MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  coftr Structured version   Visualization version   GIF version

Theorem coftr 10311
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 10312. (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 6746 . . . . . . . 8 (𝑔:𝐶𝐴 → dom 𝑔 = 𝐶)
2 vex 3482 . . . . . . . . 9 𝑔 ∈ V
32dmex 7932 . . . . . . . 8 dom 𝑔 ∈ V
41, 3eqeltrrdi 2848 . . . . . . 7 (𝑔:𝐶𝐴𝐶 ∈ V)
5 coftr.1 . . . . . . . . 9 𝐻 = (𝑡𝐶 {𝑛𝐵 ∣ (𝑔𝑡) ⊆ (𝑓𝑛)})
6 fveq2 6907 . . . . . . . . . . . . 13 (𝑡 = 𝑤 → (𝑔𝑡) = (𝑔𝑤))
76sseq1d 4027 . . . . . . . . . . . 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 5261 . . . . . . . . 9 (𝑡𝐶 {𝑛𝐵 ∣ (𝑔𝑡) ⊆ (𝑓𝑛)}) = (𝑤𝐶 {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)})
115, 10eqtri 2763 . . . . . . . 8 𝐻 = (𝑤𝐶 {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)})
12 mptexg 7241 . . . . . . . 8 (𝐶 ∈ V → (𝑤𝐶 {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)}) ∈ V)
1311, 12eqeltrid 2843 . . . . . . 7 (𝐶 ∈ V → 𝐻 ∈ V)
144, 13syl 17 . . . . . 6 (𝑔:𝐶𝐴𝐻 ∈ V)
1514ad2antrl 728 . . . . 5 (((𝑓:𝐵𝐴 ∧ Smo 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦)) ∧ (𝑔:𝐶𝐴 ∧ ∀𝑧𝐴𝑤𝐶 𝑧 ⊆ (𝑔𝑤))) → 𝐻 ∈ V)
16 ffn 6737 . . . . . . . . 9 (𝑓:𝐵𝐴𝑓 Fn 𝐵)
17 smodm2 8394 . . . . . . . . 9 ((𝑓 Fn 𝐵 ∧ Smo 𝑓) → Ord 𝐵)
1816, 17sylan 580 . . . . . . . 8 ((𝑓:𝐵𝐴 ∧ Smo 𝑓) → Ord 𝐵)
19183adant3 1131 . . . . . . 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 1192 . . . . . 6 (((𝑓:𝐵𝐴 ∧ Smo 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦)) ∧ (𝑔:𝐶𝐴 ∧ ∀𝑧𝐴𝑤𝐶 𝑧 ⊆ (𝑔𝑤))) → ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦))
22 simprl 771 . . . . . 6 (((𝑓:𝐵𝐴 ∧ Smo 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦)) ∧ (𝑔:𝐶𝐴 ∧ ∀𝑧𝐴𝑤𝐶 𝑧 ⊆ (𝑔𝑤))) → 𝑔:𝐶𝐴)
23 simpl1 1190 . . . . . . . 8 (((Ord 𝐵 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦) ∧ 𝑔:𝐶𝐴) ∧ 𝑤𝐶) → Ord 𝐵)
24 simpl2 1191 . . . . . . . . 9 (((Ord 𝐵 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦) ∧ 𝑔:𝐶𝐴) ∧ 𝑤𝐶) → ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦))
25 ffvelcdm 7101 . . . . . . . . . 10 ((𝑔:𝐶𝐴𝑤𝐶) → (𝑔𝑤) ∈ 𝐴)
26253ad2antl3 1186 . . . . . . . . 9 (((Ord 𝐵 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦) ∧ 𝑔:𝐶𝐴) ∧ 𝑤𝐶) → (𝑔𝑤) ∈ 𝐴)
27 sseq1 4021 . . . . . . . . . . 11 (𝑥 = (𝑔𝑤) → (𝑥 ⊆ (𝑓𝑦) ↔ (𝑔𝑤) ⊆ (𝑓𝑦)))
2827rexbidv 3177 . . . . . . . . . 10 (𝑥 = (𝑔𝑤) → (∃𝑦𝐵 𝑥 ⊆ (𝑓𝑦) ↔ ∃𝑦𝐵 (𝑔𝑤) ⊆ (𝑓𝑦)))
2928rspccv 3619 . . . . . . . . 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 4090 . . . . . . . . . . . . 13 {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ⊆ 𝐵
32 ordsson 7802 . . . . . . . . . . . . 13 (Ord 𝐵𝐵 ⊆ On)
3331, 32sstrid 4007 . . . . . . . . . . . 12 (Ord 𝐵 → {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ⊆ On)
34 fveq2 6907 . . . . . . . . . . . . . . 15 (𝑛 = 𝑦 → (𝑓𝑛) = (𝑓𝑦))
3534sseq2d 4028 . . . . . . . . . . . . . 14 (𝑛 = 𝑦 → ((𝑔𝑤) ⊆ (𝑓𝑛) ↔ (𝑔𝑤) ⊆ (𝑓𝑦)))
3635rspcev 3622 . . . . . . . . . . . . 13 ((𝑦𝐵 ∧ (𝑔𝑤) ⊆ (𝑓𝑦)) → ∃𝑛𝐵 (𝑔𝑤) ⊆ (𝑓𝑛))
37 rabn0 4395 . . . . . . . . . . . . 13 ({𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ≠ ∅ ↔ ∃𝑛𝐵 (𝑔𝑤) ⊆ (𝑓𝑛))
3836, 37sylibr 234 . . . . . . . . . . . 12 ((𝑦𝐵 ∧ (𝑔𝑤) ⊆ (𝑓𝑦)) → {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ≠ ∅)
39 oninton 7815 . . . . . . . . . . . 12 (({𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ⊆ On ∧ {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ≠ ∅) → {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ∈ On)
4033, 38, 39syl2an 596 . . . . . . . . . . 11 ((Ord 𝐵 ∧ (𝑦𝐵 ∧ (𝑔𝑤) ⊆ (𝑓𝑦))) → {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ∈ On)
41 eloni 6396 . . . . . . . . . . 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 4979 . . . . . . . . . . 11 ((𝑦𝐵 ∧ (𝑔𝑤) ⊆ (𝑓𝑦)) → {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ⊆ 𝑦)
4544adantl 481 . . . . . . . . . 10 ((Ord 𝐵 ∧ (𝑦𝐵 ∧ (𝑔𝑤) ⊆ (𝑓𝑦))) → {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ⊆ 𝑦)
46 simprl 771 . . . . . . . . . 10 ((Ord 𝐵 ∧ (𝑦𝐵 ∧ (𝑔𝑤) ⊆ (𝑓𝑦))) → 𝑦𝐵)
47 ordtr2 6430 . . . . . . . . . . 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 839 . . . . . . . . 9 ((Ord 𝐵 ∧ (𝑦𝐵 ∧ (𝑔𝑤) ⊆ (𝑓𝑦))) → {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ∈ 𝐵)
5049rexlimdvaa 3154 . . . . . . . 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 7134 . . . . . 6 ((Ord 𝐵 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦) ∧ 𝑔:𝐶𝐴) → 𝐻:𝐶𝐵)
5320, 21, 22, 52syl3anc 1370 . . . . 5 (((𝑓:𝐵𝐴 ∧ Smo 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦)) ∧ (𝑔:𝐶𝐴 ∧ ∀𝑧𝐴𝑤𝐶 𝑧 ⊆ (𝑔𝑤))) → 𝐻:𝐶𝐵)
54 simprr 773 . . . . . . . 8 (((𝑓:𝐵𝐴 ∧ Smo 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦)) ∧ (𝑔:𝐶𝐴 ∧ ∀𝑧𝐴𝑤𝐶 𝑧 ⊆ (𝑔𝑤))) → ∀𝑧𝐴𝑤𝐶 𝑧 ⊆ (𝑔𝑤))
55 simpl1 1190 . . . . . . . 8 (((𝑓:𝐵𝐴 ∧ Smo 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦)) ∧ (𝑔:𝐶𝐴 ∧ ∀𝑧𝐴𝑤𝐶 𝑧 ⊆ (𝑔𝑤))) → 𝑓:𝐵𝐴)
56 ffvelcdm 7101 . . . . . . . . . 10 ((𝑓:𝐵𝐴𝑠𝐵) → (𝑓𝑠) ∈ 𝐴)
57 sseq1 4021 . . . . . . . . . . . 12 (𝑧 = (𝑓𝑠) → (𝑧 ⊆ (𝑔𝑤) ↔ (𝑓𝑠) ⊆ (𝑔𝑤)))
5857rexbidv 3177 . . . . . . . . . . 11 (𝑧 = (𝑓𝑠) → (∃𝑤𝐶 𝑧 ⊆ (𝑔𝑤) ↔ ∃𝑤𝐶 (𝑓𝑠) ⊆ (𝑔𝑤)))
5958rspccv 3619 . . . . . . . . . 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 584 . . . . . . 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 1191 . . . . . . . 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 3695 . . . . . . . . . . . . . . . . . . 19 (𝑦 ∈ {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ↔ (𝑦𝐵 ∧ (𝑔𝑤) ⊆ (𝑓𝑦)))
68 sstr2 4002 . . . . . . . . . . . . . . . . . . . . . . 23 ((𝑓𝑠) ⊆ (𝑔𝑤) → ((𝑔𝑤) ⊆ (𝑓𝑦) → (𝑓𝑠) ⊆ (𝑓𝑦)))
69 smoword 8405 . . . . . . . . . . . . . . . . . . . . . . . 24 (((𝑓 Fn 𝐵 ∧ Smo 𝑓) ∧ (𝑠𝐵𝑦𝐵)) → (𝑠𝑦 ↔ (𝑓𝑠) ⊆ (𝑓𝑦)))
7069biimprd 248 . . . . . . . . . . . . . . . . . . . . . . 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 242 . . . . . . . . . . . . . . . . . 18 ((((𝑓 Fn 𝐵 ∧ Smo 𝑓) ∧ 𝑠𝐵) ∧ (𝑓𝑠) ⊆ (𝑔𝑤)) → (𝑦 ∈ {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} → 𝑠𝑦))
7675ralrimiv 3143 . . . . . . . . . . . . . . . . 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 234 . . . . . . . . . . . . . . . 16 ((((𝑓 Fn 𝐵 ∧ Smo 𝑓) ∧ 𝑠𝐵) ∧ (𝑓𝑠) ⊆ (𝑔𝑤)) → 𝑠 {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)})
799, 5fvmptg 7014 . . . . . . . . . . . . . . . . 17 ((𝑤𝐶 {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ∈ 𝐵) → (𝐻𝑤) = {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)})
8079sseq2d 4028 . . . . . . . . . . . . . . . 16 ((𝑤𝐶 {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)} ∈ 𝐵) → (𝑠 ⊆ (𝐻𝑤) ↔ 𝑠 {𝑛𝐵 ∣ (𝑔𝑤) ⊆ (𝑓𝑛)}))
8178, 80syl5ibrcom 247 . . . . . . . . . . . . . . 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 3163 . . . . . . . . . 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 1377 . . . . . . 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 3143 . . . . 5 (((𝑓:𝐵𝐴 ∧ Smo 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦)) ∧ (𝑔:𝐶𝐴 ∧ ∀𝑧𝐴𝑤𝐶 𝑧 ⊆ (𝑔𝑤))) → ∀𝑠𝐵𝑤𝐶 𝑠 ⊆ (𝐻𝑤))
92 feq1 6717 . . . . . . . 8 ( = 𝐻 → (:𝐶𝐵𝐻:𝐶𝐵))
93 fveq1 6906 . . . . . . . . . . 11 ( = 𝐻 → (𝑤) = (𝐻𝑤))
9493sseq2d 4028 . . . . . . . . . 10 ( = 𝐻 → (𝑠 ⊆ (𝑤) ↔ 𝑠 ⊆ (𝐻𝑤)))
9594rexbidv 3177 . . . . . . . . 9 ( = 𝐻 → (∃𝑤𝐶 𝑠 ⊆ (𝑤) ↔ ∃𝑤𝐶 𝑠 ⊆ (𝐻𝑤)))
9695ralbidv 3176 . . . . . . . 8 ( = 𝐻 → (∀𝑠𝐵𝑤𝐶 𝑠 ⊆ (𝑤) ↔ ∀𝑠𝐵𝑤𝐶 𝑠 ⊆ (𝐻𝑤)))
9792, 96anbi12d 632 . . . . . . 7 ( = 𝐻 → ((:𝐶𝐵 ∧ ∀𝑠𝐵𝑤𝐶 𝑠 ⊆ (𝑤)) ↔ (𝐻:𝐶𝐵 ∧ ∀𝑠𝐵𝑤𝐶 𝑠 ⊆ (𝐻𝑤))))
9897spcegv 3597 . . . . . 6 (𝐻 ∈ V → ((𝐻:𝐶𝐵 ∧ ∀𝑠𝐵𝑤𝐶 𝑠 ⊆ (𝐻𝑤)) → ∃(:𝐶𝐵 ∧ ∀𝑠𝐵𝑤𝐶 𝑠 ⊆ (𝑤))))
99983impib 1115 . . . . 5 ((𝐻 ∈ V ∧ 𝐻:𝐶𝐵 ∧ ∀𝑠𝐵𝑤𝐶 𝑠 ⊆ (𝐻𝑤)) → ∃(:𝐶𝐵 ∧ ∀𝑠𝐵𝑤𝐶 𝑠 ⊆ (𝑤)))
10015, 53, 91, 99syl3anc 1370 . . . 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 1931 . 2 ((𝑓:𝐵𝐴 ∧ Smo 𝑓 ∧ ∀𝑥𝐴𝑦𝐵 𝑥 ⊆ (𝑓𝑦)) → (∃𝑔(𝑔:𝐶𝐴 ∧ ∀𝑧𝐴𝑤𝐶 𝑧 ⊆ (𝑔𝑤)) → ∃(:𝐶𝐵 ∧ ∀𝑠𝐵𝑤𝐶 𝑠 ⊆ (𝑤))))
103102exlimiv 1928 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 1086   = wceq 1537  wex 1776  wcel 2106  wne 2938  wral 3059  wrex 3068  {crab 3433  Vcvv 3478  wss 3963  c0 4339   cint 4951  cmpt 5231  dom cdm 5689  Ord word 6385  Oncon0 6386   Fn wfn 6558  wf 6559  cfv 6563  Smo wsmo 8384
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-rep 5285  ax-sep 5302  ax-nul 5312  ax-pr 5438  ax-un 7754
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-reu 3379  df-rab 3434  df-v 3480  df-sbc 3792  df-csb 3909  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-pss 3983  df-nul 4340  df-if 4532  df-pw 4607  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-int 4952  df-iun 4998  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5583  df-eprel 5589  df-po 5597  df-so 5598  df-fr 5641  df-we 5643  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ord 6389  df-on 6390  df-iota 6516  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-fv 6571  df-smo 8385
This theorem is referenced by:  cfcof  10312
  Copyright terms: Public domain W3C validator