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Theorem coftr 9285
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 9286. (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 6210 . . . . . . . 8 (𝑔:𝐶𝐴 → dom 𝑔 = 𝐶)
2 vex 3341 . . . . . . . . 9 𝑔 ∈ V
32dmex 7262 . . . . . . . 8 dom 𝑔 ∈ V
41, 3syl6eqelr 2846 . . . . . . 7 (𝑔:𝐶𝐴𝐶 ∈ V)
5 coftr.1 . . . . . . . . 9 𝐻 = (𝑡𝐶 {𝑛𝐵 ∣ (𝑔𝑡) ⊆ (𝑓𝑛)})
6 fveq2 6350 . . . . . . . . . . . . 13 (𝑡 = 𝑤 → (𝑔𝑡) = (𝑔𝑤))
76sseq1d 3771 . . . . . . . . . . . 12 (𝑡 = 𝑤 → ((𝑔𝑡) ⊆ (𝑓𝑛) ↔ (𝑔𝑤) ⊆ (𝑓𝑛)))
87rabbidv 3327 . . . . . . . . . . 11 (𝑡 = 𝑤 → {𝑛𝐵 ∣ (𝑔𝑡) ⊆ (𝑓𝑛)} = {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)})
98inteqd 4630 . . . . . . . . . 10 (𝑡 = 𝑤 {𝑛𝐵 ∣ (𝑔𝑡) ⊆ (𝑓𝑛)} = {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)})
109cbvmptv 4900 . . . . . . . . 9 (𝑡𝐶 {𝑛𝐵 ∣ (𝑔𝑡) ⊆ (𝑓𝑛)}) = (𝑤𝐶 {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)})
115, 10eqtri 2780 . . . . . . . 8 𝐻 = (𝑤𝐶 {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)})
12 mptexg 6646 . . . . . . . 8 (𝐶 ∈ V → (𝑤𝐶 {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)}) ∈ V)
1311, 12syl5eqel 2841 . . . . . . 7 (𝐶 ∈ V → 𝐻 ∈ V)
144, 13syl 17 . . . . . 6 (𝑔:𝐶𝐴𝐻 ∈ V)
1514ad2antrl 766 . . . . 5 (((𝑓:𝐵𝐴 ∧ Smo 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦)) ∧ (𝑔:𝐶𝐴 ∧ ∀𝑧𝐴𝑤𝐶 𝑧 ⊆ (𝑔𝑤))) → 𝐻 ∈ V)
16 ffn 6204 . . . . . . . . 9 (𝑓:𝐵𝐴𝑓 Fn 𝐵)
17 smodm2 7619 . . . . . . . . 9 ((𝑓 Fn 𝐵 ∧ Smo 𝑓) → Ord 𝐵)
1816, 17sylan 489 . . . . . . . 8 ((𝑓:𝐵𝐴 ∧ Smo 𝑓) → Ord 𝐵)
19183adant3 1127 . . . . . . 7 ((𝑓:𝐵𝐴 ∧ Smo 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦)) → Ord 𝐵)
2019adantr 472 . . . . . 6 (((𝑓:𝐵𝐴 ∧ Smo 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦)) ∧ (𝑔:𝐶𝐴 ∧ ∀𝑧𝐴𝑤𝐶 𝑧 ⊆ (𝑔𝑤))) → Ord 𝐵)
21 simpl3 1232 . . . . . 6 (((𝑓:𝐵𝐴 ∧ Smo 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦)) ∧ (𝑔:𝐶𝐴 ∧ ∀𝑧𝐴𝑤𝐶 𝑧 ⊆ (𝑔𝑤))) → ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦))
22 simprl 811 . . . . . 6 (((𝑓:𝐵𝐴 ∧ Smo 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦)) ∧ (𝑔:𝐶𝐴 ∧ ∀𝑧𝐴𝑤𝐶 𝑧 ⊆ (𝑔𝑤))) → 𝑔:𝐶𝐴)
23 simpl1 1228 . . . . . . . 8 (((Ord 𝐵 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦) ∧ 𝑔:𝐶𝐴) ∧ 𝑤𝐶) → Ord 𝐵)
24 simpl2 1230 . . . . . . . . 9 (((Ord 𝐵 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦) ∧ 𝑔:𝐶𝐴) ∧ 𝑤𝐶) → ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦))
25 ffvelrn 6518 . . . . . . . . . 10 ((𝑔:𝐶𝐴𝑤𝐶) → (𝑔𝑤) ∈ 𝐴)
26253ad2antl3 1203 . . . . . . . . 9 (((Ord 𝐵 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦) ∧ 𝑔:𝐶𝐴) ∧ 𝑤𝐶) → (𝑔𝑤) ∈ 𝐴)
27 sseq1 3765 . . . . . . . . . . 11 (𝑥 = (𝑔𝑤) → (𝑥 ⊆ (𝑓𝑦) ↔ (𝑔𝑤) ⊆ (𝑓𝑦)))
2827rexbidv 3188 . . . . . . . . . 10 (𝑥 = (𝑔𝑤) → (∃𝑦𝐵 𝑥 ⊆ (𝑓𝑦) ↔ ∃𝑦𝐵 (𝑔𝑤) ⊆ (𝑓𝑦)))
2928rspccv 3444 . . . . . . . . 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 3826 . . . . . . . . . . . . 13 {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ⊆ 𝐵
32 ordsson 7152 . . . . . . . . . . . . 13 (Ord 𝐵𝐵 ⊆ On)
3331, 32syl5ss 3753 . . . . . . . . . . . 12 (Ord 𝐵 → {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ⊆ On)
34 fveq2 6350 . . . . . . . . . . . . . . 15 (𝑛 = 𝑦 → (𝑓𝑛) = (𝑓𝑦))
3534sseq2d 3772 . . . . . . . . . . . . . 14 (𝑛 = 𝑦 → ((𝑔𝑤) ⊆ (𝑓𝑛) ↔ (𝑔𝑤) ⊆ (𝑓𝑦)))
3635rspcev 3447 . . . . . . . . . . . . 13 ((𝑦𝐵 ∧ (𝑔𝑤) ⊆ (𝑓𝑦)) → ∃𝑛𝐵 (𝑔𝑤) ⊆ (𝑓𝑛))
37 rabn0 4099 . . . . . . . . . . . . 13 ({𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ≠ ∅ ↔ ∃𝑛𝐵 (𝑔𝑤) ⊆ (𝑓𝑛))
3836, 37sylibr 224 . . . . . . . . . . . 12 ((𝑦𝐵 ∧ (𝑔𝑤) ⊆ (𝑓𝑦)) → {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ≠ ∅)
39 oninton 7163 . . . . . . . . . . . 12 (({𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ⊆ On ∧ {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ≠ ∅) → {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ∈ On)
4033, 38, 39syl2an 495 . . . . . . . . . . 11 ((Ord 𝐵 ∧ (𝑦𝐵 ∧ (𝑔𝑤) ⊆ (𝑓𝑦))) → {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ∈ On)
41 eloni 5892 . . . . . . . . . . 11 ( {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ∈ On → Ord {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)})
4240, 41syl 17 . . . . . . . . . 10 ((Ord 𝐵 ∧ (𝑦𝐵 ∧ (𝑔𝑤) ⊆ (𝑓𝑦))) → Ord {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)})
43 simpl 474 . . . . . . . . . 10 ((Ord 𝐵 ∧ (𝑦𝐵 ∧ (𝑔𝑤) ⊆ (𝑓𝑦))) → Ord 𝐵)
4435intminss 4653 . . . . . . . . . . 11 ((𝑦𝐵 ∧ (𝑔𝑤) ⊆ (𝑓𝑦)) → {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ⊆ 𝑦)
4544adantl 473 . . . . . . . . . 10 ((Ord 𝐵 ∧ (𝑦𝐵 ∧ (𝑔𝑤) ⊆ (𝑓𝑦))) → {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ⊆ 𝑦)
46 simprl 811 . . . . . . . . . 10 ((Ord 𝐵 ∧ (𝑦𝐵 ∧ (𝑔𝑤) ⊆ (𝑓𝑦))) → 𝑦𝐵)
47 ordtr2 5927 . . . . . . . . . . 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 1478 . . . . . . . . 9 ((Ord 𝐵 ∧ (𝑦𝐵 ∧ (𝑔𝑤) ⊆ (𝑓𝑦))) → {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ∈ 𝐵)
5049rexlimdvaa 3168 . . . . . . . 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 6546 . . . . . 6 ((Ord 𝐵 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦) ∧ 𝑔:𝐶𝐴) → 𝐻:𝐶𝐵)
5320, 21, 22, 52syl3anc 1477 . . . . 5 (((𝑓:𝐵𝐴 ∧ Smo 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦)) ∧ (𝑔:𝐶𝐴 ∧ ∀𝑧𝐴𝑤𝐶 𝑧 ⊆ (𝑔𝑤))) → 𝐻:𝐶𝐵)
54 simprr 813 . . . . . . . 8 (((𝑓:𝐵𝐴 ∧ Smo 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦)) ∧ (𝑔:𝐶𝐴 ∧ ∀𝑧𝐴𝑤𝐶 𝑧 ⊆ (𝑔𝑤))) → ∀𝑧𝐴𝑤𝐶 𝑧 ⊆ (𝑔𝑤))
55 simpl1 1228 . . . . . . . 8 (((𝑓:𝐵𝐴 ∧ Smo 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦)) ∧ (𝑔:𝐶𝐴 ∧ ∀𝑧𝐴𝑤𝐶 𝑧 ⊆ (𝑔𝑤))) → 𝑓:𝐵𝐴)
56 ffvelrn 6518 . . . . . . . . . 10 ((𝑓:𝐵𝐴𝑠𝐵) → (𝑓𝑠) ∈ 𝐴)
57 sseq1 3765 . . . . . . . . . . . 12 (𝑧 = (𝑓𝑠) → (𝑧 ⊆ (𝑔𝑤) ↔ (𝑓𝑠) ⊆ (𝑔𝑤)))
5857rexbidv 3188 . . . . . . . . . . 11 (𝑧 = (𝑓𝑠) → (∃𝑤𝐶 𝑧 ⊆ (𝑔𝑤) ↔ ∃𝑤𝐶 (𝑓𝑠) ⊆ (𝑔𝑤)))
5958rspccv 3444 . . . . . . . . . 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 696 . . . . . . 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 1230 . . . . . . . 8 (((𝑓:𝐵𝐴 ∧ Smo 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦)) ∧ (𝑔:𝐶𝐴 ∧ ∀𝑧𝐴𝑤𝐶 𝑧 ⊆ (𝑔𝑤))) → Smo 𝑓)
65 simpr 479 . . . . . . . . . . . . . . . 16 (((Ord 𝐵 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦) ∧ 𝑔:𝐶𝐴) ∧ 𝑤𝐶) → 𝑤𝐶)
6665, 51jca 555 . . . . . . . . . . . . . . 15 (((Ord 𝐵 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦) ∧ 𝑔:𝐶𝐴) ∧ 𝑤𝐶) → (𝑤𝐶 {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ∈ 𝐵))
6735elrab 3502 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ↔ (𝑦𝐵 ∧ (𝑔𝑤) ⊆ (𝑓𝑦)))
68 sstr2 3749 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑓𝑠) ⊆ (𝑔𝑤) → ((𝑔𝑤) ⊆ (𝑓𝑦) → (𝑓𝑠) ⊆ (𝑓𝑦)))
69 smoword 7630 . . . . . . . . . . . . . . . . . . . . . . . 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 644 . . . . . . . . . . . . . . . . . . . . 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 614 . . . . . . . . . . . . . . . . . . 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 3101 . . . . . . . . . . . . . . . . 17 ((((𝑓 Fn 𝐵 ∧ Smo 𝑓) ∧ 𝑠𝐵) ∧ (𝑓𝑠) ⊆ (𝑔𝑤)) → ∀𝑦 ∈ {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)}𝑠𝑦)
77 ssint 4643 . . . . . . . . . . . . . . . . 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 6440 . . . . . . . . . . . . . . . . 17 ((𝑤𝐶 {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ∈ 𝐵) → (𝐻𝑤) = {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)})
8079sseq2d 3772 . . . . . . . . . . . . . . . 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 3151 . . . . . . . . . 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 644 . . . . . . . 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 1485 . . . . . . 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 3101 . . . . 5 (((𝑓:𝐵𝐴 ∧ Smo 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦)) ∧ (𝑔:𝐶𝐴 ∧ ∀𝑧𝐴𝑤𝐶 𝑧 ⊆ (𝑔𝑤))) → ∀𝑠𝐵𝑤𝐶 𝑠 ⊆ (𝐻𝑤))
92 feq1 6185 . . . . . . . 8 ( = 𝐻 → (:𝐶𝐵𝐻:𝐶𝐵))
93 fveq1 6349 . . . . . . . . . . 11 ( = 𝐻 → (𝑤) = (𝐻𝑤))
9493sseq2d 3772 . . . . . . . . . 10 ( = 𝐻 → (𝑠 ⊆ (𝑤) ↔ 𝑠 ⊆ (𝐻𝑤)))
9594rexbidv 3188 . . . . . . . . 9 ( = 𝐻 → (∃𝑤𝐶 𝑠 ⊆ (𝑤) ↔ ∃𝑤𝐶 𝑠 ⊆ (𝐻𝑤)))
9695ralbidv 3122 . . . . . . . 8 ( = 𝐻 → (∀𝑠𝐵𝑤𝐶 𝑠 ⊆ (𝑤) ↔ ∀𝑠𝐵𝑤𝐶 𝑠 ⊆ (𝐻𝑤)))
9792, 96anbi12d 749 . . . . . . 7 ( = 𝐻 → ((:𝐶𝐵 ∧ ∀𝑠𝐵𝑤𝐶 𝑠 ⊆ (𝑤)) ↔ (𝐻:𝐶𝐵 ∧ ∀𝑠𝐵𝑤𝐶 𝑠 ⊆ (𝐻𝑤))))
9897spcegv 3432 . . . . . 6 (𝐻 ∈ V → ((𝐻:𝐶𝐵 ∧ ∀𝑠𝐵𝑤𝐶 𝑠 ⊆ (𝐻𝑤)) → ∃(:𝐶𝐵 ∧ ∀𝑠𝐵𝑤𝐶 𝑠 ⊆ (𝑤))))
99983impib 1109 . . . . 5 ((𝐻 ∈ V ∧ 𝐻:𝐶𝐵 ∧ ∀𝑠𝐵𝑤𝐶 𝑠 ⊆ (𝐻𝑤)) → ∃(:𝐶𝐵 ∧ ∀𝑠𝐵𝑤𝐶 𝑠 ⊆ (𝑤)))
10015, 53, 91, 99syl3anc 1477 . . . 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 2008 . 2 ((𝑓:𝐵𝐴 ∧ Smo 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦)) → (∃𝑔(𝑔:𝐶𝐴 ∧ ∀𝑧𝐴𝑤𝐶 𝑧 ⊆ (𝑔𝑤)) → ∃(:𝐶𝐵 ∧ ∀𝑠𝐵𝑤𝐶 𝑠 ⊆ (𝑤))))
103102exlimiv 2005 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 1072   = wceq 1630  wex 1851  wcel 2137  wne 2930  wral 3048  wrex 3049  {crab 3052  Vcvv 3338  wss 3713  c0 4056   cint 4625  cmpt 4879  dom cdm 5264  Ord word 5881  Oncon0 5882   Fn wfn 6042  wf 6043  cfv 6047  Smo wsmo 7609
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1986  ax-6 2052  ax-7 2088  ax-8 2139  ax-9 2146  ax-10 2166  ax-11 2181  ax-12 2194  ax-13 2389  ax-ext 2738  ax-rep 4921  ax-sep 4931  ax-nul 4939  ax-pow 4990  ax-pr 5053  ax-un 7112
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2045  df-eu 2609  df-mo 2610  df-clab 2745  df-cleq 2751  df-clel 2754  df-nfc 2889  df-ne 2931  df-ral 3053  df-rex 3054  df-reu 3055  df-rab 3057  df-v 3340  df-sbc 3575  df-csb 3673  df-dif 3716  df-un 3718  df-in 3720  df-ss 3727  df-pss 3729  df-nul 4057  df-if 4229  df-pw 4302  df-sn 4320  df-pr 4322  df-tp 4324  df-op 4326  df-uni 4587  df-int 4626  df-iun 4672  df-br 4803  df-opab 4863  df-mpt 4880  df-tr 4903  df-id 5172  df-eprel 5177  df-po 5185  df-so 5186  df-fr 5223  df-we 5225  df-xp 5270  df-rel 5271  df-cnv 5272  df-co 5273  df-dm 5274  df-rn 5275  df-res 5276  df-ima 5277  df-ord 5885  df-on 5886  df-iota 6010  df-fun 6049  df-fn 6050  df-f 6051  df-f1 6052  df-fo 6053  df-f1o 6054  df-fv 6055  df-smo 7610
This theorem is referenced by:  cfcof  9286
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