Theorem tfrlem3a 6409

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 5376	. . . 4 ⊢ ((𝑓 = 𝐺 ∧ 𝑥 = 𝑧) → (𝑓 Fn 𝑥 ↔ 𝐺 Fn 𝑧))
3		simpll 527	. . . . . . 7 ⊢ (((𝑓 = 𝐺 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → 𝑓 = 𝐺)
4		simpr 110	. . . . . . 7 ⊢ (((𝑓 = 𝐺 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → 𝑦 = 𝑤)
5	3, 4	fveq12d 5596	. . . . . 6 ⊢ (((𝑓 = 𝐺 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → (𝑓‘𝑦) = (𝐺‘𝑤))
6	3, 4	reseq12d 4969	. . . . . . 7 ⊢ (((𝑓 = 𝐺 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → (𝑓 ↾ 𝑦) = (𝐺 ↾ 𝑤))
7	6	fveq2d 5593	. . . . . 6 ⊢ (((𝑓 = 𝐺 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → (𝐹‘(𝑓 ↾ 𝑦)) = (𝐹‘(𝐺 ↾ 𝑤)))
8	5, 7	eqeq12d 2221	. . . . 5 ⊢ (((𝑓 = 𝐺 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → ((𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)) ↔ (𝐺‘𝑤) = (𝐹‘(𝐺 ↾ 𝑤))))
9		simpr 110	. . . . . 6 ⊢ ((𝑓 = 𝐺 ∧ 𝑥 = 𝑧) → 𝑥 = 𝑧)
10	9	adantr 276	. . . . 5 ⊢ (((𝑓 = 𝐺 ∧ 𝑥 = 𝑧) ∧ 𝑦 = 𝑤) → 𝑥 = 𝑧)
11	8, 10	cbvraldva2 2746	. . . 4 ⊢ ((𝑓 = 𝐺 ∧ 𝑥 = 𝑧) → (∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)) ↔ ∀𝑤 ∈ 𝑧 (𝐺‘𝑤) = (𝐹‘(𝐺 ↾ 𝑤))))
12	2, 11	anbi12d 473	. . 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 2749	. 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 2925	1 ⊢ (𝐺 ∈ 𝐴 ↔ ∃𝑧 ∈ On (𝐺 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝐺‘𝑤) = (𝐹‘(𝐺 ↾ 𝑤))))

Syntax hints:   ∧ wa 104   ↔ wb 105   = wceq 1373   ∈ wcel 2177  {cab 2192  ∀wral 2485  ∃wrex 2486  Vcvv 2773  Oncon0 4418   ↾ cres 4685   Fn wfn 5275  ‘cfv 5280

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2188

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ral 2490  df-rex 2491  df-v 2775  df-un 3174  df-in 3176  df-ss 3183  df-sn 3644  df-pr 3645  df-op 3647  df-uni 3857  df-br 4052  df-opab 4114  df-xp 4689  df-rel 4690  df-cnv 4691  df-co 4692  df-dm 4693  df-res 4695  df-iota 5241  df-fun 5282  df-fn 5283  df-fv 5288

This theorem is referenced by:  tfrlem3  6410  tfrlem5  6413  tfrlemisucaccv  6424  tfrlemibxssdm  6426  tfrlemi14d  6432  tfrexlem  6433