝐵𝑧 ⊆ (𝑓𝐾)) ∧ 𝑤𝐾) → Ord 𝐴)
120119, 50syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴𝐾𝐵𝑧 ⊆ (𝑓𝐾)) ∧ 𝑤𝐾) → 𝐴 ⊆ On)
121118, 120fssd 6752 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴𝐾𝐵𝑧 ⊆ (𝑓𝐾)) ∧ 𝑤𝐾) → 𝑓:𝐵⟶On)
122 simp22 1207 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴𝐾𝐵𝑧 ⊆ (𝑓𝐾)) ∧ 𝑤𝐾) → 𝐾𝐵)
123121, 122ffvelcdmd 7104 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴𝐾𝐵𝑧 ⊆ (𝑓𝐾)) ∧ 𝑤𝐾) → (𝑓𝐾) ∈ On)
124 simp1r 1198 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴𝐾𝐵𝑧 ⊆ (𝑓𝐾)) ∧ 𝑤𝐾) → 𝐵 ∈ On)
125 ontr1 6429 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝐵 ∈ On → ((𝑤𝐾𝐾𝐵) → 𝑤𝐵))
1261253impib 1116 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 ((𝐵 ∈ On ∧ 𝑤𝐾𝐾𝐵) → 𝑤𝐵)
127124, 116, 122, 126syl3anc 1372 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴𝐾𝐵𝑧 ⊆ (𝑓𝐾)) ∧ 𝑤𝐾) → 𝑤𝐵)
128121, 127ffvelcdmd 7104 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴𝐾𝐵𝑧 ⊆ (𝑓𝐾)) ∧ 𝑤𝐾) → (𝑓𝑤) ∈ On)
129 ontri1 6417 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝑓𝐾) ∈ On ∧ (𝑓𝑤) ∈ On) → ((𝑓𝐾) ⊆ (𝑓𝑤) ↔ ¬ (𝑓𝑤) ∈ (𝑓𝐾)))
130123, 128, 129syl2anc 584 . . . . . . . . . . . . . . . . . . . . . . . 24 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴𝐾𝐵𝑧 ⊆ (𝑓𝐾)) ∧ 𝑤𝐾) → ((𝑓𝐾) ⊆ (𝑓𝑤) ↔ ¬ (𝑓𝑤) ∈ (𝑓𝐾)))
131 simp23 1208 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴𝐾𝐵𝑧 ⊆ (𝑓𝐾)) ∧ 𝑤𝐾) → 𝑧 ⊆ (𝑓𝐾))
132 simpl1 1191 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝐵 ∈ On ∧ 𝑤𝐾𝐾𝐵) ∧ (𝑧 ⊆ (𝑓𝐾) ∧ (𝑓𝐾) ⊆ (𝑓𝑤))) → 𝐵 ∈ On)
133132, 98syl 17 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝐵 ∈ On ∧ 𝑤𝐾𝐾𝐵) ∧ (𝑧 ⊆ (𝑓𝐾) ∧ (𝑓𝐾) ⊆ (𝑓𝑤))) → {𝑤𝐵𝑧 ⊆ (𝑓𝑤)} ⊆ On)
134 sstr 3991 . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 ((𝑧 ⊆ (𝑓𝐾) ∧ (𝑓𝐾) ⊆ (𝑓𝑤)) → 𝑧 ⊆ (𝑓𝑤))
135126, 134anim12i 613 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (((𝐵 ∈ On ∧ 𝑤𝐾𝐾𝐵) ∧ (𝑧 ⊆ (𝑓𝐾) ∧ (𝑓𝐾) ⊆ (𝑓𝑤))) → (𝑤𝐵𝑧 ⊆ (𝑓𝑤)))
136 rabid 3457 . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 (𝑤 ∈ {𝑤𝐵𝑧 ⊆ (𝑓𝑤)} ↔ (𝑤𝐵𝑧 ⊆ (𝑓𝑤)))
137135, 136sylibr 234 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (((𝐵 ∈ On ∧ 𝑤𝐾𝐾𝐵) ∧ (𝑧 ⊆ (𝑓𝐾) ∧ (𝑓𝐾) ⊆ (𝑓𝑤))) → 𝑤 ∈ {𝑤𝐵𝑧 ⊆ (𝑓𝑤)})
138 onnmin 7819 . . . . . . . . . . . . . . . . . . . . . . . . . . 27 (({𝑤𝐵𝑧 ⊆ (𝑓𝑤)} ⊆ On ∧ 𝑤 ∈ {𝑤𝐵𝑧 ⊆ (𝑓𝑤)}) → ¬ 𝑤 {𝑤𝐵𝑧 ⊆ (𝑓𝑤)})
139133, 137, 138syl2anc 584 . . . . . . . . . . . . . . . . . . . . . . . . . 26 (((𝐵 ∈ On ∧ 𝑤𝐾𝐾𝐵) ∧ (𝑧 ⊆ (𝑓𝐾) ∧ (𝑓𝐾) ⊆ (𝑓𝑤))) → ¬ 𝑤 {𝑤𝐵𝑧 ⊆ (𝑓𝑤)})
140139expr 456 . . . . . . . . . . . . . . . . . . . . . . . . 25 (((𝐵 ∈ On ∧ 𝑤𝐾𝐾𝐵) ∧ 𝑧 ⊆ (𝑓𝐾)) → ((𝑓𝐾) ⊆ (𝑓𝑤) → ¬ 𝑤 {𝑤𝐵𝑧 ⊆ (𝑓𝑤)}))
141124, 116, 122, 131, 140syl31anc 1374 . . . . . . . . . . . . . . . . . . . . . . . 24 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴𝐾𝐵𝑧 ⊆ (𝑓𝐾)) ∧ 𝑤𝐾) → ((𝑓𝐾) ⊆ (𝑓𝑤) → ¬ 𝑤 {𝑤𝐵𝑧 ⊆ (𝑓𝑤)}))
142130, 141sylbird 260 . . . . . . . . . . . . . . . . . . . . . . 23 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴𝐾𝐵𝑧 ⊆ (𝑓𝐾)) ∧ 𝑤𝐾) → (¬ (𝑓𝑤) ∈ (𝑓𝐾) → ¬ 𝑤 {𝑤𝐵𝑧 ⊆ (𝑓𝑤)}))
143142con4d 115 . . . . . . . . . . . . . . . . . . . . . 22 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴𝐾𝐵𝑧 ⊆ (𝑓𝐾)) ∧ 𝑤𝐾) → (𝑤 {𝑤𝐵𝑧 ⊆ (𝑓𝑤)} → (𝑓𝑤) ∈ (𝑓𝐾)))
144117, 143biimtrid 242 . . . . . . . . . . . . . . . . . . . . 21 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴𝐾𝐵𝑧 ⊆ (𝑓𝐾)) ∧ 𝑤𝐾) → (𝑤𝐾 → (𝑓𝑤) ∈ (𝑓𝐾)))
145116, 144mpd 15 . . . . . . . . . . . . . . . . . . . 20 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴𝐾𝐵𝑧 ⊆ (𝑓𝐾)) ∧ 𝑤𝐾) → (𝑓𝑤) ∈ (𝑓𝐾))
1461453expia 1121 . . . . . . . . . . . . . . . . . . 19 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴𝐾𝐵𝑧 ⊆ (𝑓𝐾))) → (𝑤𝐾 → (𝑓𝑤) ∈ (𝑓𝐾)))
147146ralrimiv 3144 . . . . . . . . . . . . . . . . . 18 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴𝐾𝐵𝑧 ⊆ (𝑓𝐾))) → ∀𝑤𝐾 (𝑓𝑤) ∈ (𝑓𝐾))
148 fveq2 6905 . . . . . . . . . . . . . . . . . . . . 21 (𝑦 = 𝐾 → (𝑓𝑦) = (𝑓𝐾))
149148eleq2d 2826 . . . . . . . . . . . . . . . . . . . 20 (𝑦 = 𝐾 → ((𝑓𝑤) ∈ (𝑓𝑦) ↔ (𝑓𝑤) ∈ (𝑓𝐾)))
150149raleqbi1dv 3337 . . . . . . . . . . . . . . . . . . 19 (𝑦 = 𝐾 → (∀𝑤𝑦 (𝑓𝑤) ∈ (𝑓𝑦) ↔ ∀𝑤𝐾 (𝑓𝑤) ∈ (𝑓𝐾)))
151150, 1elrab2 3694 . . . . . . . . . . . . . . . . . 18 (𝐾𝐶 ↔ (𝐾𝐵 ∧ ∀𝑤𝐾 (𝑓𝑤) ∈ (𝑓𝐾)))
152115, 147, 151sylanbrc 583 . . . . . . . . . . . . . . . . 17 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴𝐾𝐵𝑧 ⊆ (𝑓𝐾))) → 𝐾𝐶)
153152expcom 413 . . . . . . . . . . . . . . . 16 ((𝑓:𝐵𝐴𝐾𝐵𝑧 ⊆ (𝑓𝐾)) → ((Ord 𝐴𝐵 ∈ On) → 𝐾𝐶))
1541533expib 1122 . . . . . . . . . . . . . . 15 (𝑓:𝐵𝐴 → ((𝐾𝐵𝑧 ⊆ (𝑓𝐾)) → ((Ord 𝐴𝐵 ∈ On) → 𝐾𝐶)))
155154com13 88 . . . . . . . . . . . . . 14 ((Ord 𝐴𝐵 ∈ On) → ((𝐾𝐵𝑧 ⊆ (𝑓𝐾)) → (𝑓:𝐵𝐴𝐾𝐶)))
156114, 155syld 47 . . . . . . . . . . . . 13 ((Ord 𝐴𝐵 ∈ On) → (∃𝑤𝐵 𝑧 ⊆ (𝑓𝑤) → (𝑓:𝐵𝐴𝐾𝐶)))
157156com23 86 . . . . . . . . . . . 12 ((Ord 𝐴𝐵 ∈ On) → (𝑓:𝐵𝐴 → (∃𝑤𝐵 𝑧 ⊆ (𝑓𝑤) → 𝐾𝐶)))
158157imp31 417 . . . . . . . . . . 11 ((((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) ∧ ∃𝑤𝐵 𝑧 ⊆ (𝑓𝑤)) → 𝐾𝐶)
159 foelrn 7126 . . . . . . . . . . 11 ((𝑂:dom 𝑂onto𝐶𝐾𝐶) → ∃𝑣 ∈ dom 𝑂 𝐾 = (𝑂𝑣))
16016, 158, 159syl2an2r 685 . . . . . . . . . 10 ((((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) ∧ ∃𝑤𝐵 𝑧 ⊆ (𝑓𝑤)) → ∃𝑣 ∈ dom 𝑂 𝐾 = (𝑂𝑣))
161 eleq1 2828 . . . . . . . . . . . . . . . 16 (𝐾 = (𝑂𝑣) → (𝐾 ∈ {𝑤𝐵𝑧 ⊆ (𝑓𝑤)} ↔ (𝑂𝑣) ∈ {𝑤𝐵𝑧 ⊆ (𝑓𝑤)}))
162161biimpcd 249 . . . . . . . . . . . . . . 15 (𝐾 ∈ {𝑤𝐵𝑧 ⊆ (𝑓𝑤)} → (𝐾 = (𝑂𝑣) → (𝑂𝑣) ∈ {𝑤𝐵𝑧 ⊆ (𝑓𝑤)}))
163 fveq2 6905 . . . . . . . . . . . . . . . . . 18 (𝑥 = (𝑂𝑣) → (𝑓𝑥) = (𝑓‘(𝑂𝑣)))
164163sseq2d 4015 . . . . . . . . . . . . . . . . 17 (𝑥 = (𝑂𝑣) → (𝑧 ⊆ (𝑓𝑥) ↔ 𝑧 ⊆ (𝑓‘(𝑂𝑣))))
16565sseq2d 4015 . . . . . . . . . . . . . . . . . 18 (𝑤 = 𝑥 → (𝑧 ⊆ (𝑓𝑤) ↔ 𝑧 ⊆ (𝑓𝑥)))
166165cbvrabv 3446 . . . . . . . . . . . . . . . . 17 {𝑤𝐵𝑧 ⊆ (𝑓𝑤)} = {𝑥𝐵𝑧 ⊆ (𝑓𝑥)}
167164, 166elrab2 3694 . . . . . . . . . . . . . . . 16 ((𝑂𝑣) ∈ {𝑤𝐵𝑧 ⊆ (𝑓𝑤)} ↔ ((𝑂𝑣) ∈ 𝐵𝑧 ⊆ (𝑓‘(𝑂𝑣))))
168167simprbi 496 . . . . . . . . . . . . . . 15 ((𝑂𝑣) ∈ {𝑤𝐵𝑧 ⊆ (𝑓𝑤)} → 𝑧 ⊆ (𝑓‘(𝑂𝑣)))
169162, 168syl6 35 . . . . . . . . . . . . . 14 (𝐾 ∈ {𝑤𝐵𝑧 ⊆ (𝑓𝑤)} → (𝐾 = (𝑂𝑣) → 𝑧 ⊆ (𝑓‘(𝑂𝑣))))
170108, 169syl 17 . . . . . . . . . . . . 13 ((𝐵 ∈ On ∧ ∃𝑤𝐵 𝑧 ⊆ (𝑓𝑤)) → (𝐾 = (𝑂𝑣) → 𝑧 ⊆ (𝑓‘(𝑂𝑣))))
171170ad5ant24 760 . . . . . . . . . . . 12 (((((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) ∧ ∃𝑤𝐵 𝑧 ⊆ (𝑓𝑤)) ∧ 𝑣 ∈ dom 𝑂) → (𝐾 = (𝑂𝑣) → 𝑧 ⊆ (𝑓‘(𝑂𝑣))))
17220ad2antrr 726 . . . . . . . . . . . . . 14 (((((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) ∧ ∃𝑤𝐵 𝑧 ⊆ (𝑓𝑤)) ∧ 𝑣 ∈ dom 𝑂) → 𝑂:dom 𝑂𝐵)
173 fvco3 7007 . . . . . . . . . . . . . 14 ((𝑂:dom 𝑂𝐵𝑣 ∈ dom 𝑂) → ((𝑓𝑂)‘𝑣) = (𝑓‘(𝑂𝑣)))
174172, 173sylancom 588 . . . . . . . . . . . . 13 (((((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) ∧ ∃𝑤𝐵 𝑧 ⊆ (𝑓𝑤)) ∧ 𝑣 ∈ dom 𝑂) → ((𝑓𝑂)‘𝑣) = (𝑓‘(𝑂𝑣)))
175174sseq2d 4015 . . . . . . . . . . . 12 (((((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) ∧ ∃𝑤𝐵 𝑧 ⊆ (𝑓𝑤)) ∧ 𝑣 ∈ dom 𝑂) → (𝑧 ⊆ ((𝑓𝑂)‘𝑣) ↔ 𝑧 ⊆ (𝑓‘(𝑂𝑣))))
176171, 175sylibrd 259 . . . . . . . . . . 11 (((((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) ∧ ∃𝑤𝐵 𝑧 ⊆ (𝑓𝑤)) ∧ 𝑣 ∈ dom 𝑂) → (𝐾 = (𝑂𝑣) → 𝑧 ⊆ ((𝑓𝑂)‘𝑣)))
177176reximdva 3167 . . . . . . . . . 10 ((((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) ∧ ∃𝑤𝐵 𝑧 ⊆ (𝑓𝑤)) → (∃𝑣 ∈ dom 𝑂 𝐾 = (𝑂𝑣) → ∃𝑣 ∈ dom 𝑂 𝑧 ⊆ ((𝑓𝑂)‘𝑣)))
178160, 177mpd 15 . . . . . . . . 9 ((((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) ∧ ∃𝑤𝐵 𝑧 ⊆ (𝑓𝑤)) → ∃𝑣 ∈ dom 𝑂 𝑧 ⊆ ((𝑓𝑂)‘𝑣))
179178ex 412 . . . . . . . 8 (((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) → (∃𝑤𝐵 𝑧 ⊆ (𝑓𝑤) → ∃𝑣 ∈ dom 𝑂 𝑧 ⊆ ((𝑓𝑂)‘𝑣)))
180179ralimdv 3168 . . . . . . 7 (((Ord 𝐴𝐵 ∈ On) ∧ 𝑓:𝐵𝐴) → (∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤) → ∀𝑧𝐴𝑣 ∈ dom 𝑂 𝑧 ⊆ ((𝑓𝑂)‘𝑣)))
181180impr 454 . . . . . 6 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤))) → ∀𝑧𝐴𝑣 ∈ dom 𝑂 𝑧 ⊆ ((𝑓𝑂)‘𝑣))
18247, 95, 1813jca 1128 . . . . 5 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤))) → ((𝑓𝑂):dom 𝑂𝐴 ∧ Smo (𝑓𝑂) ∧ ∀𝑧𝐴𝑣 ∈ dom 𝑂 𝑧 ⊆ ((𝑓𝑂)‘𝑣)))
183 feq1 6715 . . . . . 6 (𝑔 = (𝑓𝑂) → (𝑔:dom 𝑂𝐴 ↔ (𝑓𝑂):dom 𝑂𝐴))
184 smoeq 8391 . . . . . 6 (𝑔 = (𝑓𝑂) → (Smo 𝑔 ↔ Smo (𝑓𝑂)))
185 fveq1 6904 . . . . . . . . 9 (𝑔 = (𝑓𝑂) → (𝑔𝑣) = ((𝑓𝑂)‘𝑣))
186185sseq2d 4015 . . . . . . . 8 (𝑔 = (𝑓𝑂) → (𝑧 ⊆ (𝑔𝑣) ↔ 𝑧 ⊆ ((𝑓𝑂)‘𝑣)))
187186rexbidv 3178 . . . . . . 7 (𝑔 = (𝑓𝑂) → (∃𝑣 ∈ dom 𝑂 𝑧 ⊆ (𝑔𝑣) ↔ ∃𝑣 ∈ dom 𝑂 𝑧 ⊆ ((𝑓𝑂)‘𝑣)))
188187ralbidv 3177 . . . . . 6 (𝑔 = (𝑓𝑂) → (∀𝑧𝐴𝑣 ∈ dom 𝑂 𝑧 ⊆ (𝑔𝑣) ↔ ∀𝑧𝐴𝑣 ∈ dom 𝑂 𝑧 ⊆ ((𝑓𝑂)‘𝑣)))
189183, 184, 1883anbi123d 1437 . . . . 5 (𝑔 = (𝑓𝑂) → ((𝑔:dom 𝑂𝐴 ∧ Smo 𝑔 ∧ ∀𝑧𝐴𝑣 ∈ dom 𝑂 𝑧 ⊆ (𝑔𝑣)) ↔ ((𝑓𝑂):dom 𝑂𝐴 ∧ Smo (𝑓𝑂) ∧ ∀𝑧𝐴𝑣 ∈ dom 𝑂 𝑧 ⊆ ((𝑓𝑂)‘𝑣))))
19044, 182, 189spcedv 3597 . . . 4 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤))) → ∃𝑔(𝑔:dom 𝑂𝐴 ∧ Smo 𝑔 ∧ ∀𝑧𝐴𝑣 ∈ dom 𝑂 𝑧 ⊆ (𝑔𝑣)))
191 feq2 6716 . . . . . . 7 (𝑥 = dom 𝑂 → (𝑔:𝑥𝐴𝑔:dom 𝑂𝐴))
192 rexeq 3321 . . . . . . . 8 (𝑥 = dom 𝑂 → (∃𝑣𝑥 𝑧 ⊆ (𝑔𝑣) ↔ ∃𝑣 ∈ dom 𝑂 𝑧 ⊆ (𝑔𝑣)))
193192ralbidv 3177 . . . . . . 7 (𝑥 = dom 𝑂 → (∀𝑧𝐴𝑣𝑥 𝑧 ⊆ (𝑔𝑣) ↔ ∀𝑧𝐴𝑣 ∈ dom 𝑂 𝑧 ⊆ (𝑔𝑣)))
194191, 1933anbi13d 1439 . . . . . 6 (𝑥 = dom 𝑂 → ((𝑔:𝑥𝐴 ∧ Smo 𝑔 ∧ ∀𝑧𝐴𝑣𝑥 𝑧 ⊆ (𝑔𝑣)) ↔ (𝑔:dom 𝑂𝐴 ∧ Smo 𝑔 ∧ ∀𝑧𝐴𝑣 ∈ dom 𝑂 𝑧 ⊆ (𝑔𝑣))))
195194exbidv 1920 . . . . 5 (𝑥 = dom 𝑂 → (∃𝑔(𝑔:𝑥𝐴 ∧ Smo 𝑔 ∧ ∀𝑧𝐴𝑣𝑥 𝑧 ⊆ (𝑔𝑣)) ↔ ∃𝑔(𝑔:dom 𝑂𝐴 ∧ Smo 𝑔 ∧ ∀𝑧𝐴𝑣 ∈ dom 𝑂 𝑧 ⊆ (𝑔𝑣))))
196195rspcev 3621 . . . 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)))
19738, 190, 196syl2anc 584 . . 3 (((Ord 𝐴𝐵 ∈ On) ∧ (𝑓:𝐵𝐴 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤))) → ∃𝑥 ∈ suc 𝐵𝑔(𝑔:𝑥𝐴 ∧ Smo 𝑔 ∧ ∀𝑧𝐴𝑣𝑥 𝑧 ⊆ (𝑔𝑣)))
198197ex 412 . 2 ((Ord 𝐴𝐵 ∈ On) → ((𝑓:𝐵𝐴 ∧ ∀𝑧𝐴𝑤𝐵 𝑧 ⊆ (𝑓𝑤)) → ∃𝑥 ∈ suc 𝐵𝑔(𝑔:𝑥𝐴 ∧ Smo 𝑔 ∧ ∀𝑧𝐴𝑣𝑥 𝑧 ⊆ (𝑔𝑣))))
199198exlimdv 1932 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 206  wa 395  w3a 1086   = wceq 1539  wex 1778  wcel 2107  wne 2939  wral 3060  wrex 3069  {crab 3435  Vcvv 3479  wss 3950  c0 4332   cint 4945   E cep 5582   We wwe 5635  dom cdm 5684  ccom 5688  Ord word 6382  Oncon0 6383  suc csuc 6385   Fn wfn 6555  wf 6556  ontowfo 6558  1-1-ontowf1o 6559  cfv 6560   Isom wiso 6561  Smo wsmo 8386  OrdIsocoi 9550
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707  ax-rep 5278  ax-sep 5295  ax-nul 5305  ax-pow 5364  ax-pr 5431  ax-un 7756
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ne 2940  df-ral 3061  df-rex 3070  df-rmo 3379  df-reu 3380  df-rab 3436  df-v 3481  df-sbc 3788  df-csb 3899  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-pss 3970  df-nul 4333  df-if 4525  df-pw 4601  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-int 4946  df-iun 4992  df-br 5143  df-opab 5205  df-mpt 5225  df-tr 5259  df-id 5577  df-eprel 5583  df-po 5591  df-so 5592  df-fr 5636  df-se 5637  df-we 5638  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-ord 6386  df-on 6387  df-lim 6388  df-suc 6389  df-iota 6513  df-fun 6562  df-fn 6563  df-f 6564  df-f1 6565  df-fo 6566  df-f1o 6567  df-fv 6568  df-isom 6569  df-riota 7389  df-ov 7435  df-2nd 8016  df-frecs 8307  df-wrecs 8338  df-smo 8387  df-recs 8412  df-oi 9551
This theorem is referenced by:  cfcof  10315
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