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Mirrors > Home > MPE Home > Th. List > Mathboxes > frrlem5c | Structured version Visualization version GIF version |
Description: Lemma for founded recursion. The union of a subclass of 𝐵 is a function. (Contributed by Paul Chapman, 29-Apr-2012.) |
Ref | Expression |
---|---|
frrlem5.1 | ⊢ 𝑅 Fr 𝐴 |
frrlem5.2 | ⊢ 𝑅 Se 𝐴 |
frrlem5.3 | ⊢ 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} |
Ref | Expression |
---|---|
frrlem5c | ⊢ (𝐶 ⊆ 𝐵 → Fun ∪ 𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uniss 4595 | . 2 ⊢ (𝐶 ⊆ 𝐵 → ∪ 𝐶 ⊆ ∪ 𝐵) | |
2 | ssid 3773 | . . . 4 ⊢ 𝐵 ⊆ 𝐵 | |
3 | frrlem5.1 | . . . . 5 ⊢ 𝑅 Fr 𝐴 | |
4 | frrlem5.2 | . . . . 5 ⊢ 𝑅 Se 𝐴 | |
5 | frrlem5.3 | . . . . 5 ⊢ 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))} | |
6 | 3, 4, 5 | frrlem5b 32122 | . . . 4 ⊢ (𝐵 ⊆ 𝐵 → Rel ∪ 𝐵) |
7 | 2, 6 | ax-mp 5 | . . 3 ⊢ Rel ∪ 𝐵 |
8 | eluni 4577 | . . . . . . . . 9 ⊢ (〈𝑥, 𝑢〉 ∈ ∪ 𝐵 ↔ ∃𝑔(〈𝑥, 𝑢〉 ∈ 𝑔 ∧ 𝑔 ∈ 𝐵)) | |
9 | df-br 4787 | . . . . . . . . 9 ⊢ (𝑥∪ 𝐵𝑢 ↔ 〈𝑥, 𝑢〉 ∈ ∪ 𝐵) | |
10 | df-br 4787 | . . . . . . . . . . 11 ⊢ (𝑥𝑔𝑢 ↔ 〈𝑥, 𝑢〉 ∈ 𝑔) | |
11 | 10 | anbi1i 610 | . . . . . . . . . 10 ⊢ ((𝑥𝑔𝑢 ∧ 𝑔 ∈ 𝐵) ↔ (〈𝑥, 𝑢〉 ∈ 𝑔 ∧ 𝑔 ∈ 𝐵)) |
12 | 11 | exbii 1924 | . . . . . . . . 9 ⊢ (∃𝑔(𝑥𝑔𝑢 ∧ 𝑔 ∈ 𝐵) ↔ ∃𝑔(〈𝑥, 𝑢〉 ∈ 𝑔 ∧ 𝑔 ∈ 𝐵)) |
13 | 8, 9, 12 | 3bitr4i 292 | . . . . . . . 8 ⊢ (𝑥∪ 𝐵𝑢 ↔ ∃𝑔(𝑥𝑔𝑢 ∧ 𝑔 ∈ 𝐵)) |
14 | eluni 4577 | . . . . . . . . 9 ⊢ (〈𝑥, 𝑣〉 ∈ ∪ 𝐵 ↔ ∃ℎ(〈𝑥, 𝑣〉 ∈ ℎ ∧ ℎ ∈ 𝐵)) | |
15 | df-br 4787 | . . . . . . . . 9 ⊢ (𝑥∪ 𝐵𝑣 ↔ 〈𝑥, 𝑣〉 ∈ ∪ 𝐵) | |
16 | df-br 4787 | . . . . . . . . . . 11 ⊢ (𝑥ℎ𝑣 ↔ 〈𝑥, 𝑣〉 ∈ ℎ) | |
17 | 16 | anbi1i 610 | . . . . . . . . . 10 ⊢ ((𝑥ℎ𝑣 ∧ ℎ ∈ 𝐵) ↔ (〈𝑥, 𝑣〉 ∈ ℎ ∧ ℎ ∈ 𝐵)) |
18 | 17 | exbii 1924 | . . . . . . . . 9 ⊢ (∃ℎ(𝑥ℎ𝑣 ∧ ℎ ∈ 𝐵) ↔ ∃ℎ(〈𝑥, 𝑣〉 ∈ ℎ ∧ ℎ ∈ 𝐵)) |
19 | 14, 15, 18 | 3bitr4i 292 | . . . . . . . 8 ⊢ (𝑥∪ 𝐵𝑣 ↔ ∃ℎ(𝑥ℎ𝑣 ∧ ℎ ∈ 𝐵)) |
20 | 13, 19 | anbi12i 612 | . . . . . . 7 ⊢ ((𝑥∪ 𝐵𝑢 ∧ 𝑥∪ 𝐵𝑣) ↔ (∃𝑔(𝑥𝑔𝑢 ∧ 𝑔 ∈ 𝐵) ∧ ∃ℎ(𝑥ℎ𝑣 ∧ ℎ ∈ 𝐵))) |
21 | eeanv 2344 | . . . . . . 7 ⊢ (∃𝑔∃ℎ((𝑥𝑔𝑢 ∧ 𝑔 ∈ 𝐵) ∧ (𝑥ℎ𝑣 ∧ ℎ ∈ 𝐵)) ↔ (∃𝑔(𝑥𝑔𝑢 ∧ 𝑔 ∈ 𝐵) ∧ ∃ℎ(𝑥ℎ𝑣 ∧ ℎ ∈ 𝐵))) | |
22 | 20, 21 | bitr4i 267 | . . . . . 6 ⊢ ((𝑥∪ 𝐵𝑢 ∧ 𝑥∪ 𝐵𝑣) ↔ ∃𝑔∃ℎ((𝑥𝑔𝑢 ∧ 𝑔 ∈ 𝐵) ∧ (𝑥ℎ𝑣 ∧ ℎ ∈ 𝐵))) |
23 | 3, 4, 5 | frrlem5 32121 | . . . . . . . . 9 ⊢ ((𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) → ((𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣) → 𝑢 = 𝑣)) |
24 | 23 | impcom 394 | . . . . . . . 8 ⊢ (((𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣) ∧ (𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵)) → 𝑢 = 𝑣) |
25 | 24 | an4s 639 | . . . . . . 7 ⊢ (((𝑥𝑔𝑢 ∧ 𝑔 ∈ 𝐵) ∧ (𝑥ℎ𝑣 ∧ ℎ ∈ 𝐵)) → 𝑢 = 𝑣) |
26 | 25 | exlimivv 2012 | . . . . . 6 ⊢ (∃𝑔∃ℎ((𝑥𝑔𝑢 ∧ 𝑔 ∈ 𝐵) ∧ (𝑥ℎ𝑣 ∧ ℎ ∈ 𝐵)) → 𝑢 = 𝑣) |
27 | 22, 26 | sylbi 207 | . . . . 5 ⊢ ((𝑥∪ 𝐵𝑢 ∧ 𝑥∪ 𝐵𝑣) → 𝑢 = 𝑣) |
28 | 27 | ax-gen 1870 | . . . 4 ⊢ ∀𝑣((𝑥∪ 𝐵𝑢 ∧ 𝑥∪ 𝐵𝑣) → 𝑢 = 𝑣) |
29 | 28 | gen2 1871 | . . 3 ⊢ ∀𝑥∀𝑢∀𝑣((𝑥∪ 𝐵𝑢 ∧ 𝑥∪ 𝐵𝑣) → 𝑢 = 𝑣) |
30 | dffun2 6041 | . . 3 ⊢ (Fun ∪ 𝐵 ↔ (Rel ∪ 𝐵 ∧ ∀𝑥∀𝑢∀𝑣((𝑥∪ 𝐵𝑢 ∧ 𝑥∪ 𝐵𝑣) → 𝑢 = 𝑣))) | |
31 | 7, 29, 30 | mpbir2an 690 | . 2 ⊢ Fun ∪ 𝐵 |
32 | funss 6050 | . 2 ⊢ (∪ 𝐶 ⊆ ∪ 𝐵 → (Fun ∪ 𝐵 → Fun ∪ 𝐶)) | |
33 | 1, 31, 32 | mpisyl 21 | 1 ⊢ (𝐶 ⊆ 𝐵 → Fun ∪ 𝐶) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 382 ∧ w3a 1071 ∀wal 1629 = wceq 1631 ∃wex 1852 ∈ wcel 2145 {cab 2757 ∀wral 3061 ⊆ wss 3723 〈cop 4322 ∪ cuni 4574 class class class wbr 4786 Fr wfr 5205 Se wse 5206 ↾ cres 5251 Rel wrel 5254 Predcpred 5822 Fun wfun 6025 Fn wfn 6026 ‘cfv 6031 (class class class)co 6793 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1870 ax-4 1885 ax-5 1991 ax-6 2057 ax-7 2093 ax-8 2147 ax-9 2154 ax-10 2174 ax-11 2190 ax-12 2203 ax-13 2408 ax-ext 2751 ax-rep 4904 ax-sep 4915 ax-nul 4923 ax-pow 4974 ax-pr 5034 ax-un 7096 ax-inf2 8702 |
This theorem depends on definitions: df-bi 197 df-an 383 df-or 835 df-3or 1072 df-3an 1073 df-tru 1634 df-ex 1853 df-nf 1858 df-sb 2050 df-eu 2622 df-mo 2623 df-clab 2758 df-cleq 2764 df-clel 2767 df-nfc 2902 df-ne 2944 df-ral 3066 df-rex 3067 df-reu 3068 df-rab 3070 df-v 3353 df-sbc 3588 df-csb 3683 df-dif 3726 df-un 3728 df-in 3730 df-ss 3737 df-pss 3739 df-nul 4064 df-if 4226 df-pw 4299 df-sn 4317 df-pr 4319 df-tp 4321 df-op 4323 df-uni 4575 df-iun 4656 df-br 4787 df-opab 4847 df-mpt 4864 df-tr 4887 df-id 5157 df-eprel 5162 df-po 5170 df-so 5171 df-fr 5208 df-se 5209 df-we 5210 df-xp 5255 df-rel 5256 df-cnv 5257 df-co 5258 df-dm 5259 df-rn 5260 df-res 5261 df-ima 5262 df-pred 5823 df-ord 5869 df-on 5870 df-lim 5871 df-suc 5872 df-iota 5994 df-fun 6033 df-fn 6034 df-f 6035 df-f1 6036 df-fo 6037 df-f1o 6038 df-fv 6039 df-ov 6796 df-om 7213 df-wrecs 7559 df-recs 7621 df-rdg 7659 df-trpred 32054 |
This theorem is referenced by: frrlem10 32128 |
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