Theorem wfr3OLD 8368

Description: Obsolete form of wfr3 8367 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 469	. 2 ⊢ (𝑅 We 𝐴 ∧ 𝑅 Se 𝐴)
4		wfr3OLD.3	. . . . 5 ⊢ 𝐹 = wrecs(𝑅, 𝐴, 𝐺)
5	4	wfr1 8365	. . . 4 ⊢ ((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) → 𝐹 Fn 𝐴)
6	1, 2, 5	mp2an 690	. . 3 ⊢ 𝐹 Fn 𝐴
7	4	wfr2 8366	. . . . 5 ⊢ (((𝑅 We 𝐴 ∧ 𝑅 Se 𝐴) ∧ 𝑧 ∈ 𝐴) → (𝐹‘𝑧) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))))
8	1, 2, 7	mpanl12 700	. . . 4 ⊢ (𝑧 ∈ 𝐴 → (𝐹‘𝑧) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))))
9	8	rgen 3053	. . 3 ⊢ ∀𝑧 ∈ 𝐴 (𝐹‘𝑧) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧)))
10	6, 9	pm3.2i 469	. 2 ⊢ (𝐹 Fn 𝐴 ∧ ∀𝑧 ∈ 𝐴 (𝐹‘𝑧) = (𝐺‘(𝐹 ↾ Pred(𝑅, 𝐴, 𝑧))))
11		wfr3g 8337	. 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 1448	1 ⊢ ((𝐻 Fn 𝐴 ∧ ∀𝑧 ∈ 𝐴 (𝐻‘𝑧) = (𝐺‘(𝐻 ↾ Pred(𝑅, 𝐴, 𝑧)))) → 𝐹 = 𝐻)

Syntax hints:   → wi 4   ∧ wa 394   = wceq 1534   ∈ wcel 2099  ∀wral 3051   Se wse 5635   We wwe 5636   ↾ cres 5684  Predcpred 6311   Fn wfn 6549  ‘cfv 6554  wrecscwrecs 8326

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-rep 5290  ax-sep 5304  ax-nul 5311  ax-pr 5433  ax-un 7746

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-mo 2529  df-eu 2558  df-clab 2704  df-cleq 2718  df-clel 2803  df-nfc 2878  df-ne 2931  df-ral 3052  df-rex 3061  df-reu 3365  df-rab 3420  df-v 3464  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4326  df-if 4534  df-pw 4609  df-sn 4634  df-pr 4636  df-op 4640  df-uni 4914  df-iun 5003  df-br 5154  df-opab 5216  df-mpt 5237  df-id 5580  df-po 5594  df-so 5595  df-fr 5637  df-se 5638  df-we 5639  df-xp 5688  df-rel 5689  df-cnv 5690  df-co 5691  df-dm 5692  df-rn 5693  df-res 5694  df-ima 5695  df-pred 6312  df-iota 6506  df-fun 6556  df-fn 6557  df-f 6558  df-f1 6559  df-fo 6560  df-f1o 6561  df-fv 6562  df-ov 7427  df-2nd 8004  df-frecs 8296  df-wrecs 8327

This theorem is referenced by: (None)