Theorem tfrlem3a 5866

Description: Lemma for transfinite recursion. Let A be the class of "acceptable" functions. The final thing we're interested in is the union of all these acceptable functions. This lemma just changes some bound variables in A for later use. (Contributed by NM, 9-Apr-1995.)

Hypotheses

Ref	Expression
tfrlem3.1	⊢ A = {f ∣ ∃x ∈ On (f Fn x ∧ ∀y ∈ x (f‘y) = (𝐹‘(f ↾ y)))}
tfrlem3.2	⊢ 𝐺 ∈ V

Assertion

Ref	Expression
tfrlem3a	⊢ (𝐺 ∈ A ↔ ∃z ∈ On (𝐺 Fn z ∧ ∀w ∈ z (𝐺‘w) = (𝐹‘(𝐺 ↾ w))))

Distinct variable groups:   w,f,x,y,z,𝐹   f,𝐺,w,x,y,z

Allowed substitution hints:   A(x,y,z,w,f)

Proof of Theorem tfrlem3a

Step	Hyp	Ref	Expression
1		tfrlem3.2	. 2 ⊢ 𝐺 ∈ V
2		fneq12 4935	. . . 4 ⊢ ((f = 𝐺 ∧ x = z) → (f Fn x ↔ 𝐺 Fn z))
3		simpll 481	. . . . . . 7 ⊢ (((f = 𝐺 ∧ x = z) ∧ y = w) → f = 𝐺)
4		simpr 103	. . . . . . 7 ⊢ (((f = 𝐺 ∧ x = z) ∧ y = w) → y = w)
5	3, 4	fveq12d 5127	. . . . . 6 ⊢ (((f = 𝐺 ∧ x = z) ∧ y = w) → (f‘y) = (𝐺‘w))
6	3, 4	reseq12d 4556	. . . . . . 7 ⊢ (((f = 𝐺 ∧ x = z) ∧ y = w) → (f ↾ y) = (𝐺 ↾ w))
7	6	fveq2d 5125	. . . . . 6 ⊢ (((f = 𝐺 ∧ x = z) ∧ y = w) → (𝐹‘(f ↾ y)) = (𝐹‘(𝐺 ↾ w)))
8	5, 7	eqeq12d 2051	. . . . 5 ⊢ (((f = 𝐺 ∧ x = z) ∧ y = w) → ((f‘y) = (𝐹‘(f ↾ y)) ↔ (𝐺‘w) = (𝐹‘(𝐺 ↾ w))))
9		simpr 103	. . . . . 6 ⊢ ((f = 𝐺 ∧ x = z) → x = z)
10	9	adantr 261	. . . . 5 ⊢ (((f = 𝐺 ∧ x = z) ∧ y = w) → x = z)
11	8, 10	cbvraldva2 2531	. . . 4 ⊢ ((f = 𝐺 ∧ x = z) → (∀y ∈ x (f‘y) = (𝐹‘(f ↾ y)) ↔ ∀w ∈ z (𝐺‘w) = (𝐹‘(𝐺 ↾ w))))
12	2, 11	anbi12d 442	. . 3 ⊢ ((f = 𝐺 ∧ x = z) → ((f Fn x ∧ ∀y ∈ x (f‘y) = (𝐹‘(f ↾ y))) ↔ (𝐺 Fn z ∧ ∀w ∈ z (𝐺‘w) = (𝐹‘(𝐺 ↾ w)))))
13	12	cbvrexdva 2534	. 2 ⊢ (f = 𝐺 → (∃x ∈ On (f Fn x ∧ ∀y ∈ x (f‘y) = (𝐹‘(f ↾ y))) ↔ ∃z ∈ On (𝐺 Fn z ∧ ∀w ∈ z (𝐺‘w) = (𝐹‘(𝐺 ↾ w)))))
14		tfrlem3.1	. 2 ⊢ A = {f ∣ ∃x ∈ On (f Fn x ∧ ∀y ∈ x (f‘y) = (𝐹‘(f ↾ y)))}
15	1, 13, 14	elab2 2684	1 ⊢ (𝐺 ∈ A ↔ ∃z ∈ On (𝐺 Fn z ∧ ∀w ∈ z (𝐺‘w) = (𝐹‘(𝐺 ↾ w))))

Syntax hints:   ∧ wa 97   ↔ wb 98   = wceq 1242   ∈ wcel 1390  {cab 2023  ∀wral 2300  ∃wrex 2301  Vcvv 2551  Oncon0 4066   ↾ cres 4290   Fn wfn 4840  ‘cfv 4845

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-res 4300  df-iota 4810  df-fun 4847  df-fn 4848  df-fv 4853

This theorem is referenced by:  tfrlem3  5867  tfrlem5  5871  tfrlemisucaccv  5880  tfrlemibxssdm  5882  tfrlemi14d  5888  tfrexlem  5889