Theorem frrlem5c 32123

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 4595	. 2 ⊢ (𝐶 ⊆ 𝐵 → ∪ 𝐶 ⊆ ∪ 𝐵)
2		ssid 3773	. . . 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 32122	. . . 4 ⊢ (𝐵 ⊆ 𝐵 → Rel ∪ 𝐵)
7	2, 6	ax-mp 5	. . 3 ⊢ Rel ∪ 𝐵
8		eluni 4577	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑢⟩ ∈ ∪ 𝐵 ↔ ∃𝑔(⟨𝑥, 𝑢⟩ ∈ 𝑔 ∧ 𝑔 ∈ 𝐵))
9		df-br 4787	. . . . . . . . 9 ⊢ (𝑥∪ 𝐵𝑢 ↔ ⟨𝑥, 𝑢⟩ ∈ ∪ 𝐵)
10		df-br 4787	. . . . . . . . . . 11 ⊢ (𝑥𝑔𝑢 ↔ ⟨𝑥, 𝑢⟩ ∈ 𝑔)
11	10	anbi1i 610	. . . . . . . . . 10 ⊢ ((𝑥𝑔𝑢 ∧ 𝑔 ∈ 𝐵) ↔ (⟨𝑥, 𝑢⟩ ∈ 𝑔 ∧ 𝑔 ∈ 𝐵))
12	11	exbii 1924	. . . . . . . . 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 4577	. . . . . . . . 9 ⊢ (⟨𝑥, 𝑣⟩ ∈ ∪ 𝐵 ↔ ∃ℎ(⟨𝑥, 𝑣⟩ ∈ ℎ ∧ ℎ ∈ 𝐵))
15		df-br 4787	. . . . . . . . 9 ⊢ (𝑥∪ 𝐵𝑣 ↔ ⟨𝑥, 𝑣⟩ ∈ ∪ 𝐵)
16		df-br 4787	. . . . . . . . . . 11 ⊢ (𝑥ℎ𝑣 ↔ ⟨𝑥, 𝑣⟩ ∈ ℎ)
17	16	anbi1i 610	. . . . . . . . . 10 ⊢ ((𝑥ℎ𝑣 ∧ ℎ ∈ 𝐵) ↔ (⟨𝑥, 𝑣⟩ ∈ ℎ ∧ ℎ ∈ 𝐵))
18	17	exbii 1924	. . . . . . . . 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 612	. . . . . . 7 ⊢ ((𝑥∪ 𝐵𝑢 ∧ 𝑥∪ 𝐵𝑣) ↔ (∃𝑔(𝑥𝑔𝑢 ∧ 𝑔 ∈ 𝐵) ∧ ∃ℎ(𝑥ℎ𝑣 ∧ ℎ ∈ 𝐵)))
21		eeanv 2344	. . . . . . 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 32121	. . . . . . . . 9 ⊢ ((𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵) → ((𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣) → 𝑢 = 𝑣))
24	23	impcom 394	. . . . . . . 8 ⊢ (((𝑥𝑔𝑢 ∧ 𝑥ℎ𝑣) ∧ (𝑔 ∈ 𝐵 ∧ ℎ ∈ 𝐵)) → 𝑢 = 𝑣)
25	24	an4s 639	. . . . . . 7 ⊢ (((𝑥𝑔𝑢 ∧ 𝑔 ∈ 𝐵) ∧ (𝑥ℎ𝑣 ∧ ℎ ∈ 𝐵)) → 𝑢 = 𝑣)
26	25	exlimivv 2012	. . . . . 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 1870	. . . 4 ⊢ ∀𝑣((𝑥∪ 𝐵𝑢 ∧ 𝑥∪ 𝐵𝑣) → 𝑢 = 𝑣)
29	28	gen2 1871	. . 3 ⊢ ∀𝑥∀𝑢∀𝑣((𝑥∪ 𝐵𝑢 ∧ 𝑥∪ 𝐵𝑣) → 𝑢 = 𝑣)
30		dffun2 6041	. . 3 ⊢ (Fun ∪ 𝐵 ↔ (Rel ∪ 𝐵 ∧ ∀𝑥∀𝑢∀𝑣((𝑥∪ 𝐵𝑢 ∧ 𝑥∪ 𝐵𝑣) → 𝑢 = 𝑣)))
31	7, 29, 30	mpbir2an 690	. 2 ⊢ Fun ∪ 𝐵
32		funss 6050	. 2 ⊢ (∪ 𝐶 ⊆ ∪ 𝐵 → (Fun ∪ 𝐵 → Fun ∪ 𝐶))
33	1, 31, 32	mpisyl 21	1 ⊢ (𝐶 ⊆ 𝐵 → Fun ∪ 𝐶)

Syntax hints:   → wi 4   ∧ wa 382   ∧ w3a 1071  ∀wal 1629   = wceq 1631  ∃wex 1852   ∈ wcel 2145  {cab 2757  ∀wral 3061   ⊆ wss 3723  ⟨cop 4322  ∪ cuni 4574   class class class wbr 4786   Fr wfr 5205   Se wse 5206   ↾ cres 5251  Rel wrel 5254  Predcpred 5822  Fun wfun 6025   Fn wfn 6026  ‘cfv 6031  (class class class)co 6793

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-8 2147  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-rep 4904  ax-sep 4915  ax-nul 4923  ax-pow 4974  ax-pr 5034  ax-un 7096  ax-inf2 8702

This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3or 1072  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ne 2944  df-ral 3066  df-rex 3067  df-reu 3068  df-rab 3070  df-v 3353  df-sbc 3588  df-csb 3683  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-pss 3739  df-nul 4064  df-if 4226  df-pw 4299  df-sn 4317  df-pr 4319  df-tp 4321  df-op 4323  df-uni 4575  df-iun 4656  df-br 4787  df-opab 4847  df-mpt 4864  df-tr 4887  df-id 5157  df-eprel 5162  df-po 5170  df-so 5171  df-fr 5208  df-se 5209  df-we 5210  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-pred 5823  df-ord 5869  df-on 5870  df-lim 5871  df-suc 5872  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-f1 6036  df-fo 6037  df-f1o 6038  df-fv 6039  df-ov 6796  df-om 7213  df-wrecs 7559  df-recs 7621  df-rdg 7659  df-trpred 32054

This theorem is referenced by:  frrlem10  32128