Theorem frrlem3 8287

Description: Lemma for well-founded recursion. An acceptable function's domain is a subset of 𝐴. (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
frrlem3	⊢ (𝑔 ∈ 𝐵 → dom 𝑔 ⊆ 𝐴)

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 frrlem3

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 8285	. . 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	eqabri 2872	. 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		fndm 6651	. . . . . . 7 ⊢ (𝑔 Fn 𝑧 → dom 𝑔 = 𝑧)
5	4	sseq1d 4009	. . . . . 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 716	. . . 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) = (𝑤𝐺(𝑔 ↾ Pred(𝑅, 𝐴, 𝑤)))) → dom 𝑔 ⊆ 𝐴)
9	8	exlimiv 1926	. 2 ⊢ (∃𝑧(𝑔 Fn 𝑧 ∧ (𝑧 ⊆ 𝐴 ∧ ∀𝑤 ∈ 𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧) ∧ ∀𝑤 ∈ 𝑧 (𝑔‘𝑤) = (𝑤𝐺(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 1534  ∃wex 1774   ∈ wcel 2099  {cab 2704  ∀wral 3056   ⊆ wss 3944  dom cdm 5672   ↾ cres 5674  Predcpred 6298   Fn wfn 6537  ‘cfv 6542  (class class class)co 7414

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-12 2164  ax-ext 2698

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2705  df-cleq 2719  df-clel 2805  df-ral 3057  df-rex 3066  df-rab 3428  df-v 3471  df-dif 3947  df-un 3949  df-in 3951  df-ss 3961  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5143  df-opab 5205  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-pred 6299  df-iota 6494  df-fun 6544  df-fn 6545  df-fv 6550  df-ov 7417

This theorem is referenced by:  frrlem7  8291  fprlem1  8299  frrlem15  9772