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Mirrors > Home > MPE Home > Th. List > Mathboxes > frrlem11 | Structured version Visualization version GIF version |
Description: Lemma for 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 ordering or Infinity. To begin with, we establish the functionhood 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 32622 | . . . . . 6 ⊢ (𝜑 → Fun 𝐹) |
5 | funres 6228 | . . . . . 6 ⊢ (Fun 𝐹 → Fun (𝐹 ↾ 𝑆)) | |
6 | 4, 5 | syl 17 | . . . . 5 ⊢ (𝜑 → Fun (𝐹 ↾ 𝑆)) |
7 | dmres 5718 | . . . . . 6 ⊢ dom (𝐹 ↾ 𝑆) = (𝑆 ∩ dom 𝐹) | |
8 | df-fn 6189 | . . . . . 6 ⊢ ((𝐹 ↾ 𝑆) Fn (𝑆 ∩ dom 𝐹) ↔ (Fun (𝐹 ↾ 𝑆) ∧ dom (𝐹 ↾ 𝑆) = (𝑆 ∩ dom 𝐹))) | |
9 | 7, 8 | mpbiran2 697 | . . . . 5 ⊢ ((𝐹 ↾ 𝑆) Fn (𝑆 ∩ dom 𝐹) ↔ Fun (𝐹 ↾ 𝑆)) |
10 | 6, 9 | sylibr 226 | . . . 4 ⊢ (𝜑 → (𝐹 ↾ 𝑆) Fn (𝑆 ∩ dom 𝐹)) |
11 | vex 3415 | . . . . 5 ⊢ 𝑧 ∈ V | |
12 | ovex 7006 | . . . . 5 ⊢ (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))) ∈ V | |
13 | 11, 12 | fnsn 6243 | . . . 4 ⊢ {〈𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))〉} Fn {𝑧} |
14 | 10, 13 | jctir 513 | . . 3 ⊢ (𝜑 → ((𝐹 ↾ 𝑆) Fn (𝑆 ∩ dom 𝐹) ∧ {〈𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))〉} Fn {𝑧})) |
15 | eldifn 3993 | . . . . 5 ⊢ (𝑧 ∈ (𝐴 ∖ dom 𝐹) → ¬ 𝑧 ∈ dom 𝐹) | |
16 | elinel2 4060 | . . . . 5 ⊢ (𝑧 ∈ (𝑆 ∩ dom 𝐹) → 𝑧 ∈ dom 𝐹) | |
17 | 15, 16 | nsyl 138 | . . . 4 ⊢ (𝑧 ∈ (𝐴 ∖ dom 𝐹) → ¬ 𝑧 ∈ (𝑆 ∩ dom 𝐹)) |
18 | disjsn 4519 | . . . 4 ⊢ (((𝑆 ∩ dom 𝐹) ∩ {𝑧}) = ∅ ↔ ¬ 𝑧 ∈ (𝑆 ∩ dom 𝐹)) | |
19 | 17, 18 | sylibr 226 | . . 3 ⊢ (𝑧 ∈ (𝐴 ∖ dom 𝐹) → ((𝑆 ∩ dom 𝐹) ∩ {𝑧}) = ∅) |
20 | fnun 6294 | . . 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})) | |
21 | 14, 19, 20 | syl2an 586 | . 2 ⊢ ((𝜑 ∧ 𝑧 ∈ (𝐴 ∖ dom 𝐹)) → ((𝐹 ↾ 𝑆) ∪ {〈𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))〉}) Fn ((𝑆 ∩ dom 𝐹) ∪ {𝑧})) |
22 | frrlem11.4 | . . 3 ⊢ 𝐶 = ((𝐹 ↾ 𝑆) ∪ {〈𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))〉}) | |
23 | 22 | fneq1i 6281 | . 2 ⊢ (𝐶 Fn ((𝑆 ∩ dom 𝐹) ∪ {𝑧}) ↔ ((𝐹 ↾ 𝑆) ∪ {〈𝑧, (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))〉}) Fn ((𝑆 ∩ dom 𝐹) ∪ {𝑧})) |
24 | 21, 23 | sylibr 226 | 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 387 ∧ w3a 1068 = wceq 1507 ∃wex 1742 ∈ wcel 2048 {cab 2755 ∀wral 3085 ∖ cdif 3825 ∪ cun 3826 ∩ cin 3827 ⊆ wss 3828 ∅c0 4177 {csn 4439 〈cop 4445 class class class wbr 4927 dom cdm 5404 ↾ cres 5406 Predcpred 5983 Fun wfun 6180 Fn wfn 6181 ‘cfv 6186 (class class class)co 6974 frecscfrecs 32608 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1964 ax-8 2050 ax-9 2057 ax-10 2077 ax-11 2091 ax-12 2104 ax-13 2299 ax-ext 2747 ax-sep 5058 ax-nul 5065 ax-pr 5184 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2014 df-mo 2544 df-eu 2580 df-clab 2756 df-cleq 2768 df-clel 2843 df-nfc 2915 df-ral 3090 df-rex 3091 df-rab 3094 df-v 3414 df-sbc 3681 df-dif 3831 df-un 3833 df-in 3835 df-ss 3842 df-nul 4178 df-if 4349 df-sn 4440 df-pr 4442 df-op 4446 df-uni 4711 df-iun 4792 df-br 4928 df-opab 4990 df-id 5309 df-xp 5410 df-rel 5411 df-cnv 5412 df-co 5413 df-dm 5414 df-rn 5415 df-res 5416 df-ima 5417 df-pred 5984 df-iota 6150 df-fun 6188 df-fn 6189 df-fv 6194 df-ov 6977 df-frecs 32609 |
This theorem is referenced by: frrlem12 32625 frrlem13 32626 |
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