Theorem frrlem7 8232

Description: Lemma for well-founded recursion. The well-founded recursion generator's domain is a subclass of 𝐴. (Contributed by Scott Fenton, 27-Aug-2022.)

Hypotheses

Ref	Expression
frrlem5.1	⊢ 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))))}
frrlem5.2	⊢ 𝐹 = frecs(𝑅, 𝐴, 𝐺)

Assertion

Ref	Expression
frrlem7	⊢ dom 𝐹 ⊆ 𝐴

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

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

Proof of Theorem frrlem7

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		frrlem5.1	. . . . . . 7 ⊢ 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		frrlem5.2	. . . . . . 7 ⊢ 𝐹 = frecs(𝑅, 𝐴, 𝐺)
3	1, 2	frrlem5 8230	. . . . . 6 ⊢ 𝐹 = ∪ 𝐵
4	3	dmeqi 5851	. . . . 5 ⊢ dom 𝐹 = dom ∪ 𝐵
5		dmuni 5861	. . . . 5 ⊢ dom ∪ 𝐵 = ∪ 𝑔 ∈ 𝐵 dom 𝑔
6	4, 5	eqtri 2757	. . . 4 ⊢ dom 𝐹 = ∪ 𝑔 ∈ 𝐵 dom 𝑔
7	6	sseq1i 3960	. . 3 ⊢ (dom 𝐹 ⊆ 𝐴 ↔ ∪ 𝑔 ∈ 𝐵 dom 𝑔 ⊆ 𝐴)
8		iunss 4998	. . 3 ⊢ (∪ 𝑔 ∈ 𝐵 dom 𝑔 ⊆ 𝐴 ↔ ∀𝑔 ∈ 𝐵 dom 𝑔 ⊆ 𝐴)
9	7, 8	bitri 275	. 2 ⊢ (dom 𝐹 ⊆ 𝐴 ↔ ∀𝑔 ∈ 𝐵 dom 𝑔 ⊆ 𝐴)
10	1	frrlem3 8228	. 2 ⊢ (𝑔 ∈ 𝐵 → dom 𝑔 ⊆ 𝐴)
11	9, 10	mprgbir 3056	1 ⊢ dom 𝐹 ⊆ 𝐴

Syntax hints:   ∧ wa 395   ∧ w3a 1086   = wceq 1541  ∃wex 1780  {cab 2712  ∀wral 3049   ⊆ wss 3899  ∪ cuni 4861  ∪ ciun 4944  dom cdm 5622   ↾ cres 5624  Predcpred 6256   Fn wfn 6485  ‘cfv 6490  (class class class)co 7356  frecscfrecs 8220

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-11 2162  ax-12 2182  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-clab 2713  df-cleq 2726  df-clel 2809  df-ral 3050  df-rex 3059  df-rab 3398  df-v 3440  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-nul 4284  df-if 4478  df-sn 4579  df-pr 4581  df-op 4585  df-uni 4862  df-iun 4946  df-br 5097  df-opab 5159  df-xp 5628  df-rel 5629  df-cnv 5630  df-co 5631  df-dm 5632  df-rn 5633  df-res 5634  df-ima 5635  df-pred 6257  df-iota 6446  df-fun 6492  df-fn 6493  df-fv 6498  df-ov 7359  df-frecs 8221

This theorem is referenced by:  frrlem14  8239  frrdmss  8247