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Mirrors > Home > MPE Home > Th. List > wfr2 | Structured version Visualization version GIF version |
Description: The Principle of Well-Founded Recursion, part 2 of 3. Next, we show that the value of 𝐹 at any 𝑋 ∈ 𝐴 is 𝐺 recursively applied to all "previous" values of 𝐹. (Contributed by Scott Fenton, 18-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.) |
Ref | Expression |
---|---|
wfr2.1 | ⊢ 𝑅 We 𝐴 |
wfr2.2 | ⊢ 𝑅 Se 𝐴 |
wfr2.3 | ⊢ 𝐹 = wrecs(𝑅, 𝐴, 𝐺) |
Ref | Expression |
---|---|
wfr2 | ⊢ (𝑋 ∈ 𝐴 → (𝐹‘𝑋) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋)))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | wfr2.1 | . . . 4 ⊢ 𝑅 We 𝐴 | |
2 | wfr2.2 | . . . 4 ⊢ 𝑅 Se 𝐴 | |
3 | wfr2.3 | . . . 4 ⊢ 𝐹 = wrecs(𝑅, 𝐴, 𝐺) | |
4 | eqid 2821 | . . . 4 ⊢ (𝐹 ∪ {〈𝑥, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑥)))〉}) = (𝐹 ∪ {〈𝑥, (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑥)))〉}) | |
5 | 1, 2, 3, 4 | wfrlem16 7969 | . . 3 ⊢ dom 𝐹 = 𝐴 |
6 | 5 | eleq2i 2904 | . 2 ⊢ (𝑋 ∈ dom 𝐹 ↔ 𝑋 ∈ 𝐴) |
7 | 1, 2, 3 | wfr2a 7971 | . 2 ⊢ (𝑋 ∈ dom 𝐹 → (𝐹‘𝑋) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋)))) |
8 | 6, 7 | sylbir 237 | 1 ⊢ (𝑋 ∈ 𝐴 → (𝐹‘𝑋) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋)))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 = wceq 1533 ∈ wcel 2110 ∪ cun 3933 {csn 4566 〈cop 4572 Se wse 5511 We wwe 5512 dom cdm 5554 ↾ cres 5556 Predcpred 6146 ‘cfv 6354 wrecscwrecs 7945 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 ax-rep 5189 ax-sep 5202 ax-nul 5209 ax-pow 5265 ax-pr 5329 ax-un 7460 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3or 1084 df-3an 1085 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-mo 2618 df-eu 2650 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ne 3017 df-ral 3143 df-rex 3144 df-reu 3145 df-rmo 3146 df-rab 3147 df-v 3496 df-sbc 3772 df-csb 3883 df-dif 3938 df-un 3940 df-in 3942 df-ss 3951 df-nul 4291 df-if 4467 df-sn 4567 df-pr 4569 df-op 4573 df-uni 4838 df-iun 4920 df-br 5066 df-opab 5128 df-mpt 5146 df-id 5459 df-po 5473 df-so 5474 df-fr 5513 df-se 5514 df-we 5515 df-xp 5560 df-rel 5561 df-cnv 5562 df-co 5563 df-dm 5564 df-rn 5565 df-res 5566 df-ima 5567 df-pred 6147 df-iota 6313 df-fun 6356 df-fn 6357 df-f 6358 df-f1 6359 df-fo 6360 df-f1o 6361 df-fv 6362 df-wrecs 7946 |
This theorem is referenced by: wfr3 7974 tfr2ALT 8036 bpolylem 15401 |
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