Theorem tfr3ALT 7668

Description: Alternate proof of tfr3 7665 using well-founded recursion. (Contributed by Scott Fenton, 3-Aug-2020.) (New usage is discouraged.) (Proof modification is discouraged.)

Hypothesis

Ref	Expression
tfrALT.1	⊢ 𝐹 = recs(𝐺)

Assertion

Ref	Expression
tfr3ALT	⊢ ((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥))) → 𝐵 = 𝐹)

Distinct variable groups:   𝑥,𝐵   𝑥,𝐺   𝑥,𝐹

Proof of Theorem tfr3ALT

Step	Hyp	Ref	Expression
1		predon 7157	. . . . . . 7 ⊢ (𝑥 ∈ On → Pred( E , On, 𝑥) = 𝑥)
2	1	reseq2d 5551	. . . . . 6 ⊢ (𝑥 ∈ On → (𝐵 ↾ Pred( E , On, 𝑥)) = (𝐵 ↾ 𝑥))
3	2	fveq2d 6357	. . . . 5 ⊢ (𝑥 ∈ On → (𝐺‘(𝐵 ↾ Pred( E , On, 𝑥))) = (𝐺‘(𝐵 ↾ 𝑥)))
4	3	eqeq2d 2770	. . . 4 ⊢ (𝑥 ∈ On → ((𝐵‘𝑥) = (𝐺‘(𝐵 ↾ Pred( E , On, 𝑥))) ↔ (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥))))
5	4	ralbiia 3117	. . 3 ⊢ (∀𝑥 ∈ On (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ Pred( E , On, 𝑥))) ↔ ∀𝑥 ∈ On (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥)))
6		epweon 7149	. . . 4 ⊢ E We On
7		epse 5249	. . . 4 ⊢ E Se On
8		tfrALT.1	. . . . 5 ⊢ 𝐹 = recs(𝐺)
9		df-recs 7638	. . . . 5 ⊢ recs(𝐺) = wrecs( E , On, 𝐺)
10	8, 9	eqtri 2782	. . . 4 ⊢ 𝐹 = wrecs( E , On, 𝐺)
11	6, 7, 10	wfr3 7605	. . 3 ⊢ ((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ Pred( E , On, 𝑥)))) → 𝐹 = 𝐵)
12	5, 11	sylan2br 494	. 2 ⊢ ((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥))) → 𝐹 = 𝐵)
13	12	eqcomd 2766	1 ⊢ ((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥))) → 𝐵 = 𝐹)

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1632   ∈ wcel 2139  ∀wral 3050   E cep 5178   ↾ cres 5268  Predcpred 5840  Oncon0 5884   Fn wfn 6044  ‘cfv 6049  wrecscwrecs 7576  recscrecs 7637

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-rep 4923  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-reu 3057  df-rmo 3058  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-se 5226  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-wrecs 7577  df-recs 7638

This theorem is referenced by: (None)