Theorem tfrlem3 8008

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.)

Hypothesis

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

Assertion

Ref	Expression
tfrlem3	⊢ 𝐴 = {𝑔 ∣ ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ 𝑤)))}

Distinct variable groups:   𝐴,𝑔   𝑓,𝑔,𝑤,𝑥,𝑦,𝑧,𝐹

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

Proof of Theorem tfrlem3

Step	Hyp	Ref	Expression
1		tfrlem3.1	. . 3 ⊢ 𝐴 = {𝑓 ∣ ∃𝑥 ∈ On (𝑓 Fn 𝑥 ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ 𝑦)))}
2		vex 3497	. . 3 ⊢ 𝑔 ∈ V
3	1, 2	tfrlem3a 8007	. 2 ⊢ (𝑔 ∈ 𝐴 ↔ ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ 𝑤))))
4	3	abbi2i 2953	1 ⊢ 𝐴 = {𝑔 ∣ ∃𝑧 ∈ On (𝑔 Fn 𝑧 ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ 𝑤)))}

Syntax hints:   ∧ wa 398   = wceq 1533  {cab 2799  ∀wral 3138  ∃wrex 3139   ↾ cres 5551  Oncon0 6185   Fn wfn 6344  ‘cfv 6349

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2157  ax-12 2173  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-dif 3938  df-un 3940  df-in 3942  df-ss 3951  df-nul 4291  df-if 4467  df-sn 4561  df-pr 4563  df-op 4567  df-uni 4832  df-br 5059  df-opab 5121  df-xp 5555  df-rel 5556  df-cnv 5557  df-co 5558  df-dm 5559  df-res 5561  df-iota 6308  df-fun 6351  df-fn 6352  df-fv 6357

This theorem is referenced by:  tfrlem4  8009  tfrlem8  8014  rdglem1  8045