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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 7320 and wfr2 7321 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 470 | . 2 ⊢ (𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) |
4 | wfr3.3 | . . . 4 ⊢ 𝐹 = wrecs(𝑅, 𝐴, 𝐺) | |
5 | 1, 2, 4 | wfr1 7320 | . . 3 ⊢ 𝐹 Fn 𝐴 |
6 | 1, 2, 4 | wfr2 7321 | . . . 4 ⊢ (𝑧 ∈ 𝐴 → (𝐹‘𝑧) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))) |
7 | 6 | rgen 2906 | . . 3 ⊢ ∀𝑧 ∈ 𝐴 (𝐹‘𝑧) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))) |
8 | 5, 7 | pm3.2i 470 | . 2 ⊢ (𝐹 Fn 𝐴 ∧ ∀𝑧 ∈ 𝐴 (𝐹‘𝑧) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))) |
9 | wfr3g 7300 | . 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 1406 | 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 383 = wceq 1475 ∀wral 2896 Se wse 4995 We wwe 4996 ↾ cres 5040 Predcpred 5596 Fn wfn 5799 ‘cfv 5804 wrecscwrecs 7293 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-rep 4699 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 |
This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-ral 2901 df-rex 2902 df-reu 2903 df-rmo 2904 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-iun 4457 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-fr 4997 df-se 4998 df-we 4999 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-pred 5597 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-wrecs 7294 |
This theorem is referenced by: tfr3ALT 7385 |
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