Theorem frrlem7 8318

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 8316	. . . . . 6 ⊢ 𝐹 = ∪ 𝐵
4	3	dmeqi 5914	. . . . 5 ⊢ dom 𝐹 = dom ∪ 𝐵
5		dmuni 5924	. . . . 5 ⊢ dom ∪ 𝐵 = ∪ 𝑔 ∈ 𝐵 dom 𝑔
6	4, 5	eqtri 2764	. . . 4 ⊢ dom 𝐹 = ∪ 𝑔 ∈ 𝐵 dom 𝑔
7	6	sseq1i 4011	. . 3 ⊢ (dom 𝐹 ⊆ 𝐴 ↔ ∪ 𝑔 ∈ 𝐵 dom 𝑔 ⊆ 𝐴)
8		iunss 5044	. . 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 8314	. 2 ⊢ (𝑔 ∈ 𝐵 → dom 𝑔 ⊆ 𝐴)
11	9, 10	mprgbir 3067	1 ⊢ dom 𝐹 ⊆ 𝐴

Syntax hints:   ∧ wa 395   ∧ w3a 1086   = wceq 1539  ∃wex 1778  {cab 2713  ∀wral 3060   ⊆ wss 3950  ∪ cuni 4906  ∪ ciun 4990  dom cdm 5684   ↾ cres 5686  Predcpred 6319   Fn wfn 6555  ‘cfv 6560  (class class class)co 7432  frecscfrecs 8306

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-11 2156  ax-12 2176  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-nfc 2891  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-iun 4992  df-br 5143  df-opab 5205  df-xp 5690  df-rel 5691  df-cnv 5692  df-co 5693  df-dm 5694  df-rn 5695  df-res 5696  df-ima 5697  df-pred 6320  df-iota 6513  df-fun 6562  df-fn 6563  df-fv 6568  df-ov 7435  df-frecs 8307

This theorem is referenced by:  frrlem14  8325  frrdmss  8333