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Theorem onfununi 8341
Description: A property of functions on ordinal numbers. Generalization of Theorem Schema 8E of [Enderton] p. 218. (Contributed by Eric Schmidt, 26-May-2009.)
Hypotheses
Ref Expression
onfununi.1 (Lim 𝑦 → (𝐹𝑦) = 𝑥𝑦 (𝐹𝑥))
onfununi.2 ((𝑥 ∈ On ∧ 𝑦 ∈ On ∧ 𝑥𝑦) → (𝐹𝑥) ⊆ (𝐹𝑦))
Assertion
Ref Expression
onfununi ((𝑆𝑇𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → (𝐹 𝑆) = 𝑥𝑆 (𝐹𝑥))
Distinct variable groups:   𝑥,𝑦,𝑆   𝑥,𝐹,𝑦   𝑥,𝑇
Allowed substitution hint:   𝑇(𝑦)

Proof of Theorem onfununi
StepHypRef Expression
1 ssorduni 7766 . . . . . . . . . 10 (𝑆 ⊆ On → Ord 𝑆)
21ad2antrr 725 . . . . . . . . 9 (((𝑆 ⊆ On ∧ ¬ 𝑆𝑆) ∧ 𝑆 ≠ ∅) → Ord 𝑆)
3 nelneq 2858 . . . . . . . . . . . . . . . 16 ((𝑥𝑆 ∧ ¬ 𝑆𝑆) → ¬ 𝑥 = 𝑆)
4 elssuni 4942 . . . . . . . . . . . . . . . . . . . 20 (𝑥𝑆𝑥 𝑆)
54adantl 483 . . . . . . . . . . . . . . . . . . 19 ((𝑆 ⊆ On ∧ 𝑥𝑆) → 𝑥 𝑆)
6 ssel 3976 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑆 ⊆ On → (𝑥𝑆𝑥 ∈ On))
7 eloni 6375 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 ∈ On → Ord 𝑥)
86, 7syl6 35 . . . . . . . . . . . . . . . . . . . . . 22 (𝑆 ⊆ On → (𝑥𝑆 → Ord 𝑥))
98imp 408 . . . . . . . . . . . . . . . . . . . . 21 ((𝑆 ⊆ On ∧ 𝑥𝑆) → Ord 𝑥)
10 ordsseleq 6394 . . . . . . . . . . . . . . . . . . . . 21 ((Ord 𝑥 ∧ Ord 𝑆) → (𝑥 𝑆 ↔ (𝑥 𝑆𝑥 = 𝑆)))
119, 1, 10syl2an 597 . . . . . . . . . . . . . . . . . . . 20 (((𝑆 ⊆ On ∧ 𝑥𝑆) ∧ 𝑆 ⊆ On) → (𝑥 𝑆 ↔ (𝑥 𝑆𝑥 = 𝑆)))
1211anabss1 665 . . . . . . . . . . . . . . . . . . 19 ((𝑆 ⊆ On ∧ 𝑥𝑆) → (𝑥 𝑆 ↔ (𝑥 𝑆𝑥 = 𝑆)))
135, 12mpbid 231 . . . . . . . . . . . . . . . . . 18 ((𝑆 ⊆ On ∧ 𝑥𝑆) → (𝑥 𝑆𝑥 = 𝑆))
1413ord 863 . . . . . . . . . . . . . . . . 17 ((𝑆 ⊆ On ∧ 𝑥𝑆) → (¬ 𝑥 𝑆𝑥 = 𝑆))
1514con1d 145 . . . . . . . . . . . . . . . 16 ((𝑆 ⊆ On ∧ 𝑥𝑆) → (¬ 𝑥 = 𝑆𝑥 𝑆))
163, 15syl5 34 . . . . . . . . . . . . . . 15 ((𝑆 ⊆ On ∧ 𝑥𝑆) → ((𝑥𝑆 ∧ ¬ 𝑆𝑆) → 𝑥 𝑆))
1716exp4b 432 . . . . . . . . . . . . . 14 (𝑆 ⊆ On → (𝑥𝑆 → (𝑥𝑆 → (¬ 𝑆𝑆𝑥 𝑆))))
1817pm2.43d 53 . . . . . . . . . . . . 13 (𝑆 ⊆ On → (𝑥𝑆 → (¬ 𝑆𝑆𝑥 𝑆)))
1918com23 86 . . . . . . . . . . . 12 (𝑆 ⊆ On → (¬ 𝑆𝑆 → (𝑥𝑆𝑥 𝑆)))
2019imp 408 . . . . . . . . . . 11 ((𝑆 ⊆ On ∧ ¬ 𝑆𝑆) → (𝑥𝑆𝑥 𝑆))
2120ssrdv 3989 . . . . . . . . . 10 ((𝑆 ⊆ On ∧ ¬ 𝑆𝑆) → 𝑆 𝑆)
22 ssn0 4401 . . . . . . . . . 10 ((𝑆 𝑆𝑆 ≠ ∅) → 𝑆 ≠ ∅)
2321, 22sylan 581 . . . . . . . . 9 (((𝑆 ⊆ On ∧ ¬ 𝑆𝑆) ∧ 𝑆 ≠ ∅) → 𝑆 ≠ ∅)
2421unissd 4919 . . . . . . . . . . 11 ((𝑆 ⊆ On ∧ ¬ 𝑆𝑆) → 𝑆 𝑆)
25 orduniss 6462 . . . . . . . . . . . . 13 (Ord 𝑆 𝑆 𝑆)
261, 25syl 17 . . . . . . . . . . . 12 (𝑆 ⊆ On → 𝑆 𝑆)
2726adantr 482 . . . . . . . . . . 11 ((𝑆 ⊆ On ∧ ¬ 𝑆𝑆) → 𝑆 𝑆)
2824, 27eqssd 4000 . . . . . . . . . 10 ((𝑆 ⊆ On ∧ ¬ 𝑆𝑆) → 𝑆 = 𝑆)
2928adantr 482 . . . . . . . . 9 (((𝑆 ⊆ On ∧ ¬ 𝑆𝑆) ∧ 𝑆 ≠ ∅) → 𝑆 = 𝑆)
30 df-lim 6370 . . . . . . . . 9 (Lim 𝑆 ↔ (Ord 𝑆 𝑆 ≠ ∅ ∧ 𝑆 = 𝑆))
312, 23, 29, 30syl3anbrc 1344 . . . . . . . 8 (((𝑆 ⊆ On ∧ ¬ 𝑆𝑆) ∧ 𝑆 ≠ ∅) → Lim 𝑆)
3231an32s 651 . . . . . . 7 (((𝑆 ⊆ On ∧ 𝑆 ≠ ∅) ∧ ¬ 𝑆𝑆) → Lim 𝑆)
33323adantl1 1167 . . . . . 6 (((𝑆𝑇𝑆 ⊆ On ∧ 𝑆 ≠ ∅) ∧ ¬ 𝑆𝑆) → Lim 𝑆)
34 ssonuni 7767 . . . . . . . . . 10 (𝑆𝑇 → (𝑆 ⊆ On → 𝑆 ∈ On))
35 limeq 6377 . . . . . . . . . . . 12 (𝑦 = 𝑆 → (Lim 𝑦 ↔ Lim 𝑆))
36 fveq2 6892 . . . . . . . . . . . . 13 (𝑦 = 𝑆 → (𝐹𝑦) = (𝐹 𝑆))
37 iuneq1 5014 . . . . . . . . . . . . 13 (𝑦 = 𝑆 𝑥𝑦 (𝐹𝑥) = 𝑥 𝑆(𝐹𝑥))
3836, 37eqeq12d 2749 . . . . . . . . . . . 12 (𝑦 = 𝑆 → ((𝐹𝑦) = 𝑥𝑦 (𝐹𝑥) ↔ (𝐹 𝑆) = 𝑥 𝑆(𝐹𝑥)))
3935, 38imbi12d 345 . . . . . . . . . . 11 (𝑦 = 𝑆 → ((Lim 𝑦 → (𝐹𝑦) = 𝑥𝑦 (𝐹𝑥)) ↔ (Lim 𝑆 → (𝐹 𝑆) = 𝑥 𝑆(𝐹𝑥))))
40 onfununi.1 . . . . . . . . . . 11 (Lim 𝑦 → (𝐹𝑦) = 𝑥𝑦 (𝐹𝑥))
4139, 40vtoclg 3557 . . . . . . . . . 10 ( 𝑆 ∈ On → (Lim 𝑆 → (𝐹 𝑆) = 𝑥 𝑆(𝐹𝑥)))
4234, 41syl6 35 . . . . . . . . 9 (𝑆𝑇 → (𝑆 ⊆ On → (Lim 𝑆 → (𝐹 𝑆) = 𝑥 𝑆(𝐹𝑥))))
4342imp 408 . . . . . . . 8 ((𝑆𝑇𝑆 ⊆ On) → (Lim 𝑆 → (𝐹 𝑆) = 𝑥 𝑆(𝐹𝑥)))
44433adant3 1133 . . . . . . 7 ((𝑆𝑇𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → (Lim 𝑆 → (𝐹 𝑆) = 𝑥 𝑆(𝐹𝑥)))
4544adantr 482 . . . . . 6 (((𝑆𝑇𝑆 ⊆ On ∧ 𝑆 ≠ ∅) ∧ ¬ 𝑆𝑆) → (Lim 𝑆 → (𝐹 𝑆) = 𝑥 𝑆(𝐹𝑥)))
4633, 45mpd 15 . . . . 5 (((𝑆𝑇𝑆 ⊆ On ∧ 𝑆 ≠ ∅) ∧ ¬ 𝑆𝑆) → (𝐹 𝑆) = 𝑥 𝑆(𝐹𝑥))
47 eluni2 4913 . . . . . . . . . . . 12 (𝑥 𝑆 ↔ ∃𝑦𝑆 𝑥𝑦)
48 ssel 3976 . . . . . . . . . . . . . . . . . 18 (𝑆 ⊆ On → (𝑦𝑆𝑦 ∈ On))
4948anim1d 612 . . . . . . . . . . . . . . . . 17 (𝑆 ⊆ On → ((𝑦𝑆𝑥𝑦) → (𝑦 ∈ On ∧ 𝑥𝑦)))
50 onelon 6390 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ On ∧ 𝑥𝑦) → 𝑥 ∈ On)
5149, 50syl6 35 . . . . . . . . . . . . . . . 16 (𝑆 ⊆ On → ((𝑦𝑆𝑥𝑦) → 𝑥 ∈ On))
5248adantrd 493 . . . . . . . . . . . . . . . 16 (𝑆 ⊆ On → ((𝑦𝑆𝑥𝑦) → 𝑦 ∈ On))
53 eloni 6375 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ On → Ord 𝑦)
5448, 53syl6 35 . . . . . . . . . . . . . . . . 17 (𝑆 ⊆ On → (𝑦𝑆 → Ord 𝑦))
55 ordelss 6381 . . . . . . . . . . . . . . . . . 18 ((Ord 𝑦𝑥𝑦) → 𝑥𝑦)
5655a1i 11 . . . . . . . . . . . . . . . . 17 (𝑆 ⊆ On → ((Ord 𝑦𝑥𝑦) → 𝑥𝑦))
5754, 56syland 604 . . . . . . . . . . . . . . . 16 (𝑆 ⊆ On → ((𝑦𝑆𝑥𝑦) → 𝑥𝑦))
5851, 52, 573jcad 1130 . . . . . . . . . . . . . . 15 (𝑆 ⊆ On → ((𝑦𝑆𝑥𝑦) → (𝑥 ∈ On ∧ 𝑦 ∈ On ∧ 𝑥𝑦)))
59 onfununi.2 . . . . . . . . . . . . . . 15 ((𝑥 ∈ On ∧ 𝑦 ∈ On ∧ 𝑥𝑦) → (𝐹𝑥) ⊆ (𝐹𝑦))
6058, 59syl6 35 . . . . . . . . . . . . . 14 (𝑆 ⊆ On → ((𝑦𝑆𝑥𝑦) → (𝐹𝑥) ⊆ (𝐹𝑦)))
6160expd 417 . . . . . . . . . . . . 13 (𝑆 ⊆ On → (𝑦𝑆 → (𝑥𝑦 → (𝐹𝑥) ⊆ (𝐹𝑦))))
6261reximdvai 3166 . . . . . . . . . . . 12 (𝑆 ⊆ On → (∃𝑦𝑆 𝑥𝑦 → ∃𝑦𝑆 (𝐹𝑥) ⊆ (𝐹𝑦)))
6347, 62biimtrid 241 . . . . . . . . . . 11 (𝑆 ⊆ On → (𝑥 𝑆 → ∃𝑦𝑆 (𝐹𝑥) ⊆ (𝐹𝑦)))
64 ssiun 5050 . . . . . . . . . . 11 (∃𝑦𝑆 (𝐹𝑥) ⊆ (𝐹𝑦) → (𝐹𝑥) ⊆ 𝑦𝑆 (𝐹𝑦))
6563, 64syl6 35 . . . . . . . . . 10 (𝑆 ⊆ On → (𝑥 𝑆 → (𝐹𝑥) ⊆ 𝑦𝑆 (𝐹𝑦)))
6665ralrimiv 3146 . . . . . . . . 9 (𝑆 ⊆ On → ∀𝑥 𝑆(𝐹𝑥) ⊆ 𝑦𝑆 (𝐹𝑦))
67 iunss 5049 . . . . . . . . 9 ( 𝑥 𝑆(𝐹𝑥) ⊆ 𝑦𝑆 (𝐹𝑦) ↔ ∀𝑥 𝑆(𝐹𝑥) ⊆ 𝑦𝑆 (𝐹𝑦))
6866, 67sylibr 233 . . . . . . . 8 (𝑆 ⊆ On → 𝑥 𝑆(𝐹𝑥) ⊆ 𝑦𝑆 (𝐹𝑦))
69 fveq2 6892 . . . . . . . . 9 (𝑦 = 𝑥 → (𝐹𝑦) = (𝐹𝑥))
7069cbviunv 5044 . . . . . . . 8 𝑦𝑆 (𝐹𝑦) = 𝑥𝑆 (𝐹𝑥)
7168, 70sseqtrdi 4033 . . . . . . 7 (𝑆 ⊆ On → 𝑥 𝑆(𝐹𝑥) ⊆ 𝑥𝑆 (𝐹𝑥))
72713ad2ant2 1135 . . . . . 6 ((𝑆𝑇𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → 𝑥 𝑆(𝐹𝑥) ⊆ 𝑥𝑆 (𝐹𝑥))
7372adantr 482 . . . . 5 (((𝑆𝑇𝑆 ⊆ On ∧ 𝑆 ≠ ∅) ∧ ¬ 𝑆𝑆) → 𝑥 𝑆(𝐹𝑥) ⊆ 𝑥𝑆 (𝐹𝑥))
7446, 73eqsstrd 4021 . . . 4 (((𝑆𝑇𝑆 ⊆ On ∧ 𝑆 ≠ ∅) ∧ ¬ 𝑆𝑆) → (𝐹 𝑆) ⊆ 𝑥𝑆 (𝐹𝑥))
7574ex 414 . . 3 ((𝑆𝑇𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → (¬ 𝑆𝑆 → (𝐹 𝑆) ⊆ 𝑥𝑆 (𝐹𝑥)))
76 fveq2 6892 . . . 4 (𝑥 = 𝑆 → (𝐹𝑥) = (𝐹 𝑆))
7776ssiun2s 5052 . . 3 ( 𝑆𝑆 → (𝐹 𝑆) ⊆ 𝑥𝑆 (𝐹𝑥))
7875, 77pm2.61d2 181 . 2 ((𝑆𝑇𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → (𝐹 𝑆) ⊆ 𝑥𝑆 (𝐹𝑥))
7934imp 408 . . . . . 6 ((𝑆𝑇𝑆 ⊆ On) → 𝑆 ∈ On)
80793adant3 1133 . . . . 5 ((𝑆𝑇𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → 𝑆 ∈ On)
8163ad2ant2 1135 . . . . . 6 ((𝑆𝑇𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → (𝑥𝑆𝑥 ∈ On))
8281, 4jca2 515 . . . . 5 ((𝑆𝑇𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → (𝑥𝑆 → (𝑥 ∈ On ∧ 𝑥 𝑆)))
83 sseq2 4009 . . . . . . . 8 (𝑦 = 𝑆 → (𝑥𝑦𝑥 𝑆))
8483anbi2d 630 . . . . . . 7 (𝑦 = 𝑆 → ((𝑥 ∈ On ∧ 𝑥𝑦) ↔ (𝑥 ∈ On ∧ 𝑥 𝑆)))
8536sseq2d 4015 . . . . . . 7 (𝑦 = 𝑆 → ((𝐹𝑥) ⊆ (𝐹𝑦) ↔ (𝐹𝑥) ⊆ (𝐹 𝑆)))
8684, 85imbi12d 345 . . . . . 6 (𝑦 = 𝑆 → (((𝑥 ∈ On ∧ 𝑥𝑦) → (𝐹𝑥) ⊆ (𝐹𝑦)) ↔ ((𝑥 ∈ On ∧ 𝑥 𝑆) → (𝐹𝑥) ⊆ (𝐹 𝑆))))
87593com12 1124 . . . . . . 7 ((𝑦 ∈ On ∧ 𝑥 ∈ On ∧ 𝑥𝑦) → (𝐹𝑥) ⊆ (𝐹𝑦))
88873expib 1123 . . . . . 6 (𝑦 ∈ On → ((𝑥 ∈ On ∧ 𝑥𝑦) → (𝐹𝑥) ⊆ (𝐹𝑦)))
8986, 88vtoclga 3566 . . . . 5 ( 𝑆 ∈ On → ((𝑥 ∈ On ∧ 𝑥 𝑆) → (𝐹𝑥) ⊆ (𝐹 𝑆)))
9080, 82, 89sylsyld 61 . . . 4 ((𝑆𝑇𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → (𝑥𝑆 → (𝐹𝑥) ⊆ (𝐹 𝑆)))
9190ralrimiv 3146 . . 3 ((𝑆𝑇𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → ∀𝑥𝑆 (𝐹𝑥) ⊆ (𝐹 𝑆))
92 iunss 5049 . . 3 ( 𝑥𝑆 (𝐹𝑥) ⊆ (𝐹 𝑆) ↔ ∀𝑥𝑆 (𝐹𝑥) ⊆ (𝐹 𝑆))
9391, 92sylibr 233 . 2 ((𝑆𝑇𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → 𝑥𝑆 (𝐹𝑥) ⊆ (𝐹 𝑆))
9478, 93eqssd 4000 1 ((𝑆𝑇𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → (𝐹 𝑆) = 𝑥𝑆 (𝐹𝑥))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 397  wo 846  w3a 1088   = wceq 1542  wcel 2107  wne 2941  wral 3062  wrex 3071  wss 3949  c0 4323   cuni 4909   ciun 4998  Ord word 6364  Oncon0 6365  Lim wlim 6366  cfv 6544
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-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-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  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-iun 5000  df-br 5150  df-opab 5212  df-tr 5267  df-eprel 5581  df-po 5589  df-so 5590  df-fr 5632  df-we 5634  df-ord 6368  df-on 6369  df-lim 6370  df-iota 6496  df-fv 6552
This theorem is referenced by:  onovuni  8342
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