Theorem wfr2 7974

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.)

Hypotheses

Ref	Expression
wfr2.1	⊢ 𝑅 We 𝐴
wfr2.2	⊢ 𝑅 Se 𝐴
wfr2.3	⊢ 𝐹 = wrecs(𝑅, 𝐴, 𝐺)

Assertion

Ref	Expression
wfr2	⊢ (𝑋 ∈ 𝐴 → (𝐹‘𝑋) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑋))))

Proof of Theorem wfr2

Dummy variable 𝑥 is distinct from all other variables.

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 7970	. . 3 ⊢ dom 𝐹 = 𝐴
6	5	eleq2i 2904	. 2 ⊢ (𝑋 ∈ dom 𝐹 ↔ 𝑋 ∈ 𝐴)
7	1, 2, 3	wfr2a 7972	. 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))))

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2114   ∪ cun 3934  {csn 4567  ⟨cop 4573   Se wse 5512   We wwe 5513  dom cdm 5555   ↾ cres 5557  Predcpred 6147  ‘cfv 6355  wrecscwrecs 7946

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-rep 5190  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  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 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-iun 4921  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-po 5474  df-so 5475  df-fr 5514  df-se 5515  df-we 5516  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-pred 6148  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-wrecs 7947

This theorem is referenced by:  wfr3  7975  tfr2ALT  8037  bpolylem  15402