Theorem frrlem5d 31031

Description: Lemma for founded recursion. The domain of the union of a subset of 𝐵 is a subset of 𝐴. (Contributed by Paul Chapman, 29-Apr-2012.)

Hypotheses

Ref	Expression
frrlem5.1	⊢ 𝑅 Fr 𝐴
frrlem5.2	⊢ 𝑅 Se 𝐴
frrlem5.3	⊢ 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
frrlem5d	⊢ (𝐶 ⊆ 𝐵 → dom ∪ 𝐶 ⊆ 𝐴)

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

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

Proof of Theorem frrlem5d

Dummy variable 𝑔 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		dmuni 5256	. 2 ⊢ dom ∪ 𝐶 = ∪ 𝑔 ∈ 𝐶 dom 𝑔
2		ssel 3562	. . . . 5 ⊢ (𝐶 ⊆ 𝐵 → (𝑔 ∈ 𝐶 → 𝑔 ∈ 𝐵))
3		frrlem5.3	. . . . . 6 ⊢ 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)))))}
4	3	frrlem3 31026	. . . . 5 ⊢ (𝑔 ∈ 𝐵 → dom 𝑔 ⊆ 𝐴)
5	2, 4	syl6 34	. . . 4 ⊢ (𝐶 ⊆ 𝐵 → (𝑔 ∈ 𝐶 → dom 𝑔 ⊆ 𝐴))
6	5	ralrimiv 2948	. . 3 ⊢ (𝐶 ⊆ 𝐵 → ∀𝑔 ∈ 𝐶 dom 𝑔 ⊆ 𝐴)
7		iunss 4497	. . 3 ⊢ (∪ 𝑔 ∈ 𝐶 dom 𝑔 ⊆ 𝐴 ↔ ∀𝑔 ∈ 𝐶 dom 𝑔 ⊆ 𝐴)
8	6, 7	sylibr 223	. 2 ⊢ (𝐶 ⊆ 𝐵 → ∪ 𝑔 ∈ 𝐶 dom 𝑔 ⊆ 𝐴)
9	1, 8	syl5eqss 3612	1 ⊢ (𝐶 ⊆ 𝐵 → dom ∪ 𝐶 ⊆ 𝐴)

Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977  {cab 2596  ∀wral 2896   ⊆ wss 3540  ∪ cuni 4372  ∪ ciun 4455   Fr wfr 4994   Se wse 4995  dom cdm 5038   ↾ cres 5040  Predcpred 5596   Fn wfn 5799  ‘cfv 5804  (class class class)co 6549

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-pred 5597  df-iota 5768  df-fun 5806  df-fn 5807  df-fv 5812  df-ov 6552

This theorem is referenced by: (None)