Theorem bnj1384 31100

Description: Technical lemma for bnj60 31130. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem). (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypotheses

Ref	Expression
bnj1384.1	⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1384.2	⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1384.3	⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
bnj1384.4	⊢ (𝜏 ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
bnj1384.5	⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏}
bnj1384.6	⊢ (𝜓 ↔ (𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅))
bnj1384.7	⊢ (𝜒 ↔ (𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥))
bnj1384.8	⊢ (𝜏′ ↔ [𝑦 / 𝑥]𝜏)
bnj1384.9	⊢ 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
bnj1384.10	⊢ 𝑃 = ∪ 𝐻

Assertion

Ref	Expression
bnj1384	⊢ (𝑅 FrSe 𝐴 → Fun 𝑃)

Distinct variable groups:   𝐴,𝑑,𝑓,𝑥   𝐵,𝑓   𝑦,𝐶   𝐺,𝑑,𝑓   𝑅,𝑑,𝑓,𝑥   𝑦,𝑓,𝑥

Allowed substitution hints:   𝜓(𝑥,𝑦,𝑓,𝑑)   𝜒(𝑥,𝑦,𝑓,𝑑)   𝜏(𝑥,𝑦,𝑓,𝑑)   𝐴(𝑦)   𝐵(𝑥,𝑦,𝑑)   𝐶(𝑥,𝑓,𝑑)   𝐷(𝑥,𝑦,𝑓,𝑑)   𝑃(𝑥,𝑦,𝑓,𝑑)   𝑅(𝑦)   𝐺(𝑥,𝑦)   𝐻(𝑥,𝑦,𝑓,𝑑)   𝑌(𝑥,𝑦,𝑓,𝑑)   𝜏′(𝑥,𝑦,𝑓,𝑑)

Proof of Theorem bnj1384

Dummy variables 𝑧 𝑔 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		bnj1384.1	. . . . 5 ⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
2		bnj1384.2	. . . . 5 ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
3		bnj1384.3	. . . . 5 ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
4		bnj1384.4	. . . . 5 ⊢ (𝜏 ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
5		bnj1384.5	. . . . 5 ⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏}
6		bnj1384.6	. . . . 5 ⊢ (𝜓 ↔ (𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅))
7		bnj1384.7	. . . . 5 ⊢ (𝜒 ↔ (𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥))
8		bnj1384.8	. . . . 5 ⊢ (𝜏′ ↔ [𝑦 / 𝑥]𝜏)
9		bnj1384.9	. . . . 5 ⊢ 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
10		bnj1384.10	. . . . 5 ⊢ 𝑃 = ∪ 𝐻
11	1, 2, 3, 4, 8	bnj1373 31098	. . . . 5 ⊢ (𝜏′ ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
12	1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11	bnj1371 31097	. . . 4 ⊢ (𝑓 ∈ 𝐻 → Fun 𝑓)
13	12	rgen 2922	. . 3 ⊢ ∀𝑓 ∈ 𝐻 Fun 𝑓
14		id 22	. . . . . 6 ⊢ (𝑅 FrSe 𝐴 → 𝑅 FrSe 𝐴)
15	1, 2, 3, 4, 5, 6, 7, 8, 9	bnj1374 31099	. . . . . 6 ⊢ (𝑓 ∈ 𝐻 → 𝑓 ∈ 𝐶)
16		nfab1 2766	. . . . . . . . . 10 ⊢ Ⅎ𝑓{𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
17	9, 16	nfcxfr 2762	. . . . . . . . 9 ⊢ Ⅎ𝑓𝐻
18	17	nfcri 2758	. . . . . . . 8 ⊢ Ⅎ𝑓 𝑔 ∈ 𝐻
19		nfab1 2766	. . . . . . . . . 10 ⊢ Ⅎ𝑓{𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
20	3, 19	nfcxfr 2762	. . . . . . . . 9 ⊢ Ⅎ𝑓𝐶
21	20	nfcri 2758	. . . . . . . 8 ⊢ Ⅎ𝑓 𝑔 ∈ 𝐶
22	18, 21	nfim 1825	. . . . . . 7 ⊢ Ⅎ𝑓(𝑔 ∈ 𝐻 → 𝑔 ∈ 𝐶)
23		eleq1 2689	. . . . . . . 8 ⊢ (𝑓 = 𝑔 → (𝑓 ∈ 𝐻 ↔ 𝑔 ∈ 𝐻))
24		eleq1 2689	. . . . . . . 8 ⊢ (𝑓 = 𝑔 → (𝑓 ∈ 𝐶 ↔ 𝑔 ∈ 𝐶))
25	23, 24	imbi12d 334	. . . . . . 7 ⊢ (𝑓 = 𝑔 → ((𝑓 ∈ 𝐻 → 𝑓 ∈ 𝐶) ↔ (𝑔 ∈ 𝐻 → 𝑔 ∈ 𝐶)))
26	22, 25, 15	chvar 2262	. . . . . 6 ⊢ (𝑔 ∈ 𝐻 → 𝑔 ∈ 𝐶)
27		eqid 2622	. . . . . . 7 ⊢ (dom 𝑓 ∩ dom 𝑔) = (dom 𝑓 ∩ dom 𝑔)
28	1, 2, 3, 27	bnj1326 31094	. . . . . 6 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑓 ∈ 𝐶 ∧ 𝑔 ∈ 𝐶) → (𝑓 ↾ (dom 𝑓 ∩ dom 𝑔)) = (𝑔 ↾ (dom 𝑓 ∩ dom 𝑔)))
29	14, 15, 26, 28	syl3an 1368	. . . . 5 ⊢ ((𝑅 FrSe 𝐴 ∧ 𝑓 ∈ 𝐻 ∧ 𝑔 ∈ 𝐻) → (𝑓 ↾ (dom 𝑓 ∩ dom 𝑔)) = (𝑔 ↾ (dom 𝑓 ∩ dom 𝑔)))
30	29	3expib 1268	. . . 4 ⊢ (𝑅 FrSe 𝐴 → ((𝑓 ∈ 𝐻 ∧ 𝑔 ∈ 𝐻) → (𝑓 ↾ (dom 𝑓 ∩ dom 𝑔)) = (𝑔 ↾ (dom 𝑓 ∩ dom 𝑔))))
31	30	ralrimivv 2970	. . 3 ⊢ (𝑅 FrSe 𝐴 → ∀𝑓 ∈ 𝐻 ∀𝑔 ∈ 𝐻 (𝑓 ↾ (dom 𝑓 ∩ dom 𝑔)) = (𝑔 ↾ (dom 𝑓 ∩ dom 𝑔)))
32		biid 251	. . . 4 ⊢ (∀𝑓 ∈ 𝐻 Fun 𝑓 ↔ ∀𝑓 ∈ 𝐻 Fun 𝑓)
33		biid 251	. . . 4 ⊢ ((∀𝑓 ∈ 𝐻 Fun 𝑓 ∧ ∀𝑓 ∈ 𝐻 ∀𝑔 ∈ 𝐻 (𝑓 ↾ (dom 𝑓 ∩ dom 𝑔)) = (𝑔 ↾ (dom 𝑓 ∩ dom 𝑔))) ↔ (∀𝑓 ∈ 𝐻 Fun 𝑓 ∧ ∀𝑓 ∈ 𝐻 ∀𝑔 ∈ 𝐻 (𝑓 ↾ (dom 𝑓 ∩ dom 𝑔)) = (𝑔 ↾ (dom 𝑓 ∩ dom 𝑔))))
34	9	bnj1317 30892	. . . 4 ⊢ (𝑧 ∈ 𝐻 → ∀𝑓 𝑧 ∈ 𝐻)
35	32, 27, 33, 34	bnj1386 30904	. . 3 ⊢ ((∀𝑓 ∈ 𝐻 Fun 𝑓 ∧ ∀𝑓 ∈ 𝐻 ∀𝑔 ∈ 𝐻 (𝑓 ↾ (dom 𝑓 ∩ dom 𝑔)) = (𝑔 ↾ (dom 𝑓 ∩ dom 𝑔))) → Fun ∪ 𝐻)
36	13, 31, 35	sylancr 695	. 2 ⊢ (𝑅 FrSe 𝐴 → Fun ∪ 𝐻)
37	10	funeqi 5909	. 2 ⊢ (Fun 𝑃 ↔ Fun ∪ 𝐻)
38	36, 37	sylibr 224	1 ⊢ (𝑅 FrSe 𝐴 → Fun 𝑃)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 384   ∧ w3a 1037   = wceq 1483  ∃wex 1704   ∈ wcel 1990  {cab 2608   ≠ wne 2794  ∀wral 2912  ∃wrex 2913  {crab 2916  [wsbc 3435   ∪ cun 3572   ∩ cin 3573   ⊆ wss 3574  ∅c0 3915  {csn 4177  ⟨cop 4183  ∪ cuni 4436   class class class wbr 4653  dom cdm 5114   ↾ cres 5116  Fun wfun 5882   Fn wfn 5883  ‘cfv 5888   predc-bnj14 30754   FrSe w-bnj15 30758   trClc-bnj18 30760

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-reg 8497  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-fal 1489  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-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-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-om 7066  df-1o 7560  df-bnj17 30753  df-bnj14 30755  df-bnj13 30757  df-bnj15 30759  df-bnj18 30761  df-bnj19 30763

This theorem is referenced by:  bnj1312  31126