Theorem tfr3ALT 8321

Description: Alternate proof of tfr3 8318 using well-ordered 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 7719	. . . . . . 7 ⊢ (𝑥 ∈ On → Pred( E , On, 𝑥) = 𝑥)
2	1	reseq2d 5927	. . . . . 6 ⊢ (𝑥 ∈ On → (𝐵 ↾ Pred( E , On, 𝑥)) = (𝐵 ↾ 𝑥))
3	2	fveq2d 6826	. . . . 5 ⊢ (𝑥 ∈ On → (𝐺‘(𝐵 ↾ Pred( E , On, 𝑥))) = (𝐺‘(𝐵 ↾ 𝑥)))
4	3	eqeq2d 2742	. . . 4 ⊢ (𝑥 ∈ On → ((𝐵‘𝑥) = (𝐺‘(𝐵 ↾ Pred( E , On, 𝑥))) ↔ (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥))))
5	4	ralbiia 3076	. . 3 ⊢ (∀𝑥 ∈ On (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ Pred( E , On, 𝑥))) ↔ ∀𝑥 ∈ On (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥)))
6		epweon 7708	. . . 4 ⊢ E We On
7		epse 5596	. . . 4 ⊢ E Se On
8		tfrALT.1	. . . . . 6 ⊢ 𝐹 = recs(𝐺)
9		df-recs 8291	. . . . . 6 ⊢ recs(𝐺) = wrecs( E , On, 𝐺)
10	8, 9	eqtri 2754	. . . . 5 ⊢ 𝐹 = wrecs( E , On, 𝐺)
11	10	wfr3 8258	. . . 4 ⊢ ((( E We On ∧ E Se On) ∧ (𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ Pred( E , On, 𝑥))))) → 𝐹 = 𝐵)
12	6, 7, 11	mpanl12 702	. . 3 ⊢ ((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ Pred( E , On, 𝑥)))) → 𝐹 = 𝐵)
13	5, 12	sylan2br 595	. 2 ⊢ ((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥))) → 𝐹 = 𝐵)
14	13	eqcomd 2737	1 ⊢ ((𝐵 Fn On ∧ ∀𝑥 ∈ On (𝐵‘𝑥) = (𝐺‘(𝐵 ↾ 𝑥))) → 𝐵 = 𝐹)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1541   ∈ wcel 2111  ∀wral 3047   E cep 5513   Se wse 5565   We wwe 5566   ↾ cres 5616  Predcpred 6247  Oncon0 6306   Fn wfn 6476  ‘cfv 6481  wrecscwrecs 8241  recscrecs 8290

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 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-rep 5215  ax-sep 5232  ax-nul 5242  ax-pr 5368  ax-un 7668

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-ral 3048  df-rex 3057  df-reu 3347  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-pss 3917  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-iun 4941  df-br 5090  df-opab 5152  df-mpt 5171  df-tr 5197  df-id 5509  df-eprel 5514  df-po 5522  df-so 5523  df-fr 5567  df-se 5568  df-we 5569  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-pred 6248  df-ord 6309  df-on 6310  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-ov 7349  df-2nd 7922  df-frecs 8211  df-wrecs 8242  df-recs 8291

This theorem is referenced by: (None)