Theorem frrlem2 8177

Description: Lemma for well-founded recursion. An acceptable function is a function. (Contributed by Paul Chapman, 21-Apr-2012.)

Hypothesis

Ref	Expression
frrlem1.1	⊢ 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}

Assertion

Ref	Expression
frrlem2	⊢ (𝑔 ∈ 𝐵 → Fun 𝑔)

Distinct variable groups:   𝐴,𝑓,𝑔,𝑥,𝑦   𝑓,𝐺,𝑔,𝑥,𝑦   𝑅,𝑓,𝑔,𝑥,𝑦

Allowed substitution hints:   𝐵(𝑥,𝑦,𝑓,𝑔)

Proof of Theorem frrlem2

Dummy variables 𝑤 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		frrlem1.1	. . . 4 ⊢ 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝑦𝐺(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
2	1	frrlem1 8176	. . 3 ⊢ 𝐵 = {𝑔 ∣ ∃𝑧(𝑔 Fn 𝑧 ∧ (𝑧 ⊆ 𝐴 ∧ ∀𝑤 ∈ 𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧) ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝑤𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤))))}
3	2	abeq2i 2874	. 2 ⊢ (𝑔 ∈ 𝐵 ↔ ∃𝑧(𝑔 Fn 𝑧 ∧ (𝑧 ⊆ 𝐴 ∧ ∀𝑤 ∈ 𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧) ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝑤𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤)))))
4		fnfun 6589	. . . 4 ⊢ (𝑔 Fn 𝑧 → Fun 𝑔)
5	4	3ad2ant1 1133	. . 3 ⊢ ((𝑔 Fn 𝑧 ∧ (𝑧 ⊆ 𝐴 ∧ ∀𝑤 ∈ 𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧) ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝑤𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤)))) → Fun 𝑔)
6	5	exlimiv 1933	. 2 ⊢ (∃𝑧(𝑔 Fn 𝑧 ∧ (𝑧 ⊆ 𝐴 ∧ ∀𝑤 ∈ 𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧) ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝑤𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤)))) → Fun 𝑔)
7	3, 6	sylbi 216	1 ⊢ (𝑔 ∈ 𝐵 → Fun 𝑔)

Syntax hints:   → wi 4   ∧ wa 397   ∧ w3a 1087   = wceq 1541  ∃wex 1781   ∈ wcel 2106  {cab 2714  ∀wral 3062   ⊆ wss 3901   ↾ cres 5626  Predcpred 6241  Fun wfun 6477   Fn wfn 6478  ‘cfv 6483  (class class class)co 7341

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-12 2171  ax-ext 2708

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3063  df-rab 3405  df-v 3444  df-dif 3904  df-un 3906  df-in 3908  df-ss 3918  df-nul 4274  df-if 4478  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4857  df-br 5097  df-opab 5159  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-res 5636  df-ima 5637  df-pred 6242  df-iota 6435  df-fun 6485  df-fn 6486  df-fv 6491  df-ov 7344

This theorem is referenced by:  frrlem4  8179  frrlem6  8181  fprlem1  8190  frrlem15  9618