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

Theorem onfununi 7967
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 7489 . . . . . . . . . 10 (𝑆 ⊆ On → Ord 𝑆)
21ad2antrr 722 . . . . . . . . 9 (((𝑆 ⊆ On ∧ ¬ 𝑆𝑆) ∧ 𝑆 ≠ ∅) → Ord 𝑆)
3 nelneq 2934 . . . . . . . . . . . . . . . 16 ((𝑥𝑆 ∧ ¬ 𝑆𝑆) → ¬ 𝑥 = 𝑆)
4 elssuni 4859 . . . . . . . . . . . . . . . . . . . 20 (𝑥𝑆𝑥 𝑆)
54adantl 482 . . . . . . . . . . . . . . . . . . 19 ((𝑆 ⊆ On ∧ 𝑥𝑆) → 𝑥 𝑆)
6 ssel 3958 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑆 ⊆ On → (𝑥𝑆𝑥 ∈ On))
7 eloni 6194 . . . . . . . . . . . . . . . . . . . . . . 23 (𝑥 ∈ On → Ord 𝑥)
86, 7syl6 35 . . . . . . . . . . . . . . . . . . . . . 22 (𝑆 ⊆ On → (𝑥𝑆 → Ord 𝑥))
98imp 407 . . . . . . . . . . . . . . . . . . . . 21 ((𝑆 ⊆ On ∧ 𝑥𝑆) → Ord 𝑥)
10 ordsseleq 6213 . . . . . . . . . . . . . . . . . . . . 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 662 . . . . . . . . . . . . . . . . . . 19 ((𝑆 ⊆ On ∧ 𝑥𝑆) → (𝑥 𝑆 ↔ (𝑥 𝑆𝑥 = 𝑆)))
135, 12mpbid 233 . . . . . . . . . . . . . . . . . 18 ((𝑆 ⊆ On ∧ 𝑥𝑆) → (𝑥 𝑆𝑥 = 𝑆))
1413ord 858 . . . . . . . . . . . . . . . . 17 ((𝑆 ⊆ On ∧ 𝑥𝑆) → (¬ 𝑥 𝑆𝑥 = 𝑆))
1514con1d 147 . . . . . . . . . . . . . . . 16 ((𝑆 ⊆ On ∧ 𝑥𝑆) → (¬ 𝑥 = 𝑆𝑥 𝑆))
163, 15syl5 34 . . . . . . . . . . . . . . 15 ((𝑆 ⊆ On ∧ 𝑥𝑆) → ((𝑥𝑆 ∧ ¬ 𝑆𝑆) → 𝑥 𝑆))
1716exp4b 431 . . . . . . . . . . . . . 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 407 . . . . . . . . . . 11 ((𝑆 ⊆ On ∧ ¬ 𝑆𝑆) → (𝑥𝑆𝑥 𝑆))
2120ssrdv 3970 . . . . . . . . . 10 ((𝑆 ⊆ On ∧ ¬ 𝑆𝑆) → 𝑆 𝑆)
22 ssn0 4351 . . . . . . . . . 10 ((𝑆 𝑆𝑆 ≠ ∅) → 𝑆 ≠ ∅)
2321, 22sylan 580 . . . . . . . . 9 (((𝑆 ⊆ On ∧ ¬ 𝑆𝑆) ∧ 𝑆 ≠ ∅) → 𝑆 ≠ ∅)
2421unissd 4854 . . . . . . . . . . 11 ((𝑆 ⊆ On ∧ ¬ 𝑆𝑆) → 𝑆 𝑆)
25 orduniss 6278 . . . . . . . . . . . . 13 (Ord 𝑆 𝑆 𝑆)
261, 25syl 17 . . . . . . . . . . . 12 (𝑆 ⊆ On → 𝑆 𝑆)
2726adantr 481 . . . . . . . . . . 11 ((𝑆 ⊆ On ∧ ¬ 𝑆𝑆) → 𝑆 𝑆)
2824, 27eqssd 3981 . . . . . . . . . 10 ((𝑆 ⊆ On ∧ ¬ 𝑆𝑆) → 𝑆 = 𝑆)
2928adantr 481 . . . . . . . . 9 (((𝑆 ⊆ On ∧ ¬ 𝑆𝑆) ∧ 𝑆 ≠ ∅) → 𝑆 = 𝑆)
30 df-lim 6189 . . . . . . . . 9 (Lim 𝑆 ↔ (Ord 𝑆 𝑆 ≠ ∅ ∧ 𝑆 = 𝑆))
312, 23, 29, 30syl3anbrc 1335 . . . . . . . 8 (((𝑆 ⊆ On ∧ ¬ 𝑆𝑆) ∧ 𝑆 ≠ ∅) → Lim 𝑆)
3231an32s 648 . . . . . . 7 (((𝑆 ⊆ On ∧ 𝑆 ≠ ∅) ∧ ¬ 𝑆𝑆) → Lim 𝑆)
33323adantl1 1158 . . . . . 6 (((𝑆𝑇𝑆 ⊆ On ∧ 𝑆 ≠ ∅) ∧ ¬ 𝑆𝑆) → Lim 𝑆)
34 ssonuni 7490 . . . . . . . . . 10 (𝑆𝑇 → (𝑆 ⊆ On → 𝑆 ∈ On))
35 limeq 6196 . . . . . . . . . . . 12 (𝑦 = 𝑆 → (Lim 𝑦 ↔ Lim 𝑆))
36 fveq2 6663 . . . . . . . . . . . . 13 (𝑦 = 𝑆 → (𝐹𝑦) = (𝐹 𝑆))
37 iuneq1 4926 . . . . . . . . . . . . 13 (𝑦 = 𝑆 𝑥𝑦 (𝐹𝑥) = 𝑥 𝑆(𝐹𝑥))
3836, 37eqeq12d 2834 . . . . . . . . . . . 12 (𝑦 = 𝑆 → ((𝐹𝑦) = 𝑥𝑦 (𝐹𝑥) ↔ (𝐹 𝑆) = 𝑥 𝑆(𝐹𝑥)))
3935, 38imbi12d 346 . . . . . . . . . . 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 3565 . . . . . . . . . 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 407 . . . . . . . 8 ((𝑆𝑇𝑆 ⊆ On) → (Lim 𝑆 → (𝐹 𝑆) = 𝑥 𝑆(𝐹𝑥)))
44433adant3 1124 . . . . . . 7 ((𝑆𝑇𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → (Lim 𝑆 → (𝐹 𝑆) = 𝑥 𝑆(𝐹𝑥)))
4544adantr 481 . . . . . 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 4834 . . . . . . . . . . . 12 (𝑥 𝑆 ↔ ∃𝑦𝑆 𝑥𝑦)
48 ssel 3958 . . . . . . . . . . . . . . . . . 18 (𝑆 ⊆ On → (𝑦𝑆𝑦 ∈ On))
4948anim1d 610 . . . . . . . . . . . . . . . . 17 (𝑆 ⊆ On → ((𝑦𝑆𝑥𝑦) → (𝑦 ∈ On ∧ 𝑥𝑦)))
50 onelon 6209 . . . . . . . . . . . . . . . . 17 ((𝑦 ∈ On ∧ 𝑥𝑦) → 𝑥 ∈ On)
5149, 50syl6 35 . . . . . . . . . . . . . . . 16 (𝑆 ⊆ On → ((𝑦𝑆𝑥𝑦) → 𝑥 ∈ On))
5248adantrd 492 . . . . . . . . . . . . . . . 16 (𝑆 ⊆ On → ((𝑦𝑆𝑥𝑦) → 𝑦 ∈ On))
53 eloni 6194 . . . . . . . . . . . . . . . . . 18 (𝑦 ∈ On → Ord 𝑦)
5448, 53syl6 35 . . . . . . . . . . . . . . . . 17 (𝑆 ⊆ On → (𝑦𝑆 → Ord 𝑦))
55 ordelss 6200 . . . . . . . . . . . . . . . . . 18 ((Ord 𝑦𝑥𝑦) → 𝑥𝑦)
5655a1i 11 . . . . . . . . . . . . . . . . 17 (𝑆 ⊆ On → ((Ord 𝑦𝑥𝑦) → 𝑥𝑦))
5754, 56syland 602 . . . . . . . . . . . . . . . 16 (𝑆 ⊆ On → ((𝑦𝑆𝑥𝑦) → 𝑥𝑦))
5851, 52, 573jcad 1121 . . . . . . . . . . . . . . 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 416 . . . . . . . . . . . . 13 (𝑆 ⊆ On → (𝑦𝑆 → (𝑥𝑦 → (𝐹𝑥) ⊆ (𝐹𝑦))))
6261reximdvai 3269 . . . . . . . . . . . 12 (𝑆 ⊆ On → (∃𝑦𝑆 𝑥𝑦 → ∃𝑦𝑆 (𝐹𝑥) ⊆ (𝐹𝑦)))
6347, 62syl5bi 243 . . . . . . . . . . 11 (𝑆 ⊆ On → (𝑥 𝑆 → ∃𝑦𝑆 (𝐹𝑥) ⊆ (𝐹𝑦)))
64 ssiun 4961 . . . . . . . . . . 11 (∃𝑦𝑆 (𝐹𝑥) ⊆ (𝐹𝑦) → (𝐹𝑥) ⊆ 𝑦𝑆 (𝐹𝑦))
6563, 64syl6 35 . . . . . . . . . 10 (𝑆 ⊆ On → (𝑥 𝑆 → (𝐹𝑥) ⊆ 𝑦𝑆 (𝐹𝑦)))
6665ralrimiv 3178 . . . . . . . . 9 (𝑆 ⊆ On → ∀𝑥 𝑆(𝐹𝑥) ⊆ 𝑦𝑆 (𝐹𝑦))
67 iunss 4960 . . . . . . . . 9 ( 𝑥 𝑆(𝐹𝑥) ⊆ 𝑦𝑆 (𝐹𝑦) ↔ ∀𝑥 𝑆(𝐹𝑥) ⊆ 𝑦𝑆 (𝐹𝑦))
6866, 67sylibr 235 . . . . . . . 8 (𝑆 ⊆ On → 𝑥 𝑆(𝐹𝑥) ⊆ 𝑦𝑆 (𝐹𝑦))
69 fveq2 6663 . . . . . . . . 9 (𝑦 = 𝑥 → (𝐹𝑦) = (𝐹𝑥))
7069cbviunv 4956 . . . . . . . 8 𝑦𝑆 (𝐹𝑦) = 𝑥𝑆 (𝐹𝑥)
7168, 70sseqtrdi 4014 . . . . . . 7 (𝑆 ⊆ On → 𝑥 𝑆(𝐹𝑥) ⊆ 𝑥𝑆 (𝐹𝑥))
72713ad2ant2 1126 . . . . . 6 ((𝑆𝑇𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → 𝑥 𝑆(𝐹𝑥) ⊆ 𝑥𝑆 (𝐹𝑥))
7372adantr 481 . . . . 5 (((𝑆𝑇𝑆 ⊆ On ∧ 𝑆 ≠ ∅) ∧ ¬ 𝑆𝑆) → 𝑥 𝑆(𝐹𝑥) ⊆ 𝑥𝑆 (𝐹𝑥))
7446, 73eqsstrd 4002 . . . 4 (((𝑆𝑇𝑆 ⊆ On ∧ 𝑆 ≠ ∅) ∧ ¬ 𝑆𝑆) → (𝐹 𝑆) ⊆ 𝑥𝑆 (𝐹𝑥))
7574ex 413 . . 3 ((𝑆𝑇𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → (¬ 𝑆𝑆 → (𝐹 𝑆) ⊆ 𝑥𝑆 (𝐹𝑥)))
76 fveq2 6663 . . . 4 (𝑥 = 𝑆 → (𝐹𝑥) = (𝐹 𝑆))
7776ssiun2s 4963 . . 3 ( 𝑆𝑆 → (𝐹 𝑆) ⊆ 𝑥𝑆 (𝐹𝑥))
7875, 77pm2.61d2 182 . 2 ((𝑆𝑇𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → (𝐹 𝑆) ⊆ 𝑥𝑆 (𝐹𝑥))
7934imp 407 . . . . . 6 ((𝑆𝑇𝑆 ⊆ On) → 𝑆 ∈ On)
80793adant3 1124 . . . . 5 ((𝑆𝑇𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → 𝑆 ∈ On)
8163ad2ant2 1126 . . . . . 6 ((𝑆𝑇𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → (𝑥𝑆𝑥 ∈ On))
8281, 4jca2 514 . . . . 5 ((𝑆𝑇𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → (𝑥𝑆 → (𝑥 ∈ On ∧ 𝑥 𝑆)))
83 sseq2 3990 . . . . . . . 8 (𝑦 = 𝑆 → (𝑥𝑦𝑥 𝑆))
8483anbi2d 628 . . . . . . 7 (𝑦 = 𝑆 → ((𝑥 ∈ On ∧ 𝑥𝑦) ↔ (𝑥 ∈ On ∧ 𝑥 𝑆)))
8536sseq2d 3996 . . . . . . 7 (𝑦 = 𝑆 → ((𝐹𝑥) ⊆ (𝐹𝑦) ↔ (𝐹𝑥) ⊆ (𝐹 𝑆)))
8684, 85imbi12d 346 . . . . . 6 (𝑦 = 𝑆 → (((𝑥 ∈ On ∧ 𝑥𝑦) → (𝐹𝑥) ⊆ (𝐹𝑦)) ↔ ((𝑥 ∈ On ∧ 𝑥 𝑆) → (𝐹𝑥) ⊆ (𝐹 𝑆))))
87593com12 1115 . . . . . . 7 ((𝑦 ∈ On ∧ 𝑥 ∈ On ∧ 𝑥𝑦) → (𝐹𝑥) ⊆ (𝐹𝑦))
88873expib 1114 . . . . . 6 (𝑦 ∈ On → ((𝑥 ∈ On ∧ 𝑥𝑦) → (𝐹𝑥) ⊆ (𝐹𝑦)))
8986, 88vtoclga 3571 . . . . 5 ( 𝑆 ∈ On → ((𝑥 ∈ On ∧ 𝑥 𝑆) → (𝐹𝑥) ⊆ (𝐹 𝑆)))
9080, 82, 89sylsyld 61 . . . 4 ((𝑆𝑇𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → (𝑥𝑆 → (𝐹𝑥) ⊆ (𝐹 𝑆)))
9190ralrimiv 3178 . . 3 ((𝑆𝑇𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → ∀𝑥𝑆 (𝐹𝑥) ⊆ (𝐹 𝑆))
92 iunss 4960 . . 3 ( 𝑥𝑆 (𝐹𝑥) ⊆ (𝐹 𝑆) ↔ ∀𝑥𝑆 (𝐹𝑥) ⊆ (𝐹 𝑆))
9391, 92sylibr 235 . 2 ((𝑆𝑇𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → 𝑥𝑆 (𝐹𝑥) ⊆ (𝐹 𝑆))
9478, 93eqssd 3981 1 ((𝑆𝑇𝑆 ⊆ On ∧ 𝑆 ≠ ∅) → (𝐹 𝑆) = 𝑥𝑆 (𝐹𝑥))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 207  wa 396  wo 841  w3a 1079   = wceq 1528  wcel 2105  wne 3013  wral 3135  wrex 3136  wss 3933  c0 4288   cuni 4830   ciun 4910  Ord word 6183  Oncon0 6184  Lim wlim 6185  cfv 6348
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790  ax-sep 5194  ax-nul 5201  ax-pr 5320  ax-un 7450
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-3or 1080  df-3an 1081  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-mo 2615  df-eu 2647  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-ne 3014  df-ral 3140  df-rex 3141  df-rab 3144  df-v 3494  df-sbc 3770  df-dif 3936  df-un 3938  df-in 3940  df-ss 3949  df-pss 3951  df-nul 4289  df-if 4464  df-sn 4558  df-pr 4560  df-tp 4562  df-op 4564  df-uni 4831  df-iun 4912  df-br 5058  df-opab 5120  df-tr 5164  df-eprel 5458  df-po 5467  df-so 5468  df-fr 5507  df-we 5509  df-ord 6187  df-on 6188  df-lim 6189  df-iota 6307  df-fv 6356
This theorem is referenced by:  onovuni  7968
  Copyright terms: Public domain W3C validator