Theorem wfr3OLD 8239

Description: Obsolete form of wfr3 8238 as of 18-Nov-2024. (New usage is discouraged.) (Proof modification is discouraged.) (Contributed by Scott Fenton, 18-Apr-2011.) (Revised by Mario Carneiro, 26-Jun-2015.)

Hypotheses

Ref	Expression
wfr3OLD.1	⊢ 𝑅 We 𝐴
wfr3OLD.2	⊢ 𝑅 Se 𝐴
wfr3OLD.3	⊢ 𝐹 = wrecs(𝑅, 𝐴, 𝐺)

Assertion

Ref	Expression
wfr3OLD	⊢ ((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 wfr3OLD

Step	Hyp	Ref	Expression
1		wfr3OLD.1	. . 3 ⊢ 𝑅 We 𝐴
2		wfr3OLD.2	. . 3 ⊢ 𝑅 Se 𝐴
3	1, 2	pm3.2i 471	. 2 ⊢ (𝑅 We 𝐴 ∧ 𝑅 Se 𝐴)
4		wfr3OLD.3	. . . . 5 ⊢ 𝐹 = wrecs(𝑅, 𝐴, 𝐺)
5	4	wfr1 8236	. . . 4 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → 𝐹 Fn 𝐴)
6	1, 2, 5	mp2an 689	. . 3 ⊢ 𝐹 Fn 𝐴
7	4	wfr2 8237	. . . . 5 ⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))))
8	1, 2, 7	mpanl12 699	. . . 4 ⊢ (𝑧 ∈ 𝐴 → (𝐹‘𝑧) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))))
9	8	rgen 3063	. . 3 ⊢ ∀𝑧 ∈ 𝐴 (𝐹‘𝑧) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))
10	6, 9	pm3.2i 471	. 2 ⊢ (𝐹 Fn 𝐴 ∧ ∀𝑧 ∈ 𝐴 (𝐹‘𝑧) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))))
11		wfr3g 8208	. 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)
12	3, 10, 11	mp3an12 1450	1 ⊢ ((𝐻 Fn 𝐴 ∧ ∀𝑧 ∈ 𝐴 (𝐻‘𝑧) = (𝐺‘(𝐻 ↾ Pred(𝑅, 𝐴, 𝑧)))) → 𝐹 = 𝐻)

Syntax hints:   → wi 4   ∧ wa 396   = wceq 1540   ∈ wcel 2105  ∀wral 3061   Se wse 5573   We wwe 5574   ↾ cres 5622  Predcpred 6237   Fn wfn 6474  ‘cfv 6479  wrecscwrecs 8197

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 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2707  ax-rep 5229  ax-sep 5243  ax-nul 5250  ax-pr 5372  ax-un 7650

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2886  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3350  df-rab 3404  df-v 3443  df-sbc 3728  df-csb 3844  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4270  df-if 4474  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4853  df-iun 4943  df-br 5093  df-opab 5155  df-mpt 5176  df-id 5518  df-po 5532  df-so 5533  df-fr 5575  df-se 5576  df-we 5577  df-xp 5626  df-rel 5627  df-cnv 5628  df-co 5629  df-dm 5630  df-rn 5631  df-res 5632  df-ima 5633  df-pred 6238  df-iota 6431  df-fun 6481  df-fn 6482  df-f 6483  df-f1 6484  df-fo 6485  df-f1o 6486  df-fv 6487  df-ov 7340  df-2nd 7900  df-frecs 8167  df-wrecs 8198

This theorem is referenced by: (None)