Theorem frrlem5c 31786

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 4458	. 2 ⊢ (𝐶 ⊆ 𝐵 → ∪ 𝐶 ⊆ ∪ 𝐵)
2		ssid 3624	. . . 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 31785	. . . 4 ⊢ (𝐵 ⊆ 𝐵 → Rel ∪ 𝐵)
7	2, 6	ax-mp 5	. . 3 ⊢ Rel ∪ 𝐵
8		eluni 4439	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑢⟩ ∈ ∪ 𝐵 ↔ ∃𝑔(⟨𝑥, 𝑢⟩ ∈ 𝑔 ∧ 𝑔 ∈ 𝐵))
9		df-br 4654	. . . . . . . . 9 ⊢ (𝑥∪ 𝐵𝑢 ↔ ⟨𝑥, 𝑢⟩ ∈ ∪ 𝐵)
10		df-br 4654	. . . . . . . . . . 11 ⊢ (𝑥𝑔𝑢 ↔ ⟨𝑥, 𝑢⟩ ∈ 𝑔)
11	10	anbi1i 731	. . . . . . . . . 10 ⊢ ((𝑥𝑔𝑢 ∧ 𝑔 ∈ 𝐵) ↔ (⟨𝑥, 𝑢⟩ ∈ 𝑔 ∧ 𝑔 ∈ 𝐵))
12	11	exbii 1774	. . . . . . . . 9 ⊢ (∃𝑔(𝑥𝑔𝑢 ∧ 𝑔 ∈ 𝐵) ↔ ∃𝑔(⟨𝑥, 𝑢⟩ ∈ 𝑔 ∧ 𝑔 ∈ 𝐵))
13	8, 9, 12	3bitr4i 292	. . . . . . . 8 ⊢ (𝑥∪ 𝐵𝑢 ↔ ∃𝑔(𝑥𝑔𝑢 ∧ 𝑔 ∈ 𝐵))
14		eluni 4439	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑣⟩ ∈ ∪ 𝐵 ↔ ∃ℎ(⟨𝑥, 𝑣⟩ ∈ ℎ ∧ ℎ ∈ 𝐵))
15		df-br 4654	. . . . . . . . 9 ⊢ (𝑥∪ 𝐵𝑣 ↔ ⟨𝑥, 𝑣⟩ ∈ ∪ 𝐵)
16		df-br 4654	. . . . . . . . . . 11 ⊢ (𝑥ℎ𝑣 ↔ ⟨𝑥, 𝑣⟩ ∈ ℎ)
17	16	anbi1i 731	. . . . . . . . . 10 ⊢ ((𝑥ℎ𝑣 ∧ ℎ ∈ 𝐵) ↔ (⟨𝑥, 𝑣⟩ ∈ ℎ ∧ ℎ ∈ 𝐵))
18	17	exbii 1774	. . . . . . . . 9 ⊢ (∃ℎ(𝑥ℎ𝑣 ∧ ℎ ∈ 𝐵) ↔ ∃ℎ(⟨𝑥, 𝑣⟩ ∈ ℎ ∧ ℎ ∈ 𝐵))
19	14, 15, 18	3bitr4i 292	. . . . . . . 8 ⊢ (𝑥∪ 𝐵𝑣 ↔ ∃ℎ(𝑥ℎ𝑣 ∧ ℎ ∈ 𝐵))
20	13, 19	anbi12i 733	. . . . . . 7 ⊢ ((𝑥∪ 𝐵𝑢 ∧ 𝑥∪ 𝐵𝑣) ↔ (∃𝑔(𝑥𝑔𝑢 ∧ 𝑔 ∈ 𝐵) ∧ ∃ℎ(𝑥ℎ𝑣 ∧ ℎ ∈ 𝐵)))
21		eeanv 2182	. . . . . . 7 ⊢ (∃𝑔∃ℎ((𝑥𝑔𝑢 ∧ 𝑔 ∈ 𝐵) ∧ (𝑥ℎ𝑣 ∧ ℎ ∈ 𝐵)) ↔ (∃𝑔(𝑥𝑔𝑢 ∧ 𝑔 ∈ 𝐵) ∧ ∃ℎ(𝑥ℎ𝑣 ∧ ℎ ∈ 𝐵)))
22	20, 21	bitr4i 267	. . . . . 6 ⊢ ((𝑥∪ 𝐵𝑢 ∧ 𝑥∪ 𝐵𝑣) ↔ ∃𝑔∃ℎ((𝑥𝑔𝑢 ∧ 𝑔 ∈ 𝐵) ∧ (𝑥ℎ𝑣 ∧ ℎ ∈ 𝐵)))
23	3, 4, 5	frrlem5 31784	. . . . . . . . 9 ⊢ ((𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) → ((𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣) → 𝑢 = 𝑣))
24	23	impcom 446	. . . . . . . 8 ⊢ (((𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣) ∧ (𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵)) → 𝑢 = 𝑣)
25	24	an4s 869	. . . . . . 7 ⊢ (((𝑥𝑔𝑢 ∧ 𝑔 ∈ 𝐵) ∧ (𝑥ℎ𝑣 ∧ ℎ ∈ 𝐵)) → 𝑢 = 𝑣)
26	25	exlimivv 1860	. . . . . 6 ⊢ (∃𝑔∃ℎ((𝑥𝑔𝑢 ∧ 𝑔 ∈ 𝐵) ∧ (𝑥ℎ𝑣 ∧ ℎ ∈ 𝐵)) → 𝑢 = 𝑣)
27	22, 26	sylbi 207	. . . . 5 ⊢ ((𝑥∪ 𝐵𝑢 ∧ 𝑥∪ 𝐵𝑣) → 𝑢 = 𝑣)
28	27	ax-gen 1722	. . . 4 ⊢ ∀𝑣((𝑥∪ 𝐵𝑢 ∧ 𝑥∪ 𝐵𝑣) → 𝑢 = 𝑣)
29	28	gen2 1723	. . 3 ⊢ ∀𝑥∀𝑢∀𝑣((𝑥∪ 𝐵𝑢 ∧ 𝑥∪ 𝐵𝑣) → 𝑢 = 𝑣)
30		dffun2 5898	. . 3 ⊢ (Fun ∪ 𝐵 ↔ (Rel ∪ 𝐵 ∧ ∀𝑥∀𝑢∀𝑣((𝑥∪ 𝐵𝑢 ∧ 𝑥∪ 𝐵𝑣) → 𝑢 = 𝑣)))
31	7, 29, 30	mpbir2an 955	. 2 ⊢ Fun ∪ 𝐵
32		funss 5907	. 2 ⊢ (∪ 𝐶 ⊆ ∪ 𝐵 → (Fun ∪ 𝐵 → Fun ∪ 𝐶))
33	1, 31, 32	mpisyl 21	1 ⊢ (𝐶 ⊆ 𝐵 → Fun ∪ 𝐶)

Syntax hints:   → wi 4   ∧ wa 384   ∧ w3a 1037  ∀wal 1481   = wceq 1483  ∃wex 1704   ∈ wcel 1990  {cab 2608  ∀wral 2912   ⊆ wss 3574  ⟨cop 4183  ∪ cuni 4436   class class class wbr 4653   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-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-rep 4771  ax-sep 4781  ax-nul 4789  ax-pow 4843  ax-pr 4906  ax-un 6949  ax-inf2 8538

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-reu 2919  df-rab 2921  df-v 3202  df-sbc 3436  df-csb 3534  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-pw 4160  df-sn 4178  df-pr 4180  df-tp 4182  df-op 4184  df-uni 4437  df-iun 4522  df-br 4654  df-opab 4713  df-mpt 4730  df-tr 4753  df-id 5024  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-se 5074  df-we 5075  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-ord 5726  df-on 5727  df-lim 5728  df-suc 5729  df-iota 5851  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-fv 5896  df-ov 6653  df-om 7066  df-wrecs 7407  df-recs 7468  df-rdg 7506  df-trpred 31718

This theorem is referenced by:  frrlem10  31791