Theorem wfrlem3OLD 8125

Description: Lemma for well-ordered recursion. An acceptable function's domain is a subset of 𝐴. Obsolete as of 18-Nov-2024. (New usage is discouraged.) (Proof modification is discouraged.) (Contributed by Scott Fenton, 21-Apr-2011.)

Hypothesis

Ref	Expression
wfrlem1OLD.1	⊢ 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}

Assertion

Ref	Expression
wfrlem3OLD	⊢ (𝑔 ∈ 𝐵 → dom 𝑔 ⊆ 𝐴)

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

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

Proof of Theorem wfrlem3OLD

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

Step	Hyp	Ref	Expression
1		wfrlem1OLD.1	. . . 4 ⊢ 𝐵 = {𝑓 ∣ ∃𝑥(𝑓 Fn 𝑥 ∧ (𝑥 ⊆ 𝐴 ∧ ∀𝑦 ∈ 𝑥 Pred(𝑅, 𝐴, 𝑦) ⊆ 𝑥) ∧ ∀𝑦 ∈ 𝑥 (𝑓‘𝑦) = (𝐹‘(𝑓 ↾ Pred(𝑅, 𝐴, 𝑦))))}
2	1	wfrlem1OLD 8123	. . 3 ⊢ 𝐵 = {𝑔 ∣ ∃𝑧(𝑔 Fn 𝑧 ∧ (𝑧 ⊆ 𝐴 ∧ ∀𝑤 ∈ 𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧) ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤))))}
3	2	abeq2i 2876	. 2 ⊢ (𝑔 ∈ 𝐵 ↔ ∃𝑧(𝑔 Fn 𝑧 ∧ (𝑧 ⊆ 𝐴 ∧ ∀𝑤 ∈ 𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧) ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤)))))
4		fndm 6532	. . . . . . 7 ⊢ (𝑔 Fn 𝑧 → dom 𝑔 = 𝑧)
5	4	sseq1d 3956	. . . . . 6 ⊢ (𝑔 Fn 𝑧 → (dom 𝑔 ⊆ 𝐴 ↔ 𝑧 ⊆ 𝐴))
6	5	biimpar 477	. . . . 5 ⊢ ((𝑔 Fn 𝑧 ∧ 𝑧 ⊆ 𝐴) → dom 𝑔 ⊆ 𝐴)
7	6	adantrr 713	. . . 4 ⊢ ((𝑔 Fn 𝑧 ∧ (𝑧 ⊆ 𝐴 ∧ ∀𝑤 ∈ 𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧)) → dom 𝑔 ⊆ 𝐴)
8	7	3adant3 1130	. . 3 ⊢ ((𝑔 Fn 𝑧 ∧ (𝑧 ⊆ 𝐴 ∧ ∀𝑤 ∈ 𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧) ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤)))) → dom 𝑔 ⊆ 𝐴)
9	8	exlimiv 1936	. 2 ⊢ (∃𝑧(𝑔 Fn 𝑧 ∧ (𝑧 ⊆ 𝐴 ∧ ∀𝑤 ∈ 𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧) ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝐹‘(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤)))) → dom 𝑔 ⊆ 𝐴)
10	3, 9	sylbi 216	1 ⊢ (𝑔 ∈ 𝐵 → dom 𝑔 ⊆ 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1085   = wceq 1541  ∃wex 1785   ∈ wcel 2109  {cab 2716  ∀wral 3065   ⊆ wss 3891  dom cdm 5588   ↾ cres 5590  Predcpred 6198   Fn wfn 6425  ‘cfv 6430

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1801  ax-4 1815  ax-5 1916  ax-6 1974  ax-7 2014  ax-8 2111  ax-9 2119  ax-12 2174  ax-ext 2710

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 844  df-3an 1087  df-tru 1544  df-fal 1554  df-ex 1786  df-sb 2071  df-clab 2717  df-cleq 2731  df-clel 2817  df-ral 3070  df-rab 3074  df-v 3432  df-dif 3894  df-un 3896  df-in 3898  df-ss 3908  df-nul 4262  df-if 4465  df-sn 4567  df-pr 4569  df-op 4573  df-uni 4845  df-br 5079  df-opab 5141  df-xp 5594  df-rel 5595  df-cnv 5596  df-co 5597  df-dm 5598  df-rn 5599  df-res 5600  df-ima 5601  df-pred 6199  df-iota 6388  df-fun 6432  df-fn 6433  df-fv 6438

This theorem is referenced by:  wfrlem5OLD  8128  wfrdmssOLD  8130