Theorem frrlem5b 31785

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 3597	. . . 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 31781	. . . . 5 ⊢ (𝑧 ∈ 𝐵 → Fun 𝑧)
4		funrel 5905	. . . . 5 ⊢ (Fun 𝑧 → Rel 𝑧)
5	3, 4	syl 17	. . . 4 ⊢ (𝑧 ∈ 𝐵 → Rel 𝑧)
6	1, 5	syl6 35	. . 3 ⊢ (𝐶 ⊆ 𝐵 → (𝑧 ∈ 𝐶 → Rel 𝑧))
7	6	ralrimiv 2965	. 2 ⊢ (𝐶 ⊆ 𝐵 → ∀𝑧 ∈ 𝐶 Rel 𝑧)
8		reluni 5241	. 2 ⊢ (Rel ∪ 𝐶 ↔ ∀𝑧 ∈ 𝐶 Rel 𝑧)
9	7, 8	sylibr 224	1 ⊢ (𝐶 ⊆ 𝐵 → Rel ∪ 𝐶)

Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1037   = wceq 1483  ∃wex 1704   ∈ wcel 1990  {cab 2608  ∀wral 2912   ⊆ wss 3574  ∪ cuni 4436   Fr wfr 5070   Se wse 5071   ↾ cres 5116  Rel wrel 5119  Predcpred 5679  Fun wfun 5882   Fn wfn 5883  ‘cfv 5888  (class class class)co 6650

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-res 5126  df-ima 5127  df-pred 5680  df-iota 5851  df-fun 5890  df-fn 5891  df-fv 5896  df-ov 6653

This theorem is referenced by:  frrlem5c  31786