Theorem frrlem3 8298

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 8296	. . 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 6660	. . . . . . 7 ⊢ (𝑔 Fn 𝑧 → dom 𝑔 = 𝑧)
5	4	sseq1d 4011	. . . . . 6 ⊢ (𝑔 Fn 𝑧 → (dom 𝑔 ⊆ 𝐴 ↔ 𝑧 ⊆ 𝐴))
6	5	biimpar 476	. . . . 5 ⊢ ((𝑔 Fn 𝑧 ∧ 𝑧 ⊆ 𝐴) → dom 𝑔 ⊆ 𝐴)
7	6	adantrr 715	. . . 4 ⊢ ((𝑔 Fn 𝑧 ∧ (𝑧 ⊆ 𝐴 ∧ ∀𝑤 ∈ 𝑧 Pred(𝑅, 𝐴, 𝑤) ⊆ 𝑧)) → dom 𝑔 ⊆ 𝐴)
8	7	3adant3 1129	. . 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 1925	. 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 394   ∧ w3a 1084   = wceq 1533  ∃wex 1773   ∈ wcel 2098  {cab 2704  ∀wral 3057   ⊆ wss 3947  dom cdm 5680   ↾ cres 5682  Predcpred 6307   Fn wfn 6546  ‘cfv 6551  (class class class)co 7424

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-12 2166  ax-ext 2698

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2705  df-cleq 2719  df-clel 2805  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3473  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4325  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4911  df-br 5151  df-opab 5213  df-xp 5686  df-rel 5687  df-cnv 5688  df-co 5689  df-dm 5690  df-rn 5691  df-res 5692  df-ima 5693  df-pred 6308  df-iota 6503  df-fun 6553  df-fn 6554  df-fv 6559  df-ov 7427

This theorem is referenced by:  frrlem7  8302  fprlem1  8310  frrlem15  9786