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Theorem coftr 10270
Description: If there is a cofinal map from 𝐵 to 𝐴 and another from 𝐶 to 𝐴, then there is also a cofinal map from 𝐶 to 𝐵. Proposition 11.9 of [TakeutiZaring] p. 102. A limited form of transitivity for the "cof" relation. This is really a lemma for cfcof 10271. (Contributed by Mario Carneiro, 16-Mar-2013.)
Hypothesis
Ref Expression
coftr.1 𝐻 = (𝑡𝐶 {𝑛𝐵 ∣ (𝑔𝑡) ⊆ (𝑓𝑛)})
Assertion
Ref Expression
coftr (∃𝑓(𝑓:𝐵𝐴 ∧ Smo 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦)) → (∃𝑔(𝑔:𝐶𝐴 ∧ ∀𝑧𝐴𝑤𝐶 𝑧 ⊆ (𝑔𝑤)) → ∃(:𝐶𝐵 ∧ ∀𝑠𝐵𝑤𝐶 𝑠 ⊆ (𝑤))))
Distinct variable groups:   𝐴,𝑓,𝑔,𝑠,𝑤,𝑥   𝑧,𝐴,𝑓,𝑔,𝑠,𝑤   𝐵,𝑓,𝑔,,𝑠,𝑤   𝐵,𝑛,𝑡,𝑓,𝑔,𝑤   𝑥,𝐵,𝑦,𝑓,𝑔,𝑠,𝑤   𝐶,𝑓,𝑔,,𝑠,𝑤   𝑡,𝐶   𝑧,𝐶   ,𝐻,𝑠,𝑤   𝑦,𝑛
Allowed substitution hints:   𝐴(𝑦,𝑡,,𝑛)   𝐵(𝑧)   𝐶(𝑥,𝑦,𝑛)   𝐻(𝑥,𝑦,𝑧,𝑡,𝑓,𝑔,𝑛)

Proof of Theorem coftr
StepHypRef Expression
1 fdm 6720 . . . . . . . 8 (𝑔:𝐶𝐴 → dom 𝑔 = 𝐶)
2 vex 3472 . . . . . . . . 9 𝑔 ∈ V
32dmex 7899 . . . . . . . 8 dom 𝑔 ∈ V
41, 3eqeltrrdi 2836 . . . . . . 7 (𝑔:𝐶𝐴𝐶 ∈ V)
5 coftr.1 . . . . . . . . 9 𝐻 = (𝑡𝐶 {𝑛𝐵 ∣ (𝑔𝑡) ⊆ (𝑓𝑛)})
6 fveq2 6885 . . . . . . . . . . . . 13 (𝑡 = 𝑤 → (𝑔𝑡) = (𝑔𝑤))
76sseq1d 4008 . . . . . . . . . . . 12 (𝑡 = 𝑤 → ((𝑔𝑡) ⊆ (𝑓𝑛) ↔ (𝑔𝑤) ⊆ (𝑓𝑛)))
87rabbidv 3434 . . . . . . . . . . 11 (𝑡 = 𝑤 → {𝑛𝐵 ∣ (𝑔𝑡) ⊆ (𝑓𝑛)} = {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)})
98inteqd 4948 . . . . . . . . . 10 (𝑡 = 𝑤 {𝑛𝐵 ∣ (𝑔𝑡) ⊆ (𝑓𝑛)} = {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)})
109cbvmptv 5254 . . . . . . . . 9 (𝑡𝐶 {𝑛𝐵 ∣ (𝑔𝑡) ⊆ (𝑓𝑛)}) = (𝑤𝐶 {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)})
115, 10eqtri 2754 . . . . . . . 8 𝐻 = (𝑤𝐶 {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)})
12 mptexg 7218 . . . . . . . 8 (𝐶 ∈ V → (𝑤𝐶 {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)}) ∈ V)
1311, 12eqeltrid 2831 . . . . . . 7 (𝐶 ∈ V → 𝐻 ∈ V)
144, 13syl 17 . . . . . 6 (𝑔:𝐶𝐴𝐻 ∈ V)
1514ad2antrl 725 . . . . 5 (((𝑓:𝐵𝐴 ∧ Smo 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦)) ∧ (𝑔:𝐶𝐴 ∧ ∀𝑧𝐴𝑤𝐶 𝑧 ⊆ (𝑔𝑤))) → 𝐻 ∈ V)
16 ffn 6711 . . . . . . . . 9 (𝑓:𝐵𝐴𝑓 Fn 𝐵)
17 smodm2 8356 . . . . . . . . 9 ((𝑓 Fn 𝐵 ∧ Smo 𝑓) → Ord 𝐵)
1816, 17sylan 579 . . . . . . . 8 ((𝑓:𝐵𝐴 ∧ Smo 𝑓) → Ord 𝐵)
19183adant3 1129 . . . . . . 7 ((𝑓:𝐵𝐴 ∧ Smo 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦)) → Ord 𝐵)
2019adantr 480 . . . . . 6 (((𝑓:𝐵𝐴 ∧ Smo 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦)) ∧ (𝑔:𝐶𝐴 ∧ ∀𝑧𝐴𝑤𝐶 𝑧 ⊆ (𝑔𝑤))) → Ord 𝐵)
21 simpl3 1190 . . . . . 6 (((𝑓:𝐵𝐴 ∧ Smo 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦)) ∧ (𝑔:𝐶𝐴 ∧ ∀𝑧𝐴𝑤𝐶 𝑧 ⊆ (𝑔𝑤))) → ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦))
22 simprl 768 . . . . . 6 (((𝑓:𝐵𝐴 ∧ Smo 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦)) ∧ (𝑔:𝐶𝐴 ∧ ∀𝑧𝐴𝑤𝐶 𝑧 ⊆ (𝑔𝑤))) → 𝑔:𝐶𝐴)
23 simpl1 1188 . . . . . . . 8 (((Ord 𝐵 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦) ∧ 𝑔:𝐶𝐴) ∧ 𝑤𝐶) → Ord 𝐵)
24 simpl2 1189 . . . . . . . . 9 (((Ord 𝐵 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦) ∧ 𝑔:𝐶𝐴) ∧ 𝑤𝐶) → ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦))
25 ffvelcdm 7077 . . . . . . . . . 10 ((𝑔:𝐶𝐴𝑤𝐶) → (𝑔𝑤) ∈ 𝐴)
26253ad2antl3 1184 . . . . . . . . 9 (((Ord 𝐵 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦) ∧ 𝑔:𝐶𝐴) ∧ 𝑤𝐶) → (𝑔𝑤) ∈ 𝐴)
27 sseq1 4002 . . . . . . . . . . 11 (𝑥 = (𝑔𝑤) → (𝑥 ⊆ (𝑓𝑦) ↔ (𝑔𝑤) ⊆ (𝑓𝑦)))
2827rexbidv 3172 . . . . . . . . . 10 (𝑥 = (𝑔𝑤) → (∃𝑦𝐵 𝑥 ⊆ (𝑓𝑦) ↔ ∃𝑦𝐵 (𝑔𝑤) ⊆ (𝑓𝑦)))
2928rspccv 3603 . . . . . . . . 9 (∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦) → ((𝑔𝑤) ∈ 𝐴 → ∃𝑦𝐵 (𝑔𝑤) ⊆ (𝑓𝑦)))
3024, 26, 29sylc 65 . . . . . . . 8 (((Ord 𝐵 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦) ∧ 𝑔:𝐶𝐴) ∧ 𝑤𝐶) → ∃𝑦𝐵 (𝑔𝑤) ⊆ (𝑓𝑦))
31 ssrab2 4072 . . . . . . . . . . . . 13 {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ⊆ 𝐵
32 ordsson 7767 . . . . . . . . . . . . 13 (Ord 𝐵𝐵 ⊆ On)
3331, 32sstrid 3988 . . . . . . . . . . . 12 (Ord 𝐵 → {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ⊆ On)
34 fveq2 6885 . . . . . . . . . . . . . . 15 (𝑛 = 𝑦 → (𝑓𝑛) = (𝑓𝑦))
3534sseq2d 4009 . . . . . . . . . . . . . 14 (𝑛 = 𝑦 → ((𝑔𝑤) ⊆ (𝑓𝑛) ↔ (𝑔𝑤) ⊆ (𝑓𝑦)))
3635rspcev 3606 . . . . . . . . . . . . 13 ((𝑦𝐵 ∧ (𝑔𝑤) ⊆ (𝑓𝑦)) → ∃𝑛𝐵 (𝑔𝑤) ⊆ (𝑓𝑛))
37 rabn0 4380 . . . . . . . . . . . . 13 ({𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ≠ ∅ ↔ ∃𝑛𝐵 (𝑔𝑤) ⊆ (𝑓𝑛))
3836, 37sylibr 233 . . . . . . . . . . . 12 ((𝑦𝐵 ∧ (𝑔𝑤) ⊆ (𝑓𝑦)) → {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ≠ ∅)
39 oninton 7780 . . . . . . . . . . . 12 (({𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ⊆ On ∧ {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ≠ ∅) → {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ∈ On)
4033, 38, 39syl2an 595 . . . . . . . . . . 11 ((Ord 𝐵 ∧ (𝑦𝐵 ∧ (𝑔𝑤) ⊆ (𝑓𝑦))) → {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ∈ On)
41 eloni 6368 . . . . . . . . . . 11 ( {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ∈ On → Ord {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)})
4240, 41syl 17 . . . . . . . . . 10 ((Ord 𝐵 ∧ (𝑦𝐵 ∧ (𝑔𝑤) ⊆ (𝑓𝑦))) → Ord {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)})
43 simpl 482 . . . . . . . . . 10 ((Ord 𝐵 ∧ (𝑦𝐵 ∧ (𝑔𝑤) ⊆ (𝑓𝑦))) → Ord 𝐵)
4435intminss 4971 . . . . . . . . . . 11 ((𝑦𝐵 ∧ (𝑔𝑤) ⊆ (𝑓𝑦)) → {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ⊆ 𝑦)
4544adantl 481 . . . . . . . . . 10 ((Ord 𝐵 ∧ (𝑦𝐵 ∧ (𝑔𝑤) ⊆ (𝑓𝑦))) → {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ⊆ 𝑦)
46 simprl 768 . . . . . . . . . 10 ((Ord 𝐵 ∧ (𝑦𝐵 ∧ (𝑔𝑤) ⊆ (𝑓𝑦))) → 𝑦𝐵)
47 ordtr2 6402 . . . . . . . . . . 11 ((Ord {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ∧ Ord 𝐵) → (( {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ⊆ 𝑦𝑦𝐵) → {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ∈ 𝐵))
4847imp 406 . . . . . . . . . 10 (((Ord {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ∧ Ord 𝐵) ∧ ( {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ⊆ 𝑦𝑦𝐵)) → {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ∈ 𝐵)
4942, 43, 45, 46, 48syl22anc 836 . . . . . . . . 9 ((Ord 𝐵 ∧ (𝑦𝐵 ∧ (𝑔𝑤) ⊆ (𝑓𝑦))) → {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ∈ 𝐵)
5049rexlimdvaa 3150 . . . . . . . 8 (Ord 𝐵 → (∃𝑦𝐵 (𝑔𝑤) ⊆ (𝑓𝑦) → {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ∈ 𝐵))
5123, 30, 50sylc 65 . . . . . . 7 (((Ord 𝐵 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦) ∧ 𝑔:𝐶𝐴) ∧ 𝑤𝐶) → {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ∈ 𝐵)
5251, 11fmptd 7109 . . . . . 6 ((Ord 𝐵 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦) ∧ 𝑔:𝐶𝐴) → 𝐻:𝐶𝐵)
5320, 21, 22, 52syl3anc 1368 . . . . 5 (((𝑓:𝐵𝐴 ∧ Smo 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦)) ∧ (𝑔:𝐶𝐴 ∧ ∀𝑧𝐴𝑤𝐶 𝑧 ⊆ (𝑔𝑤))) → 𝐻:𝐶𝐵)
54 simprr 770 . . . . . . . 8 (((𝑓:𝐵𝐴 ∧ Smo 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦)) ∧ (𝑔:𝐶𝐴 ∧ ∀𝑧𝐴𝑤𝐶 𝑧 ⊆ (𝑔𝑤))) → ∀𝑧𝐴𝑤𝐶 𝑧 ⊆ (𝑔𝑤))
55 simpl1 1188 . . . . . . . 8 (((𝑓:𝐵𝐴 ∧ Smo 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦)) ∧ (𝑔:𝐶𝐴 ∧ ∀𝑧𝐴𝑤𝐶 𝑧 ⊆ (𝑔𝑤))) → 𝑓:𝐵𝐴)
56 ffvelcdm 7077 . . . . . . . . . 10 ((𝑓:𝐵𝐴𝑠𝐵) → (𝑓𝑠) ∈ 𝐴)
57 sseq1 4002 . . . . . . . . . . . 12 (𝑧 = (𝑓𝑠) → (𝑧 ⊆ (𝑔𝑤) ↔ (𝑓𝑠) ⊆ (𝑔𝑤)))
5857rexbidv 3172 . . . . . . . . . . 11 (𝑧 = (𝑓𝑠) → (∃𝑤𝐶 𝑧 ⊆ (𝑔𝑤) ↔ ∃𝑤𝐶 (𝑓𝑠) ⊆ (𝑔𝑤)))
5958rspccv 3603 . . . . . . . . . 10 (∀𝑧𝐴𝑤𝐶 𝑧 ⊆ (𝑔𝑤) → ((𝑓𝑠) ∈ 𝐴 → ∃𝑤𝐶 (𝑓𝑠) ⊆ (𝑔𝑤)))
6056, 59syl5 34 . . . . . . . . 9 (∀𝑧𝐴𝑤𝐶 𝑧 ⊆ (𝑔𝑤) → ((𝑓:𝐵𝐴𝑠𝐵) → ∃𝑤𝐶 (𝑓𝑠) ⊆ (𝑔𝑤)))
6160expdimp 452 . . . . . . . 8 ((∀𝑧𝐴𝑤𝐶 𝑧 ⊆ (𝑔𝑤) ∧ 𝑓:𝐵𝐴) → (𝑠𝐵 → ∃𝑤𝐶 (𝑓𝑠) ⊆ (𝑔𝑤)))
6254, 55, 61syl2anc 583 . . . . . . 7 (((𝑓:𝐵𝐴 ∧ Smo 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦)) ∧ (𝑔:𝐶𝐴 ∧ ∀𝑧𝐴𝑤𝐶 𝑧 ⊆ (𝑔𝑤))) → (𝑠𝐵 → ∃𝑤𝐶 (𝑓𝑠) ⊆ (𝑔𝑤)))
6355, 16syl 17 . . . . . . . 8 (((𝑓:𝐵𝐴 ∧ Smo 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦)) ∧ (𝑔:𝐶𝐴 ∧ ∀𝑧𝐴𝑤𝐶 𝑧 ⊆ (𝑔𝑤))) → 𝑓 Fn 𝐵)
64 simpl2 1189 . . . . . . . 8 (((𝑓:𝐵𝐴 ∧ Smo 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦)) ∧ (𝑔:𝐶𝐴 ∧ ∀𝑧𝐴𝑤𝐶 𝑧 ⊆ (𝑔𝑤))) → Smo 𝑓)
65 simpr 484 . . . . . . . . . . . . . . . 16 (((Ord 𝐵 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦) ∧ 𝑔:𝐶𝐴) ∧ 𝑤𝐶) → 𝑤𝐶)
6665, 51jca 511 . . . . . . . . . . . . . . 15 (((Ord 𝐵 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦) ∧ 𝑔:𝐶𝐴) ∧ 𝑤𝐶) → (𝑤𝐶 {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ∈ 𝐵))
6735elrab 3678 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ↔ (𝑦𝐵 ∧ (𝑔𝑤) ⊆ (𝑓𝑦)))
68 sstr2 3984 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑓𝑠) ⊆ (𝑔𝑤) → ((𝑔𝑤) ⊆ (𝑓𝑦) → (𝑓𝑠) ⊆ (𝑓𝑦)))
69 smoword 8367 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑓 Fn 𝐵 ∧ Smo 𝑓) ∧ (𝑠𝐵𝑦𝐵)) → (𝑠𝑦 ↔ (𝑓𝑠) ⊆ (𝑓𝑦)))
7069biimprd 247 . . . . . . . . . . . . . . . . . . . . . . 23 (((𝑓 Fn 𝐵 ∧ Smo 𝑓) ∧ (𝑠𝐵𝑦𝐵)) → ((𝑓𝑠) ⊆ (𝑓𝑦) → 𝑠𝑦))
7168, 70syl9r 78 . . . . . . . . . . . . . . . . . . . . . 22 (((𝑓 Fn 𝐵 ∧ Smo 𝑓) ∧ (𝑠𝐵𝑦𝐵)) → ((𝑓𝑠) ⊆ (𝑔𝑤) → ((𝑔𝑤) ⊆ (𝑓𝑦) → 𝑠𝑦)))
7271expr 456 . . . . . . . . . . . . . . . . . . . . 21 (((𝑓 Fn 𝐵 ∧ Smo 𝑓) ∧ 𝑠𝐵) → (𝑦𝐵 → ((𝑓𝑠) ⊆ (𝑔𝑤) → ((𝑔𝑤) ⊆ (𝑓𝑦) → 𝑠𝑦))))
7372com23 86 . . . . . . . . . . . . . . . . . . . 20 (((𝑓 Fn 𝐵 ∧ Smo 𝑓) ∧ 𝑠𝐵) → ((𝑓𝑠) ⊆ (𝑔𝑤) → (𝑦𝐵 → ((𝑔𝑤) ⊆ (𝑓𝑦) → 𝑠𝑦))))
7473imp4b 421 . . . . . . . . . . . . . . . . . . 19 ((((𝑓 Fn 𝐵 ∧ Smo 𝑓) ∧ 𝑠𝐵) ∧ (𝑓𝑠) ⊆ (𝑔𝑤)) → ((𝑦𝐵 ∧ (𝑔𝑤) ⊆ (𝑓𝑦)) → 𝑠𝑦))
7567, 74biimtrid 241 . . . . . . . . . . . . . . . . . 18 ((((𝑓 Fn 𝐵 ∧ Smo 𝑓) ∧ 𝑠𝐵) ∧ (𝑓𝑠) ⊆ (𝑔𝑤)) → (𝑦 ∈ {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} → 𝑠𝑦))
7675ralrimiv 3139 . . . . . . . . . . . . . . . . 17 ((((𝑓 Fn 𝐵 ∧ Smo 𝑓) ∧ 𝑠𝐵) ∧ (𝑓𝑠) ⊆ (𝑔𝑤)) → ∀𝑦 ∈ {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)}𝑠𝑦)
77 ssint 4961 . . . . . . . . . . . . . . . . 17 (𝑠 {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ↔ ∀𝑦 ∈ {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)}𝑠𝑦)
7876, 77sylibr 233 . . . . . . . . . . . . . . . 16 ((((𝑓 Fn 𝐵 ∧ Smo 𝑓) ∧ 𝑠𝐵) ∧ (𝑓𝑠) ⊆ (𝑔𝑤)) → 𝑠 {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)})
799, 5fvmptg 6990 . . . . . . . . . . . . . . . . 17 ((𝑤𝐶 {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ∈ 𝐵) → (𝐻𝑤) = {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)})
8079sseq2d 4009 . . . . . . . . . . . . . . . 16 ((𝑤𝐶 {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ∈ 𝐵) → (𝑠 ⊆ (𝐻𝑤) ↔ 𝑠 {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)}))
8178, 80syl5ibrcom 246 . . . . . . . . . . . . . . 15 ((((𝑓 Fn 𝐵 ∧ Smo 𝑓) ∧ 𝑠𝐵) ∧ (𝑓𝑠) ⊆ (𝑔𝑤)) → ((𝑤𝐶 {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ∈ 𝐵) → 𝑠 ⊆ (𝐻𝑤)))
8266, 81syl5 34 . . . . . . . . . . . . . 14 ((((𝑓 Fn 𝐵 ∧ Smo 𝑓) ∧ 𝑠𝐵) ∧ (𝑓𝑠) ⊆ (𝑔𝑤)) → (((Ord 𝐵 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦) ∧ 𝑔:𝐶𝐴) ∧ 𝑤𝐶) → 𝑠 ⊆ (𝐻𝑤)))
8382ex 412 . . . . . . . . . . . . 13 (((𝑓 Fn 𝐵 ∧ Smo 𝑓) ∧ 𝑠𝐵) → ((𝑓𝑠) ⊆ (𝑔𝑤) → (((Ord 𝐵 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦) ∧ 𝑔:𝐶𝐴) ∧ 𝑤𝐶) → 𝑠 ⊆ (𝐻𝑤))))
8483com23 86 . . . . . . . . . . . 12 (((𝑓 Fn 𝐵 ∧ Smo 𝑓) ∧ 𝑠𝐵) → (((Ord 𝐵 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦) ∧ 𝑔:𝐶𝐴) ∧ 𝑤𝐶) → ((𝑓𝑠) ⊆ (𝑔𝑤) → 𝑠 ⊆ (𝐻𝑤))))
8584expdimp 452 . . . . . . . . . . 11 ((((𝑓 Fn 𝐵 ∧ Smo 𝑓) ∧ 𝑠𝐵) ∧ (Ord 𝐵 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦) ∧ 𝑔:𝐶𝐴)) → (𝑤𝐶 → ((𝑓𝑠) ⊆ (𝑔𝑤) → 𝑠 ⊆ (𝐻𝑤))))
8685reximdvai 3159 . . . . . . . . . 10 ((((𝑓 Fn 𝐵 ∧ Smo 𝑓) ∧ 𝑠𝐵) ∧ (Ord 𝐵 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦) ∧ 𝑔:𝐶𝐴)) → (∃𝑤𝐶 (𝑓𝑠) ⊆ (𝑔𝑤) → ∃𝑤𝐶 𝑠 ⊆ (𝐻𝑤)))
8786ancoms 458 . . . . . . . . 9 (((Ord 𝐵 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦) ∧ 𝑔:𝐶𝐴) ∧ ((𝑓 Fn 𝐵 ∧ Smo 𝑓) ∧ 𝑠𝐵)) → (∃𝑤𝐶 (𝑓𝑠) ⊆ (𝑔𝑤) → ∃𝑤𝐶 𝑠 ⊆ (𝐻𝑤)))
8887expr 456 . . . . . . . 8 (((Ord 𝐵 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦) ∧ 𝑔:𝐶𝐴) ∧ (𝑓 Fn 𝐵 ∧ Smo 𝑓)) → (𝑠𝐵 → (∃𝑤𝐶 (𝑓𝑠) ⊆ (𝑔𝑤) → ∃𝑤𝐶 𝑠 ⊆ (𝐻𝑤))))
8920, 21, 22, 63, 64, 88syl32anc 1375 . . . . . . 7 (((𝑓:𝐵𝐴 ∧ Smo 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦)) ∧ (𝑔:𝐶𝐴 ∧ ∀𝑧𝐴𝑤𝐶 𝑧 ⊆ (𝑔𝑤))) → (𝑠𝐵 → (∃𝑤𝐶 (𝑓𝑠) ⊆ (𝑔𝑤) → ∃𝑤𝐶 𝑠 ⊆ (𝐻𝑤))))
9062, 89mpdd 43 . . . . . 6 (((𝑓:𝐵𝐴 ∧ Smo 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦)) ∧ (𝑔:𝐶𝐴 ∧ ∀𝑧𝐴𝑤𝐶 𝑧 ⊆ (𝑔𝑤))) → (𝑠𝐵 → ∃𝑤𝐶 𝑠 ⊆ (𝐻𝑤)))
9190ralrimiv 3139 . . . . 5 (((𝑓:𝐵𝐴 ∧ Smo 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦)) ∧ (𝑔:𝐶𝐴 ∧ ∀𝑧𝐴𝑤𝐶 𝑧 ⊆ (𝑔𝑤))) → ∀𝑠𝐵𝑤𝐶 𝑠 ⊆ (𝐻𝑤))
92 feq1 6692 . . . . . . . 8 ( = 𝐻 → (:𝐶𝐵𝐻:𝐶𝐵))
93 fveq1 6884 . . . . . . . . . . 11 ( = 𝐻 → (𝑤) = (𝐻𝑤))
9493sseq2d 4009 . . . . . . . . . 10 ( = 𝐻 → (𝑠 ⊆ (𝑤) ↔ 𝑠 ⊆ (𝐻𝑤)))
9594rexbidv 3172 . . . . . . . . 9 ( = 𝐻 → (∃𝑤𝐶 𝑠 ⊆ (𝑤) ↔ ∃𝑤𝐶 𝑠 ⊆ (𝐻𝑤)))
9695ralbidv 3171 . . . . . . . 8 ( = 𝐻 → (∀𝑠𝐵𝑤𝐶 𝑠 ⊆ (𝑤) ↔ ∀𝑠𝐵𝑤𝐶 𝑠 ⊆ (𝐻𝑤)))
9792, 96anbi12d 630 . . . . . . 7 ( = 𝐻 → ((:𝐶𝐵 ∧ ∀𝑠𝐵𝑤𝐶 𝑠 ⊆ (𝑤)) ↔ (𝐻:𝐶𝐵 ∧ ∀𝑠𝐵𝑤𝐶 𝑠 ⊆ (𝐻𝑤))))
9897spcegv 3581 . . . . . 6 (𝐻 ∈ V → ((𝐻:𝐶𝐵 ∧ ∀𝑠𝐵𝑤𝐶 𝑠 ⊆ (𝐻𝑤)) → ∃(:𝐶𝐵 ∧ ∀𝑠𝐵𝑤𝐶 𝑠 ⊆ (𝑤))))
99983impib 1113 . . . . 5 ((𝐻 ∈ V ∧ 𝐻:𝐶𝐵 ∧ ∀𝑠𝐵𝑤𝐶 𝑠 ⊆ (𝐻𝑤)) → ∃(:𝐶𝐵 ∧ ∀𝑠𝐵𝑤𝐶 𝑠 ⊆ (𝑤)))
10015, 53, 91, 99syl3anc 1368 . . . 4 (((𝑓:𝐵𝐴 ∧ Smo 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦)) ∧ (𝑔:𝐶𝐴 ∧ ∀𝑧𝐴𝑤𝐶 𝑧 ⊆ (𝑔𝑤))) → ∃(:𝐶𝐵 ∧ ∀𝑠𝐵𝑤𝐶 𝑠 ⊆ (𝑤)))
101100ex 412 . . 3 ((𝑓:𝐵𝐴 ∧ Smo 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦)) → ((𝑔:𝐶𝐴 ∧ ∀𝑧𝐴𝑤𝐶 𝑧 ⊆ (𝑔𝑤)) → ∃(:𝐶𝐵 ∧ ∀𝑠𝐵𝑤𝐶 𝑠 ⊆ (𝑤))))
102101exlimdv 1928 . 2 ((𝑓:𝐵𝐴 ∧ Smo 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦)) → (∃𝑔(𝑔:𝐶𝐴 ∧ ∀𝑧𝐴𝑤𝐶 𝑧 ⊆ (𝑔𝑤)) → ∃(:𝐶𝐵 ∧ ∀𝑠𝐵𝑤𝐶 𝑠 ⊆ (𝑤))))
103102exlimiv 1925 1 (∃𝑓(𝑓:𝐵𝐴 ∧ Smo 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦)) → (∃𝑔(𝑔:𝐶𝐴 ∧ ∀𝑧𝐴𝑤𝐶 𝑧 ⊆ (𝑔𝑤)) → ∃(:𝐶𝐵 ∧ ∀𝑠𝐵𝑤𝐶 𝑠 ⊆ (𝑤))))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395  w3a 1084   = wceq 1533  wex 1773  wcel 2098  wne 2934  wral 3055  wrex 3064  {crab 3426  Vcvv 3468  wss 3943  c0 4317   cint 4943  cmpt 5224  dom cdm 5669  Ord word 6357  Oncon0 6358   Fn wfn 6532  wf 6533  cfv 6537  Smo wsmo 8346
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-rep 5278  ax-sep 5292  ax-nul 5299  ax-pr 5420  ax-un 7722
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-reu 3371  df-rab 3427  df-v 3470  df-sbc 3773  df-csb 3889  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-pss 3962  df-nul 4318  df-if 4524  df-pw 4599  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-int 4944  df-iun 4992  df-br 5142  df-opab 5204  df-mpt 5225  df-tr 5259  df-id 5567  df-eprel 5573  df-po 5581  df-so 5582  df-fr 5624  df-we 5626  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-ord 6361  df-on 6362  df-iota 6489  df-fun 6539  df-fn 6540  df-f 6541  df-f1 6542  df-fo 6543  df-f1o 6544  df-fv 6545  df-smo 8347
This theorem is referenced by:  cfcof  10271
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