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Theorem onfununi 8355
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 7775 . . . . . . . . . 10 (𝑆 ⊆ On → Ord 𝑆)
21ad2antrr 725 . . . . . . . . 9 (((𝑆 ⊆ On ∧ ¬ 𝑆𝑆) ∧ 𝑆 ≠ ∅) → Ord 𝑆)
3 nelneq 2852 . . . . . . . . . . . . . . . 16 ((𝑥𝑆 ∧ ¬ 𝑆𝑆) → ¬ 𝑥 = 𝑆)
4 elssuni 4935 . . . . . . . . . . . . . . . . . . . 20 (𝑥𝑆𝑥 𝑆)
54adantl 481 . . . . . . . . . . . . . . . . . . 19 ((𝑆 ⊆ On ∧ 𝑥𝑆) → 𝑥 𝑆)
6 ssel 3971 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑆 ⊆ On → (𝑥𝑆𝑥 ∈ On))
7 eloni 6373 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 ∈ On → Ord 𝑥)
86, 7syl6 35 . . . . . . . . . . . . . . . . . . . . . 22 (𝑆 ⊆ On → (𝑥𝑆 → Ord 𝑥))
98imp 406 . . . . . . . . . . . . . . . . . . . . 21 ((𝑆 ⊆ On ∧ 𝑥𝑆) → Ord 𝑥)
10 ordsseleq 6392 . . . . . . . . . . . . . . . . . . . . 21 ((Ord 𝑥 ∧ Ord 𝑆) → (𝑥 𝑆 ↔ (𝑥 𝑆𝑥 = 𝑆)))
119, 1, 10syl2an 595 . . . . . . . . . . . . . . . . . . . 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 430 . . . . . . . . . . . . . 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 406 . . . . . . . . . . 11 ((𝑆 ⊆ On ∧ ¬ 𝑆𝑆) → (𝑥𝑆𝑥 𝑆))
2120ssrdv 3984 . . . . . . . . . 10 ((𝑆 ⊆ On ∧ ¬ 𝑆𝑆) → 𝑆 𝑆)
22 ssn0 4396 . . . . . . . . . 10 ((𝑆 𝑆𝑆 ≠ ∅) → 𝑆 ≠ ∅)
2321, 22sylan 579 . . . . . . . . 9 (((𝑆 ⊆ On ∧ ¬ 𝑆𝑆) ∧ 𝑆 ≠ ∅) → 𝑆 ≠ ∅)
2421unissd 4913 . . . . . . . . . . 11 ((𝑆 ⊆ On ∧ ¬ 𝑆𝑆) → 𝑆 𝑆)
25 orduniss 6460 . . . . . . . . . . . . 13 (Ord 𝑆 𝑆 𝑆)
261, 25syl 17 . . . . . . . . . . . 12 (𝑆 ⊆ On → 𝑆 𝑆)
2726adantr 480 . . . . . . . . . . 11 ((𝑆 ⊆ On ∧ ¬ 𝑆𝑆) → 𝑆 𝑆)
2824, 27eqssd 3995 . . . . . . . . . 10 ((𝑆 ⊆ On ∧ ¬ 𝑆𝑆) → 𝑆 = 𝑆)
2928adantr 480 . . . . . . . . 9 (((𝑆 ⊆ On ∧ ¬ 𝑆𝑆) ∧ 𝑆 ≠ ∅) → 𝑆 = 𝑆)
30 df-lim 6368 . . . . . . . . 9 (Lim 𝑆 ↔ (Ord 𝑆 𝑆 ≠ ∅ ∧ 𝑆 = 𝑆))
312, 23, 29, 30syl3anbrc 1341 . . . . . . . 8 (((𝑆 ⊆ On ∧ ¬ 𝑆𝑆) ∧ 𝑆 ≠ ∅) → Lim 𝑆)
3231an32s 651 . . . . . . 7 (((𝑆 ⊆ On ∧ 𝑆 ≠ ∅) ∧ ¬ 𝑆𝑆) → Lim 𝑆)
33323adantl1 1164 . . . . . 6 (((𝑆𝑇𝑆 ⊆ On ∧ 𝑆 ≠ ∅) ∧ ¬ 𝑆𝑆) → Lim 𝑆)
34 ssonuni 7776 . . . . . . . . . 10 (𝑆𝑇 → (𝑆 ⊆ On → 𝑆 ∈ On))
35 limeq 6375 . . . . . . . . . . . 12 (𝑦 = 𝑆 → (Lim 𝑦 ↔ Lim 𝑆))
36 fveq2 6891 . . . . . . . . . . . . 13 (𝑦 = 𝑆 → (𝐹𝑦) = (𝐹 𝑆))
37 iuneq1 5007 . . . . . . . . . . . . 13 (𝑦 = 𝑆 𝑥𝑦 (𝐹𝑥) = 𝑥 𝑆(𝐹𝑥))
3836, 37eqeq12d 2743 . . . . . . . . . . . 12 (𝑦 = 𝑆 → ((𝐹𝑦) = 𝑥𝑦 (𝐹𝑥) ↔ (𝐹 𝑆) = 𝑥 𝑆(𝐹𝑥)))
3935, 38imbi12d 344 . . . . . . . . . . 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 3538 . . . . . . . . . 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 406 . . . . . . . 8 ((𝑆𝑇𝑆 ⊆ On) → (Lim 𝑆 → (𝐹 𝑆) = 𝑥 𝑆(𝐹𝑥)))
44433adant3 1130 . . . . . . 7 ((𝑆𝑇𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → (Lim 𝑆 → (𝐹 𝑆) = 𝑥 𝑆(𝐹𝑥)))
4544adantr 480 . . . . . 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 4907 . . . . . . . . . . . 12 (𝑥 𝑆 ↔ ∃𝑦𝑆 𝑥𝑦)
48 ssel 3971 . . . . . . . . . . . . . . . . . 18 (𝑆 ⊆ On → (𝑦𝑆𝑦 ∈ On))
4948anim1d 610 . . . . . . . . . . . . . . . . 17 (𝑆 ⊆ On → ((𝑦𝑆𝑥𝑦) → (𝑦 ∈ On ∧ 𝑥𝑦)))
50 onelon 6388 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ On ∧ 𝑥𝑦) → 𝑥 ∈ On)
5149, 50syl6 35 . . . . . . . . . . . . . . . 16 (𝑆 ⊆ On → ((𝑦𝑆𝑥𝑦) → 𝑥 ∈ On))
5248adantrd 491 . . . . . . . . . . . . . . . 16 (𝑆 ⊆ On → ((𝑦𝑆𝑥𝑦) → 𝑦 ∈ On))
53 eloni 6373 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ On → Ord 𝑦)
5448, 53syl6 35 . . . . . . . . . . . . . . . . 17 (𝑆 ⊆ On → (𝑦𝑆 → Ord 𝑦))
55 ordelss 6379 . . . . . . . . . . . . . . . . . 18 ((Ord 𝑦𝑥𝑦) → 𝑥𝑦)
5655a1i 11 . . . . . . . . . . . . . . . . 17 (𝑆 ⊆ On → ((Ord 𝑦𝑥𝑦) → 𝑥𝑦))
5754, 56syland 602 . . . . . . . . . . . . . . . 16 (𝑆 ⊆ On → ((𝑦𝑆𝑥𝑦) → 𝑥𝑦))
5851, 52, 573jcad 1127 . . . . . . . . . . . . . . 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 415 . . . . . . . . . . . . 13 (𝑆 ⊆ On → (𝑦𝑆 → (𝑥𝑦 → (𝐹𝑥) ⊆ (𝐹𝑦))))
6261reximdvai 3160 . . . . . . . . . . . 12 (𝑆 ⊆ On → (∃𝑦𝑆 𝑥𝑦 → ∃𝑦𝑆 (𝐹𝑥) ⊆ (𝐹𝑦)))
6347, 62biimtrid 241 . . . . . . . . . . 11 (𝑆 ⊆ On → (𝑥 𝑆 → ∃𝑦𝑆 (𝐹𝑥) ⊆ (𝐹𝑦)))
64 ssiun 5043 . . . . . . . . . . 11 (∃𝑦𝑆 (𝐹𝑥) ⊆ (𝐹𝑦) → (𝐹𝑥) ⊆ 𝑦𝑆 (𝐹𝑦))
6563, 64syl6 35 . . . . . . . . . 10 (𝑆 ⊆ On → (𝑥 𝑆 → (𝐹𝑥) ⊆ 𝑦𝑆 (𝐹𝑦)))
6665ralrimiv 3140 . . . . . . . . 9 (𝑆 ⊆ On → ∀𝑥 𝑆(𝐹𝑥) ⊆ 𝑦𝑆 (𝐹𝑦))
67 iunss 5042 . . . . . . . . 9 ( 𝑥 𝑆(𝐹𝑥) ⊆ 𝑦𝑆 (𝐹𝑦) ↔ ∀𝑥 𝑆(𝐹𝑥) ⊆ 𝑦𝑆 (𝐹𝑦))
6866, 67sylibr 233 . . . . . . . 8 (𝑆 ⊆ On → 𝑥 𝑆(𝐹𝑥) ⊆ 𝑦𝑆 (𝐹𝑦))
69 fveq2 6891 . . . . . . . . 9 (𝑦 = 𝑥 → (𝐹𝑦) = (𝐹𝑥))
7069cbviunv 5037 . . . . . . . 8 𝑦𝑆 (𝐹𝑦) = 𝑥𝑆 (𝐹𝑥)
7168, 70sseqtrdi 4028 . . . . . . 7 (𝑆 ⊆ On → 𝑥 𝑆(𝐹𝑥) ⊆ 𝑥𝑆 (𝐹𝑥))
72713ad2ant2 1132 . . . . . 6 ((𝑆𝑇𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → 𝑥 𝑆(𝐹𝑥) ⊆ 𝑥𝑆 (𝐹𝑥))
7372adantr 480 . . . . 5 (((𝑆𝑇𝑆 ⊆ On ∧ 𝑆 ≠ ∅) ∧ ¬ 𝑆𝑆) → 𝑥 𝑆(𝐹𝑥) ⊆ 𝑥𝑆 (𝐹𝑥))
7446, 73eqsstrd 4016 . . . 4 (((𝑆𝑇𝑆 ⊆ On ∧ 𝑆 ≠ ∅) ∧ ¬ 𝑆𝑆) → (𝐹 𝑆) ⊆ 𝑥𝑆 (𝐹𝑥))
7574ex 412 . . 3 ((𝑆𝑇𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → (¬ 𝑆𝑆 → (𝐹 𝑆) ⊆ 𝑥𝑆 (𝐹𝑥)))
76 fveq2 6891 . . . 4 (𝑥 = 𝑆 → (𝐹𝑥) = (𝐹 𝑆))
7776ssiun2s 5045 . . 3 ( 𝑆𝑆 → (𝐹 𝑆) ⊆ 𝑥𝑆 (𝐹𝑥))
7875, 77pm2.61d2 181 . 2 ((𝑆𝑇𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → (𝐹 𝑆) ⊆ 𝑥𝑆 (𝐹𝑥))
7934imp 406 . . . . . 6 ((𝑆𝑇𝑆 ⊆ On) → 𝑆 ∈ On)
80793adant3 1130 . . . . 5 ((𝑆𝑇𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → 𝑆 ∈ On)
8163ad2ant2 1132 . . . . . 6 ((𝑆𝑇𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → (𝑥𝑆𝑥 ∈ On))
8281, 4jca2 513 . . . . 5 ((𝑆𝑇𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → (𝑥𝑆 → (𝑥 ∈ On ∧ 𝑥 𝑆)))
83 sseq2 4004 . . . . . . . 8 (𝑦 = 𝑆 → (𝑥𝑦𝑥 𝑆))
8483anbi2d 628 . . . . . . 7 (𝑦 = 𝑆 → ((𝑥 ∈ On ∧ 𝑥𝑦) ↔ (𝑥 ∈ On ∧ 𝑥 𝑆)))
8536sseq2d 4010 . . . . . . 7 (𝑦 = 𝑆 → ((𝐹𝑥) ⊆ (𝐹𝑦) ↔ (𝐹𝑥) ⊆ (𝐹 𝑆)))
8684, 85imbi12d 344 . . . . . 6 (𝑦 = 𝑆 → (((𝑥 ∈ On ∧ 𝑥𝑦) → (𝐹𝑥) ⊆ (𝐹𝑦)) ↔ ((𝑥 ∈ On ∧ 𝑥 𝑆) → (𝐹𝑥) ⊆ (𝐹 𝑆))))
87593com12 1121 . . . . . . 7 ((𝑦 ∈ On ∧ 𝑥 ∈ On ∧ 𝑥𝑦) → (𝐹𝑥) ⊆ (𝐹𝑦))
88873expib 1120 . . . . . 6 (𝑦 ∈ On → ((𝑥 ∈ On ∧ 𝑥𝑦) → (𝐹𝑥) ⊆ (𝐹𝑦)))
8986, 88vtoclga 3561 . . . . 5 ( 𝑆 ∈ On → ((𝑥 ∈ On ∧ 𝑥 𝑆) → (𝐹𝑥) ⊆ (𝐹 𝑆)))
9080, 82, 89sylsyld 61 . . . 4 ((𝑆𝑇𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → (𝑥𝑆 → (𝐹𝑥) ⊆ (𝐹 𝑆)))
9190ralrimiv 3140 . . 3 ((𝑆𝑇𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → ∀𝑥𝑆 (𝐹𝑥) ⊆ (𝐹 𝑆))
92 iunss 5042 . . 3 ( 𝑥𝑆 (𝐹𝑥) ⊆ (𝐹 𝑆) ↔ ∀𝑥𝑆 (𝐹𝑥) ⊆ (𝐹 𝑆))
9391, 92sylibr 233 . 2 ((𝑆𝑇𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → 𝑥𝑆 (𝐹𝑥) ⊆ (𝐹 𝑆))
9478, 93eqssd 3995 1 ((𝑆𝑇𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → (𝐹 𝑆) = 𝑥𝑆 (𝐹𝑥))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 395  wo 846  w3a 1085   = wceq 1534  wcel 2099  wne 2935  wral 3056  wrex 3065  wss 3944  c0 4318   cuni 4903   ciun 4991  Ord word 6362  Oncon0 6363  Lim wlim 6364  cfv 6542
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2164  ax-ext 2698  ax-sep 5293  ax-nul 5300  ax-pr 5423  ax-un 7734
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-nfc 2880  df-ne 2936  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-pss 3963  df-nul 4319  df-if 4525  df-pw 4600  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-iun 4993  df-br 5143  df-opab 5205  df-tr 5260  df-eprel 5576  df-po 5584  df-so 5585  df-fr 5627  df-we 5629  df-ord 6366  df-on 6367  df-lim 6368  df-iota 6494  df-fv 6550
This theorem is referenced by:  onovuni  8356
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