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Mirrors > Home > MPE Home > Th. List > frrlem9 | Structured version Visualization version GIF version |
Description: Lemma for well-founded recursion. Show that the well-founded recursive generator produces a function. Hypothesis three will be eliminated using different induction rules depending on if we use partial orders or the axiom of infinity. (Contributed by Scott Fenton, 27-Aug-2022.) |
Ref | Expression |
---|---|
frrlem9.1 | ⊢ 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))))} |
frrlem9.2 | ⊢ 𝐹 = frecs(𝑅, 𝐴, 𝐺) |
frrlem9.3 | ⊢ ((𝜑 ∧ (𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵)) → ((𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣) → 𝑢 = 𝑣)) |
Ref | Expression |
---|---|
frrlem9 | ⊢ (𝜑 → Fun 𝐹) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eluni2 4916 | . . . . . . . 8 ⊢ (〈𝑥, 𝑢〉 ∈ ∪ 𝐵 ↔ ∃𝑔 ∈ 𝐵 〈𝑥, 𝑢〉 ∈ 𝑔) | |
2 | df-br 5149 | . . . . . . . . 9 ⊢ (𝑥𝐹𝑢 ↔ 〈𝑥, 𝑢〉 ∈ 𝐹) | |
3 | frrlem9.1 | . . . . . . . . . . 11 ⊢ 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))))} | |
4 | frrlem9.2 | . . . . . . . . . . 11 ⊢ 𝐹 = frecs(𝑅, 𝐴, 𝐺) | |
5 | 3, 4 | frrlem5 8314 | . . . . . . . . . 10 ⊢ 𝐹 = ∪ 𝐵 |
6 | 5 | eleq2i 2831 | . . . . . . . . 9 ⊢ (〈𝑥, 𝑢〉 ∈ 𝐹 ↔ 〈𝑥, 𝑢〉 ∈ ∪ 𝐵) |
7 | 2, 6 | bitri 275 | . . . . . . . 8 ⊢ (𝑥𝐹𝑢 ↔ 〈𝑥, 𝑢〉 ∈ ∪ 𝐵) |
8 | df-br 5149 | . . . . . . . . 9 ⊢ (𝑥𝑔𝑢 ↔ 〈𝑥, 𝑢〉 ∈ 𝑔) | |
9 | 8 | rexbii 3092 | . . . . . . . 8 ⊢ (∃𝑔 ∈ 𝐵 𝑥𝑔𝑢 ↔ ∃𝑔 ∈ 𝐵 〈𝑥, 𝑢〉 ∈ 𝑔) |
10 | 1, 7, 9 | 3bitr4i 303 | . . . . . . 7 ⊢ (𝑥𝐹𝑢 ↔ ∃𝑔 ∈ 𝐵 𝑥𝑔𝑢) |
11 | eluni2 4916 | . . . . . . . 8 ⊢ (〈𝑥, 𝑣〉 ∈ ∪ 𝐵 ↔ ∃ℎ ∈ 𝐵 〈𝑥, 𝑣〉 ∈ ℎ) | |
12 | df-br 5149 | . . . . . . . . 9 ⊢ (𝑥𝐹𝑣 ↔ 〈𝑥, 𝑣〉 ∈ 𝐹) | |
13 | 5 | eleq2i 2831 | . . . . . . . . 9 ⊢ (〈𝑥, 𝑣〉 ∈ 𝐹 ↔ 〈𝑥, 𝑣〉 ∈ ∪ 𝐵) |
14 | 12, 13 | bitri 275 | . . . . . . . 8 ⊢ (𝑥𝐹𝑣 ↔ 〈𝑥, 𝑣〉 ∈ ∪ 𝐵) |
15 | df-br 5149 | . . . . . . . . 9 ⊢ (𝑥ℎ𝑣 ↔ 〈𝑥, 𝑣〉 ∈ ℎ) | |
16 | 15 | rexbii 3092 | . . . . . . . 8 ⊢ (∃ℎ ∈ 𝐵 𝑥ℎ𝑣 ↔ ∃ℎ ∈ 𝐵 〈𝑥, 𝑣〉 ∈ ℎ) |
17 | 11, 14, 16 | 3bitr4i 303 | . . . . . . 7 ⊢ (𝑥𝐹𝑣 ↔ ∃ℎ ∈ 𝐵 𝑥ℎ𝑣) |
18 | 10, 17 | anbi12i 628 | . . . . . 6 ⊢ ((𝑥𝐹𝑢 ∧ 𝑥𝐹𝑣) ↔ (∃𝑔 ∈ 𝐵 𝑥𝑔𝑢 ∧ ∃ℎ ∈ 𝐵 𝑥ℎ𝑣)) |
19 | reeanv 3227 | . . . . . 6 ⊢ (∃𝑔 ∈ 𝐵 ∃ℎ ∈ 𝐵 (𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣) ↔ (∃𝑔 ∈ 𝐵 𝑥𝑔𝑢 ∧ ∃ℎ ∈ 𝐵 𝑥ℎ𝑣)) | |
20 | 18, 19 | bitr4i 278 | . . . . 5 ⊢ ((𝑥𝐹𝑢 ∧ 𝑥𝐹𝑣) ↔ ∃𝑔 ∈ 𝐵 ∃ℎ ∈ 𝐵 (𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣)) |
21 | frrlem9.3 | . . . . . 6 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵)) → ((𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣) → 𝑢 = 𝑣)) | |
22 | 21 | rexlimdvva 3211 | . . . . 5 ⊢ (𝜑 → (∃𝑔 ∈ 𝐵 ∃ℎ ∈ 𝐵 (𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣) → 𝑢 = 𝑣)) |
23 | 20, 22 | biimtrid 242 | . . . 4 ⊢ (𝜑 → ((𝑥𝐹𝑢 ∧ 𝑥𝐹𝑣) → 𝑢 = 𝑣)) |
24 | 23 | alrimiv 1925 | . . 3 ⊢ (𝜑 → ∀𝑣((𝑥𝐹𝑢 ∧ 𝑥𝐹𝑣) → 𝑢 = 𝑣)) |
25 | 24 | alrimivv 1926 | . 2 ⊢ (𝜑 → ∀𝑥∀𝑢∀𝑣((𝑥𝐹𝑢 ∧ 𝑥𝐹𝑣) → 𝑢 = 𝑣)) |
26 | 3, 4 | frrlem6 8315 | . . 3 ⊢ Rel 𝐹 |
27 | dffun2 6573 | . . 3 ⊢ (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥∀𝑢∀𝑣((𝑥𝐹𝑢 ∧ 𝑥𝐹𝑣) → 𝑢 = 𝑣))) | |
28 | 26, 27 | mpbiran 709 | . 2 ⊢ (Fun 𝐹 ↔ ∀𝑥∀𝑢∀𝑣((𝑥𝐹𝑢 ∧ 𝑥𝐹𝑣) → 𝑢 = 𝑣)) |
29 | 25, 28 | sylibr 234 | 1 ⊢ (𝜑 → Fun 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 ∀wal 1535 = wceq 1537 ∃wex 1776 ∈ wcel 2106 {cab 2712 ∀wral 3059 ∃wrex 3068 ⊆ wss 3963 〈cop 4637 ∪ cuni 4912 class class class wbr 5148 ↾ cres 5691 Rel wrel 5694 Predcpred 6322 Fun wfun 6557 Fn wfn 6558 ‘cfv 6563 (class class class)co 7431 frecscfrecs 8304 |
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-sep 5302 ax-nul 5312 ax-pr 5438 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ral 3060 df-rex 3069 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-iun 4998 df-br 5149 df-opab 5211 df-id 5583 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-pred 6323 df-iota 6516 df-fun 6565 df-fn 6566 df-fv 6571 df-ov 7434 df-frecs 8305 |
This theorem is referenced by: frrlem10 8319 frrlem11 8320 frrlem12 8321 frrlem13 8322 fpr1 8327 fprfung 8333 frr1 9797 |
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