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Mirrors > Home > MPE Home > Th. List > wfr1 | Structured version Visualization version GIF version |
Description: The Principle of Well-Founded Recursion, part 1 of 3. We start with an arbitrary function 𝐺. Then, using a base class 𝐴 and a well-ordering 𝑅 of 𝐴, we define a function 𝐹. This function is said to be defined by "well-founded recursion." The purpose of these three theorems is to demonstrate the properties of 𝐹. We begin by showing that 𝐹 is a function over 𝐴. (Contributed by Scott Fenton, 22-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.) |
Ref | Expression |
---|---|
wfr1.1 | ⊢ 𝑅 We 𝐴 |
wfr1.2 | ⊢ 𝑅 Se 𝐴 |
wfr1.3 | ⊢ 𝐹 = wrecs(𝑅, 𝐴, 𝐺) |
Ref | Expression |
---|---|
wfr1 | ⊢ 𝐹 Fn 𝐴 |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wfr1.1 | . . 3 ⊢ 𝑅 We 𝐴 | |
2 | wfr1.2 | . . 3 ⊢ 𝑅 Se 𝐴 | |
3 | wfr1.3 | . . 3 ⊢ 𝐹 = wrecs(𝑅, 𝐴, 𝐺) | |
4 | 1, 2, 3 | wfrfun 7312 | . 2 ⊢ Fun 𝐹 |
5 | eqid 2610 | . . 3 ⊢ (𝐹 ∪ {〈𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))〉}) = (𝐹 ∪ {〈𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))〉}) | |
6 | 1, 2, 3, 5 | wfrlem16 7317 | . 2 ⊢ dom 𝐹 = 𝐴 |
7 | df-fn 5807 | . 2 ⊢ (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴)) | |
8 | 4, 6, 7 | mpbir2an 957 | 1 ⊢ 𝐹 Fn 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1475 ∪ cun 3538 {csn 4125 〈cop 4131 Se wse 4995 We wwe 4996 dom cdm 5038 ↾ cres 5040 Predcpred 5596 Fun wfun 5798 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: wfr3 7322 tfr1ALT 7383 bpolylem 14618 |
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