Theorem wfr3 8284

Description: The principle of Well-Ordered 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 8282 and wfr2 8283 is identical to 𝐹. (Contributed by Scott Fenton, 18-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.) (Revised by Scott Fenton, 18-Nov-2024.)

Hypothesis

Ref	Expression
wfr3.3	⊢ 𝐹 = wrecs(𝑅, 𝐴, 𝐺)

Assertion

Ref	Expression
wfr3	⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐻 Fn 𝐴 ∧ ∀𝑧 ∈ 𝐴 (𝐻‘𝑧) = (𝐺‘(𝐻 ↾ Pred(𝑅, 𝐴, 𝑧))))) → 𝐹 = 𝐻)

Distinct variable groups:   𝑧,𝐴   𝑧,𝐹   𝑧,𝐺   𝑧,𝐻   𝑧,𝑅

Proof of Theorem wfr3

Step	Hyp	Ref	Expression
1		simpl 482	. 2 ⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐻 Fn 𝐴 ∧ ∀𝑧 ∈ 𝐴 (𝐻‘𝑧) = (𝐺‘(𝐻 ↾ Pred(𝑅, 𝐴, 𝑧))))) → (𝑅 We 𝐴 ∧ 𝑅 Se 𝐴))
2		wfr3.3	. . . . 5 ⊢ 𝐹 = wrecs(𝑅, 𝐴, 𝐺)
3	2	wfr1 8282	. . . 4 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → 𝐹 Fn 𝐴)
4	2	wfr2 8283	. . . . 5 ⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))))
5	4	ralrimiva 3125	. . . 4 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → ∀𝑧 ∈ 𝐴 (𝐹‘𝑧) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))))
6	3, 5	jca 511	. . 3 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → (𝐹 Fn 𝐴 ∧ ∀𝑧 ∈ 𝐴 (𝐹‘𝑧) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))))
7	6	adantr 480	. 2 ⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐻 Fn 𝐴 ∧ ∀𝑧 ∈ 𝐴 (𝐻‘𝑧) = (𝐺‘(𝐻 ↾ Pred(𝑅, 𝐴, 𝑧))))) → (𝐹 Fn 𝐴 ∧ ∀𝑧 ∈ 𝐴 (𝐹‘𝑧) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))))
8		simpr 484	. 2 ⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐻 Fn 𝐴 ∧ ∀𝑧 ∈ 𝐴 (𝐻‘𝑧) = (𝐺‘(𝐻 ↾ Pred(𝑅, 𝐴, 𝑧))))) → (𝐻 Fn 𝐴 ∧ ∀𝑧 ∈ 𝐴 (𝐻‘𝑧) = (𝐺‘(𝐻 ↾ Pred(𝑅, 𝐴, 𝑧)))))
9		wfr3g 8275	. 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	1, 7, 8, 9	syl3anc 1373	1 ⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ (𝐻 Fn 𝐴 ∧ ∀𝑧 ∈ 𝐴 (𝐻‘𝑧) = (𝐺‘(𝐻 ↾ Pred(𝑅, 𝐴, 𝑧))))) → 𝐹 = 𝐻)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1540  ∀wral 3044   Se wse 5582   We wwe 5583   ↾ cres 5633  Predcpred 6261   Fn wfn 6494  ‘cfv 6499  wrecscwrecs 8267

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-rep 5229  ax-sep 5246  ax-nul 5256  ax-pr 5382  ax-un 7691

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-reu 3352  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-po 5539  df-so 5540  df-fr 5584  df-se 5585  df-we 5586  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-pred 6262  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-ov 7372  df-2nd 7948  df-frecs 8237  df-wrecs 8268

This theorem is referenced by:  tfr3ALT  8347