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Theorem coftr 10268
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 10269. (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 6727 . . . . . . . 8 (𝑔:𝐶𝐴 → dom 𝑔 = 𝐶)
2 vex 3479 . . . . . . . . 9 𝑔 ∈ V
32dmex 7902 . . . . . . . 8 dom 𝑔 ∈ V
41, 3eqeltrrdi 2843 . . . . . . 7 (𝑔:𝐶𝐴𝐶 ∈ V)
5 coftr.1 . . . . . . . . 9 𝐻 = (𝑡𝐶 {𝑛𝐵 ∣ (𝑔𝑡) ⊆ (𝑓𝑛)})
6 fveq2 6892 . . . . . . . . . . . . 13 (𝑡 = 𝑤 → (𝑔𝑡) = (𝑔𝑤))
76sseq1d 4014 . . . . . . . . . . . 12 (𝑡 = 𝑤 → ((𝑔𝑡) ⊆ (𝑓𝑛) ↔ (𝑔𝑤) ⊆ (𝑓𝑛)))
87rabbidv 3441 . . . . . . . . . . 11 (𝑡 = 𝑤 → {𝑛𝐵 ∣ (𝑔𝑡) ⊆ (𝑓𝑛)} = {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)})
98inteqd 4956 . . . . . . . . . 10 (𝑡 = 𝑤 {𝑛𝐵 ∣ (𝑔𝑡) ⊆ (𝑓𝑛)} = {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)})
109cbvmptv 5262 . . . . . . . . 9 (𝑡𝐶 {𝑛𝐵 ∣ (𝑔𝑡) ⊆ (𝑓𝑛)}) = (𝑤𝐶 {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)})
115, 10eqtri 2761 . . . . . . . 8 𝐻 = (𝑤𝐶 {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)})
12 mptexg 7223 . . . . . . . 8 (𝐶 ∈ V → (𝑤𝐶 {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)}) ∈ V)
1311, 12eqeltrid 2838 . . . . . . 7 (𝐶 ∈ V → 𝐻 ∈ V)
144, 13syl 17 . . . . . 6 (𝑔:𝐶𝐴𝐻 ∈ V)
1514ad2antrl 727 . . . . 5 (((𝑓:𝐵𝐴 ∧ Smo 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦)) ∧ (𝑔:𝐶𝐴 ∧ ∀𝑧𝐴𝑤𝐶 𝑧 ⊆ (𝑔𝑤))) → 𝐻 ∈ V)
16 ffn 6718 . . . . . . . . 9 (𝑓:𝐵𝐴𝑓 Fn 𝐵)
17 smodm2 8355 . . . . . . . . 9 ((𝑓 Fn 𝐵 ∧ Smo 𝑓) → Ord 𝐵)
1816, 17sylan 581 . . . . . . . 8 ((𝑓:𝐵𝐴 ∧ Smo 𝑓) → Ord 𝐵)
19183adant3 1133 . . . . . . 7 ((𝑓:𝐵𝐴 ∧ Smo 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦)) → Ord 𝐵)
2019adantr 482 . . . . . 6 (((𝑓:𝐵𝐴 ∧ Smo 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦)) ∧ (𝑔:𝐶𝐴 ∧ ∀𝑧𝐴𝑤𝐶 𝑧 ⊆ (𝑔𝑤))) → Ord 𝐵)
21 simpl3 1194 . . . . . 6 (((𝑓:𝐵𝐴 ∧ Smo 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦)) ∧ (𝑔:𝐶𝐴 ∧ ∀𝑧𝐴𝑤𝐶 𝑧 ⊆ (𝑔𝑤))) → ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦))
22 simprl 770 . . . . . 6 (((𝑓:𝐵𝐴 ∧ Smo 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦)) ∧ (𝑔:𝐶𝐴 ∧ ∀𝑧𝐴𝑤𝐶 𝑧 ⊆ (𝑔𝑤))) → 𝑔:𝐶𝐴)
23 simpl1 1192 . . . . . . . 8 (((Ord 𝐵 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦) ∧ 𝑔:𝐶𝐴) ∧ 𝑤𝐶) → Ord 𝐵)
24 simpl2 1193 . . . . . . . . 9 (((Ord 𝐵 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦) ∧ 𝑔:𝐶𝐴) ∧ 𝑤𝐶) → ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦))
25 ffvelcdm 7084 . . . . . . . . . 10 ((𝑔:𝐶𝐴𝑤𝐶) → (𝑔𝑤) ∈ 𝐴)
26253ad2antl3 1188 . . . . . . . . 9 (((Ord 𝐵 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦) ∧ 𝑔:𝐶𝐴) ∧ 𝑤𝐶) → (𝑔𝑤) ∈ 𝐴)
27 sseq1 4008 . . . . . . . . . . 11 (𝑥 = (𝑔𝑤) → (𝑥 ⊆ (𝑓𝑦) ↔ (𝑔𝑤) ⊆ (𝑓𝑦)))
2827rexbidv 3179 . . . . . . . . . 10 (𝑥 = (𝑔𝑤) → (∃𝑦𝐵 𝑥 ⊆ (𝑓𝑦) ↔ ∃𝑦𝐵 (𝑔𝑤) ⊆ (𝑓𝑦)))
2928rspccv 3610 . . . . . . . . 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 4078 . . . . . . . . . . . . 13 {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ⊆ 𝐵
32 ordsson 7770 . . . . . . . . . . . . 13 (Ord 𝐵𝐵 ⊆ On)
3331, 32sstrid 3994 . . . . . . . . . . . 12 (Ord 𝐵 → {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ⊆ On)
34 fveq2 6892 . . . . . . . . . . . . . . 15 (𝑛 = 𝑦 → (𝑓𝑛) = (𝑓𝑦))
3534sseq2d 4015 . . . . . . . . . . . . . 14 (𝑛 = 𝑦 → ((𝑔𝑤) ⊆ (𝑓𝑛) ↔ (𝑔𝑤) ⊆ (𝑓𝑦)))
3635rspcev 3613 . . . . . . . . . . . . 13 ((𝑦𝐵 ∧ (𝑔𝑤) ⊆ (𝑓𝑦)) → ∃𝑛𝐵 (𝑔𝑤) ⊆ (𝑓𝑛))
37 rabn0 4386 . . . . . . . . . . . . 13 ({𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ≠ ∅ ↔ ∃𝑛𝐵 (𝑔𝑤) ⊆ (𝑓𝑛))
3836, 37sylibr 233 . . . . . . . . . . . 12 ((𝑦𝐵 ∧ (𝑔𝑤) ⊆ (𝑓𝑦)) → {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ≠ ∅)
39 oninton 7783 . . . . . . . . . . . 12 (({𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ⊆ On ∧ {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ≠ ∅) → {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ∈ On)
4033, 38, 39syl2an 597 . . . . . . . . . . 11 ((Ord 𝐵 ∧ (𝑦𝐵 ∧ (𝑔𝑤) ⊆ (𝑓𝑦))) → {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ∈ On)
41 eloni 6375 . . . . . . . . . . 11 ( {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ∈ On → Ord {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)})
4240, 41syl 17 . . . . . . . . . 10 ((Ord 𝐵 ∧ (𝑦𝐵 ∧ (𝑔𝑤) ⊆ (𝑓𝑦))) → Ord {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)})
43 simpl 484 . . . . . . . . . 10 ((Ord 𝐵 ∧ (𝑦𝐵 ∧ (𝑔𝑤) ⊆ (𝑓𝑦))) → Ord 𝐵)
4435intminss 4979 . . . . . . . . . . 11 ((𝑦𝐵 ∧ (𝑔𝑤) ⊆ (𝑓𝑦)) → {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ⊆ 𝑦)
4544adantl 483 . . . . . . . . . 10 ((Ord 𝐵 ∧ (𝑦𝐵 ∧ (𝑔𝑤) ⊆ (𝑓𝑦))) → {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ⊆ 𝑦)
46 simprl 770 . . . . . . . . . 10 ((Ord 𝐵 ∧ (𝑦𝐵 ∧ (𝑔𝑤) ⊆ (𝑓𝑦))) → 𝑦𝐵)
47 ordtr2 6409 . . . . . . . . . . 11 ((Ord {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ∧ Ord 𝐵) → (( {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ⊆ 𝑦𝑦𝐵) → {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ∈ 𝐵))
4847imp 408 . . . . . . . . . 10 (((Ord {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ∧ Ord 𝐵) ∧ ( {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ⊆ 𝑦𝑦𝐵)) → {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ∈ 𝐵)
4942, 43, 45, 46, 48syl22anc 838 . . . . . . . . 9 ((Ord 𝐵 ∧ (𝑦𝐵 ∧ (𝑔𝑤) ⊆ (𝑓𝑦))) → {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ∈ 𝐵)
5049rexlimdvaa 3157 . . . . . . . 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 7114 . . . . . 6 ((Ord 𝐵 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦) ∧ 𝑔:𝐶𝐴) → 𝐻:𝐶𝐵)
5320, 21, 22, 52syl3anc 1372 . . . . 5 (((𝑓:𝐵𝐴 ∧ Smo 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦)) ∧ (𝑔:𝐶𝐴 ∧ ∀𝑧𝐴𝑤𝐶 𝑧 ⊆ (𝑔𝑤))) → 𝐻:𝐶𝐵)
54 simprr 772 . . . . . . . 8 (((𝑓:𝐵𝐴 ∧ Smo 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦)) ∧ (𝑔:𝐶𝐴 ∧ ∀𝑧𝐴𝑤𝐶 𝑧 ⊆ (𝑔𝑤))) → ∀𝑧𝐴𝑤𝐶 𝑧 ⊆ (𝑔𝑤))
55 simpl1 1192 . . . . . . . 8 (((𝑓:𝐵𝐴 ∧ Smo 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦)) ∧ (𝑔:𝐶𝐴 ∧ ∀𝑧𝐴𝑤𝐶 𝑧 ⊆ (𝑔𝑤))) → 𝑓:𝐵𝐴)
56 ffvelcdm 7084 . . . . . . . . . 10 ((𝑓:𝐵𝐴𝑠𝐵) → (𝑓𝑠) ∈ 𝐴)
57 sseq1 4008 . . . . . . . . . . . 12 (𝑧 = (𝑓𝑠) → (𝑧 ⊆ (𝑔𝑤) ↔ (𝑓𝑠) ⊆ (𝑔𝑤)))
5857rexbidv 3179 . . . . . . . . . . 11 (𝑧 = (𝑓𝑠) → (∃𝑤𝐶 𝑧 ⊆ (𝑔𝑤) ↔ ∃𝑤𝐶 (𝑓𝑠) ⊆ (𝑔𝑤)))
5958rspccv 3610 . . . . . . . . . 10 (∀𝑧𝐴𝑤𝐶 𝑧 ⊆ (𝑔𝑤) → ((𝑓𝑠) ∈ 𝐴 → ∃𝑤𝐶 (𝑓𝑠) ⊆ (𝑔𝑤)))
6056, 59syl5 34 . . . . . . . . 9 (∀𝑧𝐴𝑤𝐶 𝑧 ⊆ (𝑔𝑤) → ((𝑓:𝐵𝐴𝑠𝐵) → ∃𝑤𝐶 (𝑓𝑠) ⊆ (𝑔𝑤)))
6160expdimp 454 . . . . . . . 8 ((∀𝑧𝐴𝑤𝐶 𝑧 ⊆ (𝑔𝑤) ∧ 𝑓:𝐵𝐴) → (𝑠𝐵 → ∃𝑤𝐶 (𝑓𝑠) ⊆ (𝑔𝑤)))
6254, 55, 61syl2anc 585 . . . . . . 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 1193 . . . . . . . 8 (((𝑓:𝐵𝐴 ∧ Smo 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦)) ∧ (𝑔:𝐶𝐴 ∧ ∀𝑧𝐴𝑤𝐶 𝑧 ⊆ (𝑔𝑤))) → Smo 𝑓)
65 simpr 486 . . . . . . . . . . . . . . . 16 (((Ord 𝐵 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦) ∧ 𝑔:𝐶𝐴) ∧ 𝑤𝐶) → 𝑤𝐶)
6665, 51jca 513 . . . . . . . . . . . . . . 15 (((Ord 𝐵 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦) ∧ 𝑔:𝐶𝐴) ∧ 𝑤𝐶) → (𝑤𝐶 {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ∈ 𝐵))
6735elrab 3684 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ↔ (𝑦𝐵 ∧ (𝑔𝑤) ⊆ (𝑓𝑦)))
68 sstr2 3990 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑓𝑠) ⊆ (𝑔𝑤) → ((𝑔𝑤) ⊆ (𝑓𝑦) → (𝑓𝑠) ⊆ (𝑓𝑦)))
69 smoword 8366 . . . . . . . . . . . . . . . . . . . . . . . 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 458 . . . . . . . . . . . . . . . . . . . . 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 423 . . . . . . . . . . . . . . . . . . 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 3146 . . . . . . . . . . . . . . . . 17 ((((𝑓 Fn 𝐵 ∧ Smo 𝑓) ∧ 𝑠𝐵) ∧ (𝑓𝑠) ⊆ (𝑔𝑤)) → ∀𝑦 ∈ {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)}𝑠𝑦)
77 ssint 4969 . . . . . . . . . . . . . . . . 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 6997 . . . . . . . . . . . . . . . . 17 ((𝑤𝐶 {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ∈ 𝐵) → (𝐻𝑤) = {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)})
8079sseq2d 4015 . . . . . . . . . . . . . . . 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 414 . . . . . . . . . . . . 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 454 . . . . . . . . . . 11 ((((𝑓 Fn 𝐵 ∧ Smo 𝑓) ∧ 𝑠𝐵) ∧ (Ord 𝐵 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦) ∧ 𝑔:𝐶𝐴)) → (𝑤𝐶 → ((𝑓𝑠) ⊆ (𝑔𝑤) → 𝑠 ⊆ (𝐻𝑤))))
8685reximdvai 3166 . . . . . . . . . 10 ((((𝑓 Fn 𝐵 ∧ Smo 𝑓) ∧ 𝑠𝐵) ∧ (Ord 𝐵 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦) ∧ 𝑔:𝐶𝐴)) → (∃𝑤𝐶 (𝑓𝑠) ⊆ (𝑔𝑤) → ∃𝑤𝐶 𝑠 ⊆ (𝐻𝑤)))
8786ancoms 460 . . . . . . . . 9 (((Ord 𝐵 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦) ∧ 𝑔:𝐶𝐴) ∧ ((𝑓 Fn 𝐵 ∧ Smo 𝑓) ∧ 𝑠𝐵)) → (∃𝑤𝐶 (𝑓𝑠) ⊆ (𝑔𝑤) → ∃𝑤𝐶 𝑠 ⊆ (𝐻𝑤)))
8887expr 458 . . . . . . . 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 1379 . . . . . . 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 3146 . . . . 5 (((𝑓:𝐵𝐴 ∧ Smo 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦)) ∧ (𝑔:𝐶𝐴 ∧ ∀𝑧𝐴𝑤𝐶 𝑧 ⊆ (𝑔𝑤))) → ∀𝑠𝐵𝑤𝐶 𝑠 ⊆ (𝐻𝑤))
92 feq1 6699 . . . . . . . 8 ( = 𝐻 → (:𝐶𝐵𝐻:𝐶𝐵))
93 fveq1 6891 . . . . . . . . . . 11 ( = 𝐻 → (𝑤) = (𝐻𝑤))
9493sseq2d 4015 . . . . . . . . . 10 ( = 𝐻 → (𝑠 ⊆ (𝑤) ↔ 𝑠 ⊆ (𝐻𝑤)))
9594rexbidv 3179 . . . . . . . . 9 ( = 𝐻 → (∃𝑤𝐶 𝑠 ⊆ (𝑤) ↔ ∃𝑤𝐶 𝑠 ⊆ (𝐻𝑤)))
9695ralbidv 3178 . . . . . . . 8 ( = 𝐻 → (∀𝑠𝐵𝑤𝐶 𝑠 ⊆ (𝑤) ↔ ∀𝑠𝐵𝑤𝐶 𝑠 ⊆ (𝐻𝑤)))
9792, 96anbi12d 632 . . . . . . 7 ( = 𝐻 → ((:𝐶𝐵 ∧ ∀𝑠𝐵𝑤𝐶 𝑠 ⊆ (𝑤)) ↔ (𝐻:𝐶𝐵 ∧ ∀𝑠𝐵𝑤𝐶 𝑠 ⊆ (𝐻𝑤))))
9897spcegv 3588 . . . . . 6 (𝐻 ∈ V → ((𝐻:𝐶𝐵 ∧ ∀𝑠𝐵𝑤𝐶 𝑠 ⊆ (𝐻𝑤)) → ∃(:𝐶𝐵 ∧ ∀𝑠𝐵𝑤𝐶 𝑠 ⊆ (𝑤))))
99983impib 1117 . . . . 5 ((𝐻 ∈ V ∧ 𝐻:𝐶𝐵 ∧ ∀𝑠𝐵𝑤𝐶 𝑠 ⊆ (𝐻𝑤)) → ∃(:𝐶𝐵 ∧ ∀𝑠𝐵𝑤𝐶 𝑠 ⊆ (𝑤)))
10015, 53, 91, 99syl3anc 1372 . . . 4 (((𝑓:𝐵𝐴 ∧ Smo 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦)) ∧ (𝑔:𝐶𝐴 ∧ ∀𝑧𝐴𝑤𝐶 𝑧 ⊆ (𝑔𝑤))) → ∃(:𝐶𝐵 ∧ ∀𝑠𝐵𝑤𝐶 𝑠 ⊆ (𝑤)))
101100ex 414 . . 3 ((𝑓:𝐵𝐴 ∧ Smo 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦)) → ((𝑔:𝐶𝐴 ∧ ∀𝑧𝐴𝑤𝐶 𝑧 ⊆ (𝑔𝑤)) → ∃(:𝐶𝐵 ∧ ∀𝑠𝐵𝑤𝐶 𝑠 ⊆ (𝑤))))
102101exlimdv 1937 . 2 ((𝑓:𝐵𝐴 ∧ Smo 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦)) → (∃𝑔(𝑔:𝐶𝐴 ∧ ∀𝑧𝐴𝑤𝐶 𝑧 ⊆ (𝑔𝑤)) → ∃(:𝐶𝐵 ∧ ∀𝑠𝐵𝑤𝐶 𝑠 ⊆ (𝑤))))
103102exlimiv 1934 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 397  w3a 1088   = wceq 1542  wex 1782  wcel 2107  wne 2941  wral 3062  wrex 3071  {crab 3433  Vcvv 3475  wss 3949  c0 4323   cint 4951  cmpt 5232  dom cdm 5677  Ord word 6364  Oncon0 6365   Fn wfn 6539  wf 6540  cfv 6544  Smo wsmo 8345
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5286  ax-sep 5300  ax-nul 5307  ax-pr 5428  ax-un 7725
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3779  df-csb 3895  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-pss 3968  df-nul 4324  df-if 4530  df-pw 4605  df-sn 4630  df-pr 4632  df-op 4636  df-uni 4910  df-int 4952  df-iun 5000  df-br 5150  df-opab 5212  df-mpt 5233  df-tr 5267  df-id 5575  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-ord 6368  df-on 6369  df-iota 6496  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-fv 6552  df-smo 8346
This theorem is referenced by:  cfcof  10269
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