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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 8318 | . . . . . 6 ⊢ (𝜑 → Fun 𝐹) |
5 | 4 | funresd 6611 | . . . . 5 ⊢ (𝜑 → Fun (𝐹 ↾ 𝑆)) |
6 | dmres 6032 | . . . . . 6 ⊢ dom (𝐹 ↾ 𝑆) = (𝑆 ∩ dom 𝐹) | |
7 | df-fn 6566 | . . . . . 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 3482 | . . . . 5 ⊢ 𝑧 ∈ V | |
11 | ovex 7464 | . . . . 5 ⊢ (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))) ∈ V | |
12 | 10, 11 | fnsn 6626 | . . . 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 4142 | . . . . 5 ⊢ (𝑧 ∈ (𝐴 ∖ dom 𝐹) → ¬ 𝑧 ∈ dom 𝐹) | |
15 | elinel2 4212 | . . . . 5 ⊢ (𝑧 ∈ (𝑆 ∩ dom 𝐹) → 𝑧 ∈ dom 𝐹) | |
16 | 14, 15 | nsyl 140 | . . . 4 ⊢ (𝑧 ∈ (𝐴 ∖ dom 𝐹) → ¬ 𝑧 ∈ (𝑆 ∩ dom 𝐹)) |
17 | disjsn 4716 | . . . 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 6683 | . . 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 6666 | . 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 1086 = wceq 1537 ∃wex 1776 ∈ wcel 2106 {cab 2712 ∀wral 3059 ∖ cdif 3960 ∪ cun 3961 ∩ cin 3962 ⊆ wss 3963 ∅c0 4339 {csn 4631 〈cop 4637 class class class wbr 5148 dom cdm 5689 ↾ cres 5691 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-mo 2538 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 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: frrlem12 8321 frrlem13 8322 |
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