Theorem frrlem7 8333

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 8331	. . . . . 6 ⊢ 𝐹 = ∪ 𝐵
4	3	dmeqi 5929	. . . . 5 ⊢ dom 𝐹 = dom ∪ 𝐵
5		dmuni 5939	. . . . 5 ⊢ dom ∪ 𝐵 = ∪ 𝑔 ∈ 𝐵 dom 𝑔
6	4, 5	eqtri 2768	. . . 4 ⊢ dom 𝐹 = ∪ 𝑔 ∈ 𝐵 dom 𝑔
7	6	sseq1i 4037	. . 3 ⊢ (dom 𝐹 ⊆ 𝐴 ↔ ∪ 𝑔 ∈ 𝐵 dom 𝑔 ⊆ 𝐴)
8		iunss 5068	. . 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 8329	. 2 ⊢ (𝑔 ∈ 𝐵 → dom 𝑔 ⊆ 𝐴)
11	9, 10	mprgbir 3074	1 ⊢ dom 𝐹 ⊆ 𝐴

Syntax hints:   ∧ wa 395   ∧ w3a 1087   = wceq 1537  ∃wex 1777  {cab 2717  ∀wral 3067   ⊆ wss 3976  ∪ cuni 4931  ∪ ciun 5015  dom cdm 5700   ↾ cres 5702  Predcpred 6331   Fn wfn 6568  ‘cfv 6573  (class class class)co 7448  frecscfrecs 8321

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-iun 5017  df-br 5167  df-opab 5229  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-rn 5711  df-res 5712  df-ima 5713  df-pred 6332  df-iota 6525  df-fun 6575  df-fn 6576  df-fv 6581  df-ov 7451  df-frecs 8322

This theorem is referenced by:  frrlem14  8340  frrdmss  8348