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Mirrors > Home > MPE Home > Th. List > Mathboxes > frrlem9 | Structured version Visualization version GIF version |
Description: Lemma for founded recursion. Show that the founded recursive generator produces a function. Hypothesis three will be eliminated using different induction rules depending on if we use partial ordering or 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 4842 | . . . . . . . 8 ⊢ (〈𝑥, 𝑢〉 ∈ ∪ 𝐵 ↔ ∃𝑔 ∈ 𝐵 〈𝑥, 𝑢〉 ∈ 𝑔) | |
2 | df-br 5067 | . . . . . . . . 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 33127 | . . . . . . . . . 10 ⊢ 𝐹 = ∪ 𝐵 |
6 | 5 | eleq2i 2904 | . . . . . . . . 9 ⊢ (〈𝑥, 𝑢〉 ∈ 𝐹 ↔ 〈𝑥, 𝑢〉 ∈ ∪ 𝐵) |
7 | 2, 6 | bitri 277 | . . . . . . . 8 ⊢ (𝑥𝐹𝑢 ↔ 〈𝑥, 𝑢〉 ∈ ∪ 𝐵) |
8 | df-br 5067 | . . . . . . . . 9 ⊢ (𝑥𝑔𝑢 ↔ 〈𝑥, 𝑢〉 ∈ 𝑔) | |
9 | 8 | rexbii 3247 | . . . . . . . 8 ⊢ (∃𝑔 ∈ 𝐵 𝑥𝑔𝑢 ↔ ∃𝑔 ∈ 𝐵 〈𝑥, 𝑢〉 ∈ 𝑔) |
10 | 1, 7, 9 | 3bitr4i 305 | . . . . . . 7 ⊢ (𝑥𝐹𝑢 ↔ ∃𝑔 ∈ 𝐵 𝑥𝑔𝑢) |
11 | eluni2 4842 | . . . . . . . 8 ⊢ (〈𝑥, 𝑣〉 ∈ ∪ 𝐵 ↔ ∃ℎ ∈ 𝐵 〈𝑥, 𝑣〉 ∈ ℎ) | |
12 | df-br 5067 | . . . . . . . . 9 ⊢ (𝑥𝐹𝑣 ↔ 〈𝑥, 𝑣〉 ∈ 𝐹) | |
13 | 5 | eleq2i 2904 | . . . . . . . . 9 ⊢ (〈𝑥, 𝑣〉 ∈ 𝐹 ↔ 〈𝑥, 𝑣〉 ∈ ∪ 𝐵) |
14 | 12, 13 | bitri 277 | . . . . . . . 8 ⊢ (𝑥𝐹𝑣 ↔ 〈𝑥, 𝑣〉 ∈ ∪ 𝐵) |
15 | df-br 5067 | . . . . . . . . 9 ⊢ (𝑥ℎ𝑣 ↔ 〈𝑥, 𝑣〉 ∈ ℎ) | |
16 | 15 | rexbii 3247 | . . . . . . . 8 ⊢ (∃ℎ ∈ 𝐵 𝑥ℎ𝑣 ↔ ∃ℎ ∈ 𝐵 〈𝑥, 𝑣〉 ∈ ℎ) |
17 | 11, 14, 16 | 3bitr4i 305 | . . . . . . 7 ⊢ (𝑥𝐹𝑣 ↔ ∃ℎ ∈ 𝐵 𝑥ℎ𝑣) |
18 | 10, 17 | anbi12i 628 | . . . . . 6 ⊢ ((𝑥𝐹𝑢 ∧ 𝑥𝐹𝑣) ↔ (∃𝑔 ∈ 𝐵 𝑥𝑔𝑢 ∧ ∃ℎ ∈ 𝐵 𝑥ℎ𝑣)) |
19 | reeanv 3367 | . . . . . 6 ⊢ (∃𝑔 ∈ 𝐵 ∃ℎ ∈ 𝐵 (𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣) ↔ (∃𝑔 ∈ 𝐵 𝑥𝑔𝑢 ∧ ∃ℎ ∈ 𝐵 𝑥ℎ𝑣)) | |
20 | 18, 19 | bitr4i 280 | . . . . 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 3294 | . . . . 5 ⊢ (𝜑 → (∃𝑔 ∈ 𝐵 ∃ℎ ∈ 𝐵 (𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣) → 𝑢 = 𝑣)) |
23 | 20, 22 | syl5bi 244 | . . . 4 ⊢ (𝜑 → ((𝑥𝐹𝑢 ∧ 𝑥𝐹𝑣) → 𝑢 = 𝑣)) |
24 | 23 | alrimiv 1928 | . . 3 ⊢ (𝜑 → ∀𝑣((𝑥𝐹𝑢 ∧ 𝑥𝐹𝑣) → 𝑢 = 𝑣)) |
25 | 24 | alrimivv 1929 | . 2 ⊢ (𝜑 → ∀𝑥∀𝑢∀𝑣((𝑥𝐹𝑢 ∧ 𝑥𝐹𝑣) → 𝑢 = 𝑣)) |
26 | 3, 4 | frrlem6 33128 | . . 3 ⊢ Rel 𝐹 |
27 | dffun2 6365 | . . 3 ⊢ (Fun 𝐹 ↔ (Rel 𝐹 ∧ ∀𝑥∀𝑢∀𝑣((𝑥𝐹𝑢 ∧ 𝑥𝐹𝑣) → 𝑢 = 𝑣))) | |
28 | 26, 27 | mpbiran 707 | . 2 ⊢ (Fun 𝐹 ↔ ∀𝑥∀𝑢∀𝑣((𝑥𝐹𝑢 ∧ 𝑥𝐹𝑣) → 𝑢 = 𝑣)) |
29 | 25, 28 | sylibr 236 | 1 ⊢ (𝜑 → Fun 𝐹) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∧ w3a 1083 ∀wal 1535 = wceq 1537 ∃wex 1780 ∈ wcel 2114 {cab 2799 ∀wral 3138 ∃wrex 3139 ⊆ wss 3936 〈cop 4573 ∪ cuni 4838 class class class wbr 5066 ↾ cres 5557 Rel wrel 5560 Predcpred 6147 Fun wfun 6349 Fn wfn 6350 ‘cfv 6355 (class class class)co 7156 frecscfrecs 33117 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 ax-sep 5203 ax-nul 5210 ax-pr 5330 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-mo 2622 df-eu 2654 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rex 3144 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-uni 4839 df-iun 4921 df-br 5067 df-opab 5129 df-id 5460 df-xp 5561 df-rel 5562 df-cnv 5563 df-co 5564 df-dm 5565 df-rn 5566 df-res 5567 df-ima 5568 df-pred 6148 df-iota 6314 df-fun 6357 df-fn 6358 df-fv 6363 df-ov 7159 df-frecs 33118 |
This theorem is referenced by: frrlem10 33132 frrlem11 33133 frrlem12 33134 frrlem13 33135 fpr1 33139 frr1 33144 |
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