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Mirrors > Home > MPE Home > Th. List > wfr3 | Structured version Visualization version GIF version |
Description: The principle of Well-Founded Recursion, part 3 of 3. 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 wfr1 7433 and wfr2 7434 is identical to 𝐹. (Contributed by Scott Fenton, 18-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.) |
Ref | Expression |
---|---|
wfr3.1 | ⊢ 𝑅 We 𝐴 |
wfr3.2 | ⊢ 𝑅 Se 𝐴 |
wfr3.3 | ⊢ 𝐹 = wrecs(𝑅, 𝐴, 𝐺) |
Ref | Expression |
---|---|
wfr3 | ⊢ ((𝐻 Fn 𝐴 ∧ ∀𝑧 ∈ 𝐴 (𝐻‘𝑧) = (𝐺‘(𝐻 ↾ Pred(𝑅, 𝐴, 𝑧)))) → 𝐹 = 𝐻) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wfr3.1 | . . 3 ⊢ 𝑅 We 𝐴 | |
2 | wfr3.2 | . . 3 ⊢ 𝑅 Se 𝐴 | |
3 | 1, 2 | pm3.2i 471 | . 2 ⊢ (𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) |
4 | wfr3.3 | . . . 4 ⊢ 𝐹 = wrecs(𝑅, 𝐴, 𝐺) | |
5 | 1, 2, 4 | wfr1 7433 | . . 3 ⊢ 𝐹 Fn 𝐴 |
6 | 1, 2, 4 | wfr2 7434 | . . . 4 ⊢ (𝑧 ∈ 𝐴 → (𝐹‘𝑧) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))) |
7 | 6 | rgen 2922 | . . 3 ⊢ ∀𝑧 ∈ 𝐴 (𝐹‘𝑧) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))) |
8 | 5, 7 | pm3.2i 471 | . 2 ⊢ (𝐹 Fn 𝐴 ∧ ∀𝑧 ∈ 𝐴 (𝐹‘𝑧) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))) |
9 | wfr3g 7413 | . 2 ⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐹 Fn 𝐴 ∧ ∀𝑧 ∈ 𝐴 (𝐹‘𝑧) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))) ∧ (𝐻 Fn 𝐴 ∧ ∀𝑧 ∈ 𝐴 (𝐻‘𝑧) = (𝐺‘(𝐻 ↾ Pred(𝑅, 𝐴, 𝑧))))) → 𝐹 = 𝐻) | |
10 | 3, 8, 9 | mp3an12 1414 | 1 ⊢ ((𝐻 Fn 𝐴 ∧ ∀𝑧 ∈ 𝐴 (𝐻‘𝑧) = (𝐺‘(𝐻 ↾ Pred(𝑅, 𝐴, 𝑧)))) → 𝐹 = 𝐻) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 = wceq 1483 ∀wral 2912 Se wse 5071 We wwe 5072 ↾ cres 5116 Predcpred 5679 Fn wfn 5883 ‘cfv 5888 wrecscwrecs 7406 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-8 1992 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-rep 4771 ax-sep 4781 ax-nul 4789 ax-pow 4843 ax-pr 4906 ax-un 6949 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ne 2795 df-ral 2917 df-rex 2918 df-reu 2919 df-rmo 2920 df-rab 2921 df-v 3202 df-sbc 3436 df-csb 3534 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-uni 4437 df-iun 4522 df-br 4654 df-opab 4713 df-mpt 4730 df-id 5024 df-po 5035 df-so 5036 df-fr 5073 df-se 5074 df-we 5075 df-xp 5120 df-rel 5121 df-cnv 5122 df-co 5123 df-dm 5124 df-rn 5125 df-res 5126 df-ima 5127 df-pred 5680 df-iota 5851 df-fun 5890 df-fn 5891 df-f 5892 df-f1 5893 df-fo 5894 df-f1o 5895 df-fv 5896 df-wrecs 7407 |
This theorem is referenced by: tfr3ALT 7498 |
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