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Mirrors > Home > MPE Home > Th. List > wfr1 | Structured version Visualization version GIF version |
Description: The Principle of Well-Ordered 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-ordered 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 8065 | . 2 ⊢ Fun 𝐹 |
5 | eqid 2737 | . . 3 ⊢ (𝐹 ∪ {〈𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))〉}) = (𝐹 ∪ {〈𝑧, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))〉}) | |
6 | 1, 2, 3, 5 | wfrlem16 8070 | . 2 ⊢ dom 𝐹 = 𝐴 |
7 | df-fn 6383 | . 2 ⊢ (𝐹 Fn 𝐴 ↔ (Fun 𝐹 ∧ dom 𝐹 = 𝐴)) | |
8 | 4, 6, 7 | mpbir2an 711 | 1 ⊢ 𝐹 Fn 𝐴 |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1543 ∪ cun 3864 {csn 4541 〈cop 4547 Se wse 5507 We wwe 5508 dom cdm 5551 ↾ cres 5553 Predcpred 6159 Fun wfun 6374 Fn wfn 6375 ‘cfv 6380 wrecscwrecs 8046 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1803 ax-4 1817 ax-5 1918 ax-6 1976 ax-7 2016 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2158 ax-12 2175 ax-ext 2708 ax-rep 5179 ax-sep 5192 ax-nul 5199 ax-pr 5322 ax-un 7523 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 848 df-3or 1090 df-3an 1091 df-tru 1546 df-fal 1556 df-ex 1788 df-nf 1792 df-sb 2071 df-mo 2539 df-eu 2568 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2886 df-ne 2941 df-ral 3066 df-rex 3067 df-reu 3068 df-rmo 3069 df-rab 3070 df-v 3410 df-sbc 3695 df-csb 3812 df-dif 3869 df-un 3871 df-in 3873 df-ss 3883 df-nul 4238 df-if 4440 df-sn 4542 df-pr 4544 df-op 4548 df-uni 4820 df-iun 4906 df-br 5054 df-opab 5116 df-mpt 5136 df-id 5455 df-po 5468 df-so 5469 df-fr 5509 df-se 5510 df-we 5511 df-xp 5557 df-rel 5558 df-cnv 5559 df-co 5560 df-dm 5561 df-rn 5562 df-res 5563 df-ima 5564 df-pred 6160 df-iota 6338 df-fun 6382 df-fn 6383 df-f 6384 df-f1 6385 df-fo 6386 df-f1o 6387 df-fv 6388 df-wrecs 8047 |
This theorem is referenced by: wfr3 8075 tfr1ALT 8136 bpolylem 15610 |
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