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Theorem coftr 9133
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 9134. (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 6089 . . . . . . . 8 (𝑔:𝐶𝐴 → dom 𝑔 = 𝐶)
2 vex 3234 . . . . . . . . 9 𝑔 ∈ V
32dmex 7141 . . . . . . . 8 dom 𝑔 ∈ V
41, 3syl6eqelr 2739 . . . . . . 7 (𝑔:𝐶𝐴𝐶 ∈ V)
5 coftr.1 . . . . . . . . 9 𝐻 = (𝑡𝐶 {𝑛𝐵 ∣ (𝑔𝑡) ⊆ (𝑓𝑛)})
6 fveq2 6229 . . . . . . . . . . . . 13 (𝑡 = 𝑤 → (𝑔𝑡) = (𝑔𝑤))
76sseq1d 3665 . . . . . . . . . . . 12 (𝑡 = 𝑤 → ((𝑔𝑡) ⊆ (𝑓𝑛) ↔ (𝑔𝑤) ⊆ (𝑓𝑛)))
87rabbidv 3220 . . . . . . . . . . 11 (𝑡 = 𝑤 → {𝑛𝐵 ∣ (𝑔𝑡) ⊆ (𝑓𝑛)} = {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)})
98inteqd 4512 . . . . . . . . . 10 (𝑡 = 𝑤 {𝑛𝐵 ∣ (𝑔𝑡) ⊆ (𝑓𝑛)} = {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)})
109cbvmptv 4783 . . . . . . . . 9 (𝑡𝐶 {𝑛𝐵 ∣ (𝑔𝑡) ⊆ (𝑓𝑛)}) = (𝑤𝐶 {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)})
115, 10eqtri 2673 . . . . . . . 8 𝐻 = (𝑤𝐶 {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)})
12 mptexg 6525 . . . . . . . 8 (𝐶 ∈ V → (𝑤𝐶 {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)}) ∈ V)
1311, 12syl5eqel 2734 . . . . . . 7 (𝐶 ∈ V → 𝐻 ∈ V)
144, 13syl 17 . . . . . 6 (𝑔:𝐶𝐴𝐻 ∈ V)
1514ad2antrl 764 . . . . 5 (((𝑓:𝐵𝐴 ∧ Smo 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦)) ∧ (𝑔:𝐶𝐴 ∧ ∀𝑧𝐴𝑤𝐶 𝑧 ⊆ (𝑔𝑤))) → 𝐻 ∈ V)
16 ffn 6083 . . . . . . . . 9 (𝑓:𝐵𝐴𝑓 Fn 𝐵)
17 smodm2 7497 . . . . . . . . 9 ((𝑓 Fn 𝐵 ∧ Smo 𝑓) → Ord 𝐵)
1816, 17sylan 487 . . . . . . . 8 ((𝑓:𝐵𝐴 ∧ Smo 𝑓) → Ord 𝐵)
19183adant3 1101 . . . . . . 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 1086 . . . . . 6 (((𝑓:𝐵𝐴 ∧ Smo 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦)) ∧ (𝑔:𝐶𝐴 ∧ ∀𝑧𝐴𝑤𝐶 𝑧 ⊆ (𝑔𝑤))) → ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦))
22 simprl 809 . . . . . 6 (((𝑓:𝐵𝐴 ∧ Smo 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦)) ∧ (𝑔:𝐶𝐴 ∧ ∀𝑧𝐴𝑤𝐶 𝑧 ⊆ (𝑔𝑤))) → 𝑔:𝐶𝐴)
23 simpl1 1084 . . . . . . . 8 (((Ord 𝐵 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦) ∧ 𝑔:𝐶𝐴) ∧ 𝑤𝐶) → Ord 𝐵)
24 simpl2 1085 . . . . . . . . 9 (((Ord 𝐵 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦) ∧ 𝑔:𝐶𝐴) ∧ 𝑤𝐶) → ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦))
25 ffvelrn 6397 . . . . . . . . . 10 ((𝑔:𝐶𝐴𝑤𝐶) → (𝑔𝑤) ∈ 𝐴)
26253ad2antl3 1245 . . . . . . . . 9 (((Ord 𝐵 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦) ∧ 𝑔:𝐶𝐴) ∧ 𝑤𝐶) → (𝑔𝑤) ∈ 𝐴)
27 sseq1 3659 . . . . . . . . . . 11 (𝑥 = (𝑔𝑤) → (𝑥 ⊆ (𝑓𝑦) ↔ (𝑔𝑤) ⊆ (𝑓𝑦)))
2827rexbidv 3081 . . . . . . . . . 10 (𝑥 = (𝑔𝑤) → (∃𝑦𝐵 𝑥 ⊆ (𝑓𝑦) ↔ ∃𝑦𝐵 (𝑔𝑤) ⊆ (𝑓𝑦)))
2928rspccv 3337 . . . . . . . . 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 3720 . . . . . . . . . . . . 13 {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ⊆ 𝐵
32 ordsson 7031 . . . . . . . . . . . . 13 (Ord 𝐵𝐵 ⊆ On)
3331, 32syl5ss 3647 . . . . . . . . . . . 12 (Ord 𝐵 → {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ⊆ On)
34 fveq2 6229 . . . . . . . . . . . . . . 15 (𝑛 = 𝑦 → (𝑓𝑛) = (𝑓𝑦))
3534sseq2d 3666 . . . . . . . . . . . . . 14 (𝑛 = 𝑦 → ((𝑔𝑤) ⊆ (𝑓𝑛) ↔ (𝑔𝑤) ⊆ (𝑓𝑦)))
3635rspcev 3340 . . . . . . . . . . . . 13 ((𝑦𝐵 ∧ (𝑔𝑤) ⊆ (𝑓𝑦)) → ∃𝑛𝐵 (𝑔𝑤) ⊆ (𝑓𝑛))
37 rabn0 3991 . . . . . . . . . . . . 13 ({𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ≠ ∅ ↔ ∃𝑛𝐵 (𝑔𝑤) ⊆ (𝑓𝑛))
3836, 37sylibr 224 . . . . . . . . . . . 12 ((𝑦𝐵 ∧ (𝑔𝑤) ⊆ (𝑓𝑦)) → {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ≠ ∅)
39 oninton 7042 . . . . . . . . . . . 12 (({𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ⊆ On ∧ {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ≠ ∅) → {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ∈ On)
4033, 38, 39syl2an 493 . . . . . . . . . . 11 ((Ord 𝐵 ∧ (𝑦𝐵 ∧ (𝑔𝑤) ⊆ (𝑓𝑦))) → {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ∈ On)
41 eloni 5771 . . . . . . . . . . 11 ( {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ∈ On → Ord {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)})
4240, 41syl 17 . . . . . . . . . 10 ((Ord 𝐵 ∧ (𝑦𝐵 ∧ (𝑔𝑤) ⊆ (𝑓𝑦))) → Ord {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)})
43 simpl 472 . . . . . . . . . 10 ((Ord 𝐵 ∧ (𝑦𝐵 ∧ (𝑔𝑤) ⊆ (𝑓𝑦))) → Ord 𝐵)
4435intminss 4535 . . . . . . . . . . 11 ((𝑦𝐵 ∧ (𝑔𝑤) ⊆ (𝑓𝑦)) → {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ⊆ 𝑦)
4544adantl 481 . . . . . . . . . 10 ((Ord 𝐵 ∧ (𝑦𝐵 ∧ (𝑔𝑤) ⊆ (𝑓𝑦))) → {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ⊆ 𝑦)
46 simprl 809 . . . . . . . . . 10 ((Ord 𝐵 ∧ (𝑦𝐵 ∧ (𝑔𝑤) ⊆ (𝑓𝑦))) → 𝑦𝐵)
47 ordtr2 5806 . . . . . . . . . . 11 ((Ord {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ∧ Ord 𝐵) → (( {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ⊆ 𝑦𝑦𝐵) → {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ∈ 𝐵))
4847imp 444 . . . . . . . . . 10 (((Ord {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ∧ Ord 𝐵) ∧ ( {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ⊆ 𝑦𝑦𝐵)) → {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ∈ 𝐵)
4942, 43, 45, 46, 48syl22anc 1367 . . . . . . . . 9 ((Ord 𝐵 ∧ (𝑦𝐵 ∧ (𝑔𝑤) ⊆ (𝑓𝑦))) → {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ∈ 𝐵)
5049rexlimdvaa 3061 . . . . . . . 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 6425 . . . . . 6 ((Ord 𝐵 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦) ∧ 𝑔:𝐶𝐴) → 𝐻:𝐶𝐵)
5320, 21, 22, 52syl3anc 1366 . . . . 5 (((𝑓:𝐵𝐴 ∧ Smo 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦)) ∧ (𝑔:𝐶𝐴 ∧ ∀𝑧𝐴𝑤𝐶 𝑧 ⊆ (𝑔𝑤))) → 𝐻:𝐶𝐵)
54 simprr 811 . . . . . . . 8 (((𝑓:𝐵𝐴 ∧ Smo 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦)) ∧ (𝑔:𝐶𝐴 ∧ ∀𝑧𝐴𝑤𝐶 𝑧 ⊆ (𝑔𝑤))) → ∀𝑧𝐴𝑤𝐶 𝑧 ⊆ (𝑔𝑤))
55 simpl1 1084 . . . . . . . 8 (((𝑓:𝐵𝐴 ∧ Smo 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦)) ∧ (𝑔:𝐶𝐴 ∧ ∀𝑧𝐴𝑤𝐶 𝑧 ⊆ (𝑔𝑤))) → 𝑓:𝐵𝐴)
56 ffvelrn 6397 . . . . . . . . . 10 ((𝑓:𝐵𝐴𝑠𝐵) → (𝑓𝑠) ∈ 𝐴)
57 sseq1 3659 . . . . . . . . . . . 12 (𝑧 = (𝑓𝑠) → (𝑧 ⊆ (𝑔𝑤) ↔ (𝑓𝑠) ⊆ (𝑔𝑤)))
5857rexbidv 3081 . . . . . . . . . . 11 (𝑧 = (𝑓𝑠) → (∃𝑤𝐶 𝑧 ⊆ (𝑔𝑤) ↔ ∃𝑤𝐶 (𝑓𝑠) ⊆ (𝑔𝑤)))
5958rspccv 3337 . . . . . . . . . 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 694 . . . . . . 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 1085 . . . . . . . 8 (((𝑓:𝐵𝐴 ∧ Smo 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦)) ∧ (𝑔:𝐶𝐴 ∧ ∀𝑧𝐴𝑤𝐶 𝑧 ⊆ (𝑔𝑤))) → Smo 𝑓)
65 simpr 476 . . . . . . . . . . . . . . . 16 (((Ord 𝐵 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦) ∧ 𝑔:𝐶𝐴) ∧ 𝑤𝐶) → 𝑤𝐶)
6665, 51jca 553 . . . . . . . . . . . . . . 15 (((Ord 𝐵 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦) ∧ 𝑔:𝐶𝐴) ∧ 𝑤𝐶) → (𝑤𝐶 {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ∈ 𝐵))
6735elrab 3396 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ↔ (𝑦𝐵 ∧ (𝑔𝑤) ⊆ (𝑓𝑦)))
68 sstr2 3643 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑓𝑠) ⊆ (𝑔𝑤) → ((𝑔𝑤) ⊆ (𝑓𝑦) → (𝑓𝑠) ⊆ (𝑓𝑦)))
69 smoword 7508 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑓 Fn 𝐵 ∧ Smo 𝑓) ∧ (𝑠𝐵𝑦𝐵)) → (𝑠𝑦 ↔ (𝑓𝑠) ⊆ (𝑓𝑦)))
7069biimprd 238 . . . . . . . . . . . . . . . . . . . . . . 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 642 . . . . . . . . . . . . . . . . . . . . 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 612 . . . . . . . . . . . . . . . . . . 19 ((((𝑓 Fn 𝐵 ∧ Smo 𝑓) ∧ 𝑠𝐵) ∧ (𝑓𝑠) ⊆ (𝑔𝑤)) → ((𝑦𝐵 ∧ (𝑔𝑤) ⊆ (𝑓𝑦)) → 𝑠𝑦))
7567, 74syl5bi 232 . . . . . . . . . . . . . . . . . 18 ((((𝑓 Fn 𝐵 ∧ Smo 𝑓) ∧ 𝑠𝐵) ∧ (𝑓𝑠) ⊆ (𝑔𝑤)) → (𝑦 ∈ {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} → 𝑠𝑦))
7675ralrimiv 2994 . . . . . . . . . . . . . . . . 17 ((((𝑓 Fn 𝐵 ∧ Smo 𝑓) ∧ 𝑠𝐵) ∧ (𝑓𝑠) ⊆ (𝑔𝑤)) → ∀𝑦 ∈ {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)}𝑠𝑦)
77 ssint 4525 . . . . . . . . . . . . . . . . 17 (𝑠 {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ↔ ∀𝑦 ∈ {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)}𝑠𝑦)
7876, 77sylibr 224 . . . . . . . . . . . . . . . 16 ((((𝑓 Fn 𝐵 ∧ Smo 𝑓) ∧ 𝑠𝐵) ∧ (𝑓𝑠) ⊆ (𝑔𝑤)) → 𝑠 {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)})
799, 5fvmptg 6319 . . . . . . . . . . . . . . . . 17 ((𝑤𝐶 {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ∈ 𝐵) → (𝐻𝑤) = {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)})
8079sseq2d 3666 . . . . . . . . . . . . . . . 16 ((𝑤𝐶 {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ∈ 𝐵) → (𝑠 ⊆ (𝐻𝑤) ↔ 𝑠 {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)}))
8178, 80syl5ibrcom 237 . . . . . . . . . . . . . . 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 449 . . . . . . . . . . . . 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 3044 . . . . . . . . . 10 ((((𝑓 Fn 𝐵 ∧ Smo 𝑓) ∧ 𝑠𝐵) ∧ (Ord 𝐵 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦) ∧ 𝑔:𝐶𝐴)) → (∃𝑤𝐶 (𝑓𝑠) ⊆ (𝑔𝑤) → ∃𝑤𝐶 𝑠 ⊆ (𝐻𝑤)))
8786ancoms 468 . . . . . . . . 9 (((Ord 𝐵 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦) ∧ 𝑔:𝐶𝐴) ∧ ((𝑓 Fn 𝐵 ∧ Smo 𝑓) ∧ 𝑠𝐵)) → (∃𝑤𝐶 (𝑓𝑠) ⊆ (𝑔𝑤) → ∃𝑤𝐶 𝑠 ⊆ (𝐻𝑤)))
8887expr 642 . . . . . . . 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 1374 . . . . . . 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 2994 . . . . 5 (((𝑓:𝐵𝐴 ∧ Smo 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦)) ∧ (𝑔:𝐶𝐴 ∧ ∀𝑧𝐴𝑤𝐶 𝑧 ⊆ (𝑔𝑤))) → ∀𝑠𝐵𝑤𝐶 𝑠 ⊆ (𝐻𝑤))
92 feq1 6064 . . . . . . . 8 ( = 𝐻 → (:𝐶𝐵𝐻:𝐶𝐵))
93 fveq1 6228 . . . . . . . . . . 11 ( = 𝐻 → (𝑤) = (𝐻𝑤))
9493sseq2d 3666 . . . . . . . . . 10 ( = 𝐻 → (𝑠 ⊆ (𝑤) ↔ 𝑠 ⊆ (𝐻𝑤)))
9594rexbidv 3081 . . . . . . . . 9 ( = 𝐻 → (∃𝑤𝐶 𝑠 ⊆ (𝑤) ↔ ∃𝑤𝐶 𝑠 ⊆ (𝐻𝑤)))
9695ralbidv 3015 . . . . . . . 8 ( = 𝐻 → (∀𝑠𝐵𝑤𝐶 𝑠 ⊆ (𝑤) ↔ ∀𝑠𝐵𝑤𝐶 𝑠 ⊆ (𝐻𝑤)))
9792, 96anbi12d 747 . . . . . . 7 ( = 𝐻 → ((:𝐶𝐵 ∧ ∀𝑠𝐵𝑤𝐶 𝑠 ⊆ (𝑤)) ↔ (𝐻:𝐶𝐵 ∧ ∀𝑠𝐵𝑤𝐶 𝑠 ⊆ (𝐻𝑤))))
9897spcegv 3325 . . . . . 6 (𝐻 ∈ V → ((𝐻:𝐶𝐵 ∧ ∀𝑠𝐵𝑤𝐶 𝑠 ⊆ (𝐻𝑤)) → ∃(:𝐶𝐵 ∧ ∀𝑠𝐵𝑤𝐶 𝑠 ⊆ (𝑤))))
99983impib 1281 . . . . 5 ((𝐻 ∈ V ∧ 𝐻:𝐶𝐵 ∧ ∀𝑠𝐵𝑤𝐶 𝑠 ⊆ (𝐻𝑤)) → ∃(:𝐶𝐵 ∧ ∀𝑠𝐵𝑤𝐶 𝑠 ⊆ (𝑤)))
10015, 53, 91, 99syl3anc 1366 . . . 4 (((𝑓:𝐵𝐴 ∧ Smo 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦)) ∧ (𝑔:𝐶𝐴 ∧ ∀𝑧𝐴𝑤𝐶 𝑧 ⊆ (𝑔𝑤))) → ∃(:𝐶𝐵 ∧ ∀𝑠𝐵𝑤𝐶 𝑠 ⊆ (𝑤)))
101100ex 449 . . 3 ((𝑓:𝐵𝐴 ∧ Smo 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦)) → ((𝑔:𝐶𝐴 ∧ ∀𝑧𝐴𝑤𝐶 𝑧 ⊆ (𝑔𝑤)) → ∃(:𝐶𝐵 ∧ ∀𝑠𝐵𝑤𝐶 𝑠 ⊆ (𝑤))))
102101exlimdv 1901 . 2 ((𝑓:𝐵𝐴 ∧ Smo 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦)) → (∃𝑔(𝑔:𝐶𝐴 ∧ ∀𝑧𝐴𝑤𝐶 𝑧 ⊆ (𝑔𝑤)) → ∃(:𝐶𝐵 ∧ ∀𝑠𝐵𝑤𝐶 𝑠 ⊆ (𝑤))))
103102exlimiv 1898 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 383  w3a 1054   = wceq 1523  wex 1744  wcel 2030  wne 2823  wral 2941  wrex 2942  {crab 2945  Vcvv 3231  wss 3607  c0 3948   cint 4507  cmpt 4762  dom cdm 5143  Ord word 5760  Oncon0 5761   Fn wfn 5921  wf 5922  cfv 5926  Smo wsmo 7487
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-rep 4804  ax-sep 4814  ax-nul 4822  ax-pow 4873  ax-pr 4936  ax-un 6991
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1055  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ne 2824  df-ral 2946  df-rex 2947  df-reu 2948  df-rab 2950  df-v 3233  df-sbc 3469  df-csb 3567  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-pss 3623  df-nul 3949  df-if 4120  df-pw 4193  df-sn 4211  df-pr 4213  df-tp 4215  df-op 4217  df-uni 4469  df-int 4508  df-iun 4554  df-br 4686  df-opab 4746  df-mpt 4763  df-tr 4786  df-id 5053  df-eprel 5058  df-po 5064  df-so 5065  df-fr 5102  df-we 5104  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-res 5155  df-ima 5156  df-ord 5764  df-on 5765  df-iota 5889  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-fv 5934  df-smo 7488
This theorem is referenced by:  cfcof  9134
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