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Mirrors > Home > MPE Home > Th. List > fpr3 | Structured version Visualization version GIF version |
Description: Law of well-founded recursion over a partial order, part three. Finally, we show that 𝐹 is unique. We do this by showing that any function 𝐻 with the same properties we proved of 𝐹 in fpr1 8119 and fpr2 8120 is identical to 𝐹. (Contributed by Scott Fenton, 11-Sep-2023.) |
Ref | Expression |
---|---|
fprr.1 | ⊢ 𝐹 = frecs(𝑅, 𝐴, 𝐺) |
Ref | Expression |
---|---|
fpr3 | ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐻 Fn 𝐴 ∧ ∀𝑧 ∈ 𝐴 (𝐻‘𝑧) = (𝑧𝐺(𝐻 ↾ Pred(𝑅, 𝐴, 𝑧))))) → 𝐹 = 𝐻) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simpl 483 | . 2 ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐻 Fn 𝐴 ∧ ∀𝑧 ∈ 𝐴 (𝐻‘𝑧) = (𝑧𝐺(𝐻 ↾ Pred(𝑅, 𝐴, 𝑧))))) → (𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴)) | |
2 | fprr.1 | . . . . 5 ⊢ 𝐹 = frecs(𝑅, 𝐴, 𝐺) | |
3 | 2 | fpr1 8119 | . . . 4 ⊢ ((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) → 𝐹 Fn 𝐴) |
4 | 2 | fpr2 8120 | . . . . 5 ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) = (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))) |
5 | 4 | ralrimiva 3103 | . . . 4 ⊢ ((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) → ∀𝑧 ∈ 𝐴 (𝐹‘𝑧) = (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))) |
6 | 3, 5 | jca 512 | . . 3 ⊢ ((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) → (𝐹 Fn 𝐴 ∧ ∀𝑧 ∈ 𝐴 (𝐹‘𝑧) = (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))))) |
7 | 6 | adantr 481 | . 2 ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐻 Fn 𝐴 ∧ ∀𝑧 ∈ 𝐴 (𝐻‘𝑧) = (𝑧𝐺(𝐻 ↾ Pred(𝑅, 𝐴, 𝑧))))) → (𝐹 Fn 𝐴 ∧ ∀𝑧 ∈ 𝐴 (𝐹‘𝑧) = (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))))) |
8 | simpr 485 | . 2 ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐻 Fn 𝐴 ∧ ∀𝑧 ∈ 𝐴 (𝐻‘𝑧) = (𝑧𝐺(𝐻 ↾ Pred(𝑅, 𝐴, 𝑧))))) → (𝐻 Fn 𝐴 ∧ ∀𝑧 ∈ 𝐴 (𝐻‘𝑧) = (𝑧𝐺(𝐻 ↾ Pred(𝑅, 𝐴, 𝑧))))) | |
9 | fpr3g 8101 | . 2 ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐹 Fn 𝐴 ∧ ∀𝑧 ∈ 𝐴 (𝐹‘𝑧) = (𝑧𝐺(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))) ∧ (𝐻 Fn 𝐴 ∧ ∀𝑧 ∈ 𝐴 (𝐻‘𝑧) = (𝑧𝐺(𝐻 ↾ Pred(𝑅, 𝐴, 𝑧))))) → 𝐹 = 𝐻) | |
10 | 1, 7, 8, 9 | syl3anc 1370 | 1 ⊢ (((𝑅 Fr 𝐴 ∧ 𝑅 Po 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐻 Fn 𝐴 ∧ ∀𝑧 ∈ 𝐴 (𝐻‘𝑧) = (𝑧𝐺(𝐻 ↾ Pred(𝑅, 𝐴, 𝑧))))) → 𝐹 = 𝐻) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 396 ∧ w3a 1086 = wceq 1539 ∀wral 3064 Po wpo 5501 Fr wfr 5541 Se wse 5542 ↾ cres 5591 Predcpred 6201 Fn wfn 6428 ‘cfv 6433 (class class class)co 7275 frecscfrecs 8096 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2709 ax-rep 5209 ax-sep 5223 ax-nul 5230 ax-pr 5352 ax-un 7588 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 845 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1783 df-nf 1787 df-sb 2068 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2816 df-nfc 2889 df-ne 2944 df-ral 3069 df-rex 3070 df-reu 3072 df-rab 3073 df-v 3434 df-sbc 3717 df-csb 3833 df-dif 3890 df-un 3892 df-in 3894 df-ss 3904 df-nul 4257 df-if 4460 df-pw 4535 df-sn 4562 df-pr 4564 df-op 4568 df-uni 4840 df-iun 4926 df-br 5075 df-opab 5137 df-mpt 5158 df-id 5489 df-po 5503 df-fr 5544 df-se 5545 df-xp 5595 df-rel 5596 df-cnv 5597 df-co 5598 df-dm 5599 df-rn 5600 df-res 5601 df-ima 5602 df-pred 6202 df-iota 6391 df-fun 6435 df-fn 6436 df-f 6437 df-f1 6438 df-fo 6439 df-f1o 6440 df-fv 6441 df-ov 7278 df-frecs 8097 |
This theorem is referenced by: (None) |
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