Theorem frrlem5b 31029

Description: Lemma for founded recursion. The union of a subclass of 𝐵 is a relationship. (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
frrlem5b	⊢ (𝐶 ⊆ 𝐵 → Rel ∪ 𝐶)

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 frrlem5b

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		ssel 3562	. . . 4 ⊢ (𝐶 ⊆ 𝐵 → (𝑧 ∈ 𝐶 → 𝑧 ∈ 𝐵))
2		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)))))}
3	2	frrlem2 31025	. . . . 5 ⊢ (𝑧 ∈ 𝐵 → Fun 𝑧)
4		funrel 5821	. . . . 5 ⊢ (Fun 𝑧 → Rel 𝑧)
5	3, 4	syl 17	. . . 4 ⊢ (𝑧 ∈ 𝐵 → Rel 𝑧)
6	1, 5	syl6 34	. . 3 ⊢ (𝐶 ⊆ 𝐵 → (𝑧 ∈ 𝐶 → Rel 𝑧))
7	6	ralrimiv 2948	. 2 ⊢ (𝐶 ⊆ 𝐵 → ∀𝑧 ∈ 𝐶 Rel 𝑧)
8		reluni 5164	. 2 ⊢ (Rel ∪ 𝐶 ↔ ∀𝑧 ∈ 𝐶 Rel 𝑧)
9	7, 8	sylibr 223	1 ⊢ (𝐶 ⊆ 𝐵 → Rel ∪ 𝐶)

Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031   = wceq 1475  ∃wex 1695   ∈ wcel 1977  {cab 2596  ∀wral 2896   ⊆ wss 3540  ∪ cuni 4372   Fr wfr 4994   Se wse 4995   ↾ cres 5040  Rel wrel 5043  Predcpred 5596  Fun wfun 5798   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:  frrlem5c  31030