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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 4911 | . . . . . . . 8 ⊢ (〈𝑥, 𝑢〉 ∈ ∪ 𝐵 ↔ ∃𝑔 ∈ 𝐵 〈𝑥, 𝑢〉 ∈ 𝑔) | |
| 2 | df-br 5144 | . . . . . . . . 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 8315 | . . . . . . . . . 10 ⊢ 𝐹 = ∪ 𝐵 |
| 6 | 5 | eleq2i 2833 | . . . . . . . . 9 ⊢ (〈𝑥, 𝑢〉 ∈ 𝐹 ↔ 〈𝑥, 𝑢〉 ∈ ∪ 𝐵) |
| 7 | 2, 6 | bitri 275 | . . . . . . . 8 ⊢ (𝑥𝐹𝑢 ↔ 〈𝑥, 𝑢〉 ∈ ∪ 𝐵) |
| 8 | df-br 5144 | . . . . . . . . 9 ⊢ (𝑥𝑔𝑢 ↔ 〈𝑥, 𝑢〉 ∈ 𝑔) | |
| 9 | 8 | rexbii 3094 | . . . . . . . 8 ⊢ (∃𝑔 ∈ 𝐵 𝑥𝑔𝑢 ↔ ∃𝑔 ∈ 𝐵 〈𝑥, 𝑢〉 ∈ 𝑔) |
| 10 | 1, 7, 9 | 3bitr4i 303 | . . . . . . 7 ⊢ (𝑥𝐹𝑢 ↔ ∃𝑔 ∈ 𝐵 𝑥𝑔𝑢) |
| 11 | eluni2 4911 | . . . . . . . 8 ⊢ (〈𝑥, 𝑣〉 ∈ ∪ 𝐵 ↔ ∃ℎ ∈ 𝐵 〈𝑥, 𝑣〉 ∈ ℎ) | |
| 12 | df-br 5144 | . . . . . . . . 9 ⊢ (𝑥𝐹𝑣 ↔ 〈𝑥, 𝑣〉 ∈ 𝐹) | |
| 13 | 5 | eleq2i 2833 | . . . . . . . . 9 ⊢ (〈𝑥, 𝑣〉 ∈ 𝐹 ↔ 〈𝑥, 𝑣〉 ∈ ∪ 𝐵) |
| 14 | 12, 13 | bitri 275 | . . . . . . . 8 ⊢ (𝑥𝐹𝑣 ↔ 〈𝑥, 𝑣〉 ∈ ∪ 𝐵) |
| 15 | df-br 5144 | . . . . . . . . 9 ⊢ (𝑥ℎ𝑣 ↔ 〈𝑥, 𝑣〉 ∈ ℎ) | |
| 16 | 15 | rexbii 3094 | . . . . . . . 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 3229 | . . . . . 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 3213 | . . . . 5 ⊢ (𝜑 → (∃𝑔 ∈ 𝐵 ∃ℎ ∈ 𝐵 (𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣) → 𝑢 = 𝑣)) |
| 23 | 20, 22 | biimtrid 242 | . . . 4 ⊢ (𝜑 → ((𝑥𝐹𝑢 ∧ 𝑥𝐹𝑣) → 𝑢 = 𝑣)) |
| 24 | 23 | alrimiv 1927 | . . 3 ⊢ (𝜑 → ∀𝑣((𝑥𝐹𝑢 ∧ 𝑥𝐹𝑣) → 𝑢 = 𝑣)) |
| 25 | 24 | alrimivv 1928 | . 2 ⊢ (𝜑 → ∀𝑥∀𝑢∀𝑣((𝑥𝐹𝑢 ∧ 𝑥𝐹𝑣) → 𝑢 = 𝑣)) |
| 26 | 3, 4 | frrlem6 8316 | . . 3 ⊢ Rel 𝐹 |
| 27 | dffun2 6571 | . . 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 1087 ∀wal 1538 = wceq 1540 ∃wex 1779 ∈ wcel 2108 {cab 2714 ∀wral 3061 ∃wrex 3070 ⊆ wss 3951 〈cop 4632 ∪ cuni 4907 class class class wbr 5143 ↾ cres 5687 Rel wrel 5690 Predcpred 6320 Fun wfun 6555 Fn wfn 6556 ‘cfv 6561 (class class class)co 7431 frecscfrecs 8305 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-iun 4993 df-br 5144 df-opab 5206 df-id 5578 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-pred 6321 df-iota 6514 df-fun 6563 df-fn 6564 df-fv 6569 df-ov 7434 df-frecs 8306 |
| This theorem is referenced by: frrlem10 8320 frrlem11 8321 frrlem12 8322 frrlem13 8323 fpr1 8328 fprfung 8334 frr1 9799 |
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