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| Mirrors > Home > MPE Home > Th. List > frrlem11 | Structured version Visualization version GIF version | ||
| Description: Lemma for well-founded recursion. For the next several theorems we will be aiming to prove that dom 𝐹 = 𝐴. To do this, we set up a function 𝐶 that supposedly contains an element of 𝐴 that is not in dom 𝐹 and we show that the element must be in dom 𝐹. Our choice of what to restrict 𝐹 to depends on if we assume partial orders or the axiom of infinity. To begin with, we establish the functionality of 𝐶. (Contributed by Scott Fenton, 7-Dec-2022.) |
| Ref | Expression |
|---|---|
| frrlem11.1 | ⊢ 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))))} |
| frrlem11.2 | ⊢ 𝐹 = frecs(𝑅, 𝐴, 𝐺) |
| frrlem11.3 | ⊢ ((𝜑 ∧ (𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵)) → ((𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣) → 𝑢 = 𝑣)) |
| frrlem11.4 | ⊢ 𝐶 = ((𝐹 ↾ 𝑆) ∪ {〈𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))〉}) |
| Ref | Expression |
|---|---|
| frrlem11 | ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴 ∖ dom 𝐹)) → 𝐶 Fn ((𝑆 ∩ dom 𝐹) ∪ {𝑧})) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | frrlem11.1 | . . . . . . 7 ⊢ 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))))} | |
| 2 | frrlem11.2 | . . . . . . 7 ⊢ 𝐹 = frecs(𝑅, 𝐴, 𝐺) | |
| 3 | frrlem11.3 | . . . . . . 7 ⊢ ((𝜑 ∧ (𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵)) → ((𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣) → 𝑢 = 𝑣)) | |
| 4 | 1, 2, 3 | frrlem9 8319 | . . . . . 6 ⊢ (𝜑 → Fun 𝐹) |
| 5 | 4 | funresd 6609 | . . . . 5 ⊢ (𝜑 → Fun (𝐹 ↾ 𝑆)) |
| 6 | dmres 6030 | . . . . . 6 ⊢ dom (𝐹 ↾ 𝑆) = (𝑆 ∩ dom 𝐹) | |
| 7 | df-fn 6564 | . . . . . 6 ⊢ ((𝐹 ↾ 𝑆) Fn (𝑆 ∩ dom 𝐹) ↔ (Fun (𝐹 ↾ 𝑆) ∧ dom (𝐹 ↾ 𝑆) = (𝑆 ∩ dom 𝐹))) | |
| 8 | 6, 7 | mpbiran2 710 | . . . . 5 ⊢ ((𝐹 ↾ 𝑆) Fn (𝑆 ∩ dom 𝐹) ↔ Fun (𝐹 ↾ 𝑆)) |
| 9 | 5, 8 | sylibr 234 | . . . 4 ⊢ (𝜑 → (𝐹 ↾ 𝑆) Fn (𝑆 ∩ dom 𝐹)) |
| 10 | vex 3484 | . . . . 5 ⊢ 𝑧 ∈ V | |
| 11 | ovex 7464 | . . . . 5 ⊢ (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))) ∈ V | |
| 12 | 10, 11 | fnsn 6624 | . . . 4 ⊢ {〈𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))〉} Fn {𝑧} |
| 13 | 9, 12 | jctir 520 | . . 3 ⊢ (𝜑 → ((𝐹 ↾ 𝑆) Fn (𝑆 ∩ dom 𝐹) ∧ {〈𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))〉} Fn {𝑧})) |
| 14 | eldifn 4132 | . . . . 5 ⊢ (𝑧 ∈ (𝐴 ∖ dom 𝐹) → ¬ 𝑧 ∈ dom 𝐹) | |
| 15 | elinel2 4202 | . . . . 5 ⊢ (𝑧 ∈ (𝑆 ∩ dom 𝐹) → 𝑧 ∈ dom 𝐹) | |
| 16 | 14, 15 | nsyl 140 | . . . 4 ⊢ (𝑧 ∈ (𝐴 ∖ dom 𝐹) → ¬ 𝑧 ∈ (𝑆 ∩ dom 𝐹)) |
| 17 | disjsn 4711 | . . . 4 ⊢ (((𝑆 ∩ dom 𝐹) ∩ {𝑧}) = ∅ ↔ ¬ 𝑧 ∈ (𝑆 ∩ dom 𝐹)) | |
| 18 | 16, 17 | sylibr 234 | . . 3 ⊢ (𝑧 ∈ (𝐴 ∖ dom 𝐹) → ((𝑆 ∩ dom 𝐹) ∩ {𝑧}) = ∅) |
| 19 | fnun 6682 | . . 3 ⊢ ((((𝐹 ↾ 𝑆) Fn (𝑆 ∩ dom 𝐹) ∧ {〈𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))〉} Fn {𝑧}) ∧ ((𝑆 ∩ dom 𝐹) ∩ {𝑧}) = ∅) → ((𝐹 ↾ 𝑆) ∪ {〈𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))〉}) Fn ((𝑆 ∩ dom 𝐹) ∪ {𝑧})) | |
| 20 | 13, 18, 19 | syl2an 596 | . 2 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴 ∖ dom 𝐹)) → ((𝐹 ↾ 𝑆) ∪ {〈𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))〉}) Fn ((𝑆 ∩ dom 𝐹) ∪ {𝑧})) |
| 21 | frrlem11.4 | . . 3 ⊢ 𝐶 = ((𝐹 ↾ 𝑆) ∪ {〈𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))〉}) | |
| 22 | 21 | fneq1i 6665 | . 2 ⊢ (𝐶 Fn ((𝑆 ∩ dom 𝐹) ∪ {𝑧}) ↔ ((𝐹 ↾ 𝑆) ∪ {〈𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))〉}) Fn ((𝑆 ∩ dom 𝐹) ∪ {𝑧})) |
| 23 | 20, 22 | sylibr 234 | 1 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴 ∖ dom 𝐹)) → 𝐶 Fn ((𝑆 ∩ dom 𝐹) ∪ {𝑧})) |
| Colors of variables: wff setvar class |
| Syntax hints: ¬ wn 3 → wi 4 ∧ wa 395 ∧ w3a 1087 = wceq 1540 ∃wex 1779 ∈ wcel 2108 {cab 2714 ∀wral 3061 ∖ cdif 3948 ∪ cun 3949 ∩ cin 3950 ⊆ wss 3951 ∅c0 4333 {csn 4626 〈cop 4632 class class class wbr 5143 dom cdm 5685 ↾ cres 5687 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-mo 2540 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 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: frrlem12 8322 frrlem13 8323 |
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