Theorem frrlem5c 31030

Description: Lemma for founded recursion. The union of a subclass of 𝐵 is a function. (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
frrlem5c	⊢ (𝐶 ⊆ 𝐵 → Fun ∪ 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 frrlem5c

Dummy variables 𝑔 ℎ 𝑢 𝑣 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		uniss 4394	. 2 ⊢ (𝐶 ⊆ 𝐵 → ∪ 𝐶 ⊆ ∪ 𝐵)
2		ssid 3587	. . . 4 ⊢ 𝐵 ⊆ 𝐵
3		frrlem5.1	. . . . 5 ⊢ 𝑅 Fr 𝐴
4		frrlem5.2	. . . . 5 ⊢ 𝑅 Se 𝐴
5		frrlem5.3	. . . . 5 ⊢ 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)))))}
6	3, 4, 5	frrlem5b 31029	. . . 4 ⊢ (𝐵 ⊆ 𝐵 → Rel ∪ 𝐵)
7	2, 6	ax-mp 5	. . 3 ⊢ Rel ∪ 𝐵
8		eluni 4375	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑢⟩ ∈ ∪ 𝐵 ↔ ∃𝑔(⟨𝑥, 𝑢⟩ ∈ 𝑔 ∧ 𝑔 ∈ 𝐵))
9		df-br 4584	. . . . . . . . 9 ⊢ (𝑥∪ 𝐵𝑢 ↔ ⟨𝑥, 𝑢⟩ ∈ ∪ 𝐵)
10		df-br 4584	. . . . . . . . . . 11 ⊢ (𝑥𝑔𝑢 ↔ ⟨𝑥, 𝑢⟩ ∈ 𝑔)
11	10	anbi1i 727	. . . . . . . . . 10 ⊢ ((𝑥𝑔𝑢 ∧ 𝑔 ∈ 𝐵) ↔ (⟨𝑥, 𝑢⟩ ∈ 𝑔 ∧ 𝑔 ∈ 𝐵))
12	11	exbii 1764	. . . . . . . . 9 ⊢ (∃𝑔(𝑥𝑔𝑢 ∧ 𝑔 ∈ 𝐵) ↔ ∃𝑔(⟨𝑥, 𝑢⟩ ∈ 𝑔 ∧ 𝑔 ∈ 𝐵))
13	8, 9, 12	3bitr4i 291	. . . . . . . 8 ⊢ (𝑥∪ 𝐵𝑢 ↔ ∃𝑔(𝑥𝑔𝑢 ∧ 𝑔 ∈ 𝐵))
14		eluni 4375	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑣⟩ ∈ ∪ 𝐵 ↔ ∃ℎ(⟨𝑥, 𝑣⟩ ∈ ℎ ∧ ℎ ∈ 𝐵))
15		df-br 4584	. . . . . . . . 9 ⊢ (𝑥∪ 𝐵𝑣 ↔ ⟨𝑥, 𝑣⟩ ∈ ∪ 𝐵)
16		df-br 4584	. . . . . . . . . . 11 ⊢ (𝑥ℎ𝑣 ↔ ⟨𝑥, 𝑣⟩ ∈ ℎ)
17	16	anbi1i 727	. . . . . . . . . 10 ⊢ ((𝑥ℎ𝑣 ∧ ℎ ∈ 𝐵) ↔ (⟨𝑥, 𝑣⟩ ∈ ℎ ∧ ℎ ∈ 𝐵))
18	17	exbii 1764	. . . . . . . . 9 ⊢ (∃ℎ(𝑥ℎ𝑣 ∧ ℎ ∈ 𝐵) ↔ ∃ℎ(⟨𝑥, 𝑣⟩ ∈ ℎ ∧ ℎ ∈ 𝐵))
19	14, 15, 18	3bitr4i 291	. . . . . . . 8 ⊢ (𝑥∪ 𝐵𝑣 ↔ ∃ℎ(𝑥ℎ𝑣 ∧ ℎ ∈ 𝐵))
20	13, 19	anbi12i 729	. . . . . . 7 ⊢ ((𝑥∪ 𝐵𝑢 ∧ 𝑥∪ 𝐵𝑣) ↔ (∃𝑔(𝑥𝑔𝑢 ∧ 𝑔 ∈ 𝐵) ∧ ∃ℎ(𝑥ℎ𝑣 ∧ ℎ ∈ 𝐵)))
21		eeanv 2170	. . . . . . 7 ⊢ (∃𝑔∃ℎ((𝑥𝑔𝑢 ∧ 𝑔 ∈ 𝐵) ∧ (𝑥ℎ𝑣 ∧ ℎ ∈ 𝐵)) ↔ (∃𝑔(𝑥𝑔𝑢 ∧ 𝑔 ∈ 𝐵) ∧ ∃ℎ(𝑥ℎ𝑣 ∧ ℎ ∈ 𝐵)))
22	20, 21	bitr4i 266	. . . . . 6 ⊢ ((𝑥∪ 𝐵𝑢 ∧ 𝑥∪ 𝐵𝑣) ↔ ∃𝑔∃ℎ((𝑥𝑔𝑢 ∧ 𝑔 ∈ 𝐵) ∧ (𝑥ℎ𝑣 ∧ ℎ ∈ 𝐵)))
23	3, 4, 5	frrlem5 31028	. . . . . . . . 9 ⊢ ((𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) → ((𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣) → 𝑢 = 𝑣))
24	23	impcom 445	. . . . . . . 8 ⊢ (((𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣) ∧ (𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵)) → 𝑢 = 𝑣)
25	24	an4s 865	. . . . . . 7 ⊢ (((𝑥𝑔𝑢 ∧ 𝑔 ∈ 𝐵) ∧ (𝑥ℎ𝑣 ∧ ℎ ∈ 𝐵)) → 𝑢 = 𝑣)
26	25	exlimivv 1847	. . . . . 6 ⊢ (∃𝑔∃ℎ((𝑥𝑔𝑢 ∧ 𝑔 ∈ 𝐵) ∧ (𝑥ℎ𝑣 ∧ ℎ ∈ 𝐵)) → 𝑢 = 𝑣)
27	22, 26	sylbi 206	. . . . 5 ⊢ ((𝑥∪ 𝐵𝑢 ∧ 𝑥∪ 𝐵𝑣) → 𝑢 = 𝑣)
28	27	ax-gen 1713	. . . 4 ⊢ ∀𝑣((𝑥∪ 𝐵𝑢 ∧ 𝑥∪ 𝐵𝑣) → 𝑢 = 𝑣)
29	28	gen2 1714	. . 3 ⊢ ∀𝑥∀𝑢∀𝑣((𝑥∪ 𝐵𝑢 ∧ 𝑥∪ 𝐵𝑣) → 𝑢 = 𝑣)
30		dffun2 5814	. . 3 ⊢ (Fun ∪ 𝐵 ↔ (Rel ∪ 𝐵 ∧ ∀𝑥∀𝑢∀𝑣((𝑥∪ 𝐵𝑢 ∧ 𝑥∪ 𝐵𝑣) → 𝑢 = 𝑣)))
31	7, 29, 30	mpbir2an 957	. 2 ⊢ Fun ∪ 𝐵
32		funss 5822	. 2 ⊢ (∪ 𝐶 ⊆ ∪ 𝐵 → (Fun ∪ 𝐵 → Fun ∪ 𝐶))
33	1, 31, 32	mpisyl 21	1 ⊢ (𝐶 ⊆ 𝐵 → Fun ∪ 𝐶)

Syntax hints:   → wi 4   ∧ wa 383   ∧ w3a 1031  ∀wal 1473   = wceq 1475  ∃wex 1695   ∈ wcel 1977  {cab 2596  ∀wral 2896   ⊆ wss 3540  ⟨cop 4131  ∪ cuni 4372   class class class wbr 4583   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-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-inf2 8421

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-se 4998  df-we 4999  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-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-ov 6552  df-om 6958  df-wrecs 7294  df-recs 7355  df-rdg 7393  df-trpred 30962

This theorem is referenced by:  frrlem10  31035