Theorem bnj1498 35691

Description: Technical lemma for bnj60 35692. 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
bnj1498.1	⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1498.2	⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1498.3	⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
bnj1498.4	⊢ 𝐹 = ∪ 𝐶

Assertion

Ref	Expression
bnj1498	⊢ (𝑅 FrSe 𝐴 → dom 𝐹 = 𝐴)

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

Allowed substitution hints:   𝐵(𝑥, 𝑑)   𝐶(𝑥, 𝑓, 𝑑)   𝐹(𝑥, 𝑓, 𝑑)   𝑌(𝑥, 𝑓, 𝑑)

Proof of Theorem bnj1498

Dummy variables 𝑡 𝑧 𝑤 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		eliun 4954	. . . . . . 7 ⊢ (𝑧 ∈ ∪ 𝑓 ∈ 𝐶 dom 𝑓 ↔ ∃𝑓 ∈ 𝐶 𝑧 ∈ dom 𝑓)
2		bnj1498.3	. . . . . . . . . . . . . . . 16 ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
3	2	bnj1436 35469	. . . . . . . . . . . . . . 15 ⊢ (𝑓 ∈ 𝐶 → ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌)))
4	3	bnj1299 35448	. . . . . . . . . . . . . 14 ⊢ (𝑓 ∈ 𝐶 → ∃𝑑 ∈ 𝐵 𝑓 Fn 𝑑)
5		fndm 6642	. . . . . . . . . . . . . 14 ⊢ (𝑓 Fn 𝑑 → dom 𝑓 = 𝑑)
6	4, 5	bnj31 35350	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ 𝐶 → ∃𝑑 ∈ 𝐵 dom 𝑓 = 𝑑)
7	6	bnj1196 35424	. . . . . . . . . . . 12 ⊢ (𝑓 ∈ 𝐶 → ∃𝑑(𝑑 ∈ 𝐵 ∧ dom 𝑓 = 𝑑))
8		bnj1498.1	. . . . . . . . . . . . . . 15 ⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
9	8	bnj1436 35469	. . . . . . . . . . . . . 14 ⊢ (𝑑 ∈ 𝐵 → (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑))
10	9	simpld 500	. . . . . . . . . . . . 13 ⊢ (𝑑 ∈ 𝐵 → 𝑑 ⊆ 𝐴)
11	10	anim1i 627	. . . . . . . . . . . 12 ⊢ ((𝑑 ∈ 𝐵 ∧ dom 𝑓 = 𝑑) → (𝑑 ⊆ 𝐴 ∧ dom 𝑓 = 𝑑))
12	7, 11	bnj593 35376	. . . . . . . . . . 11 ⊢ (𝑓 ∈ 𝐶 → ∃𝑑(𝑑 ⊆ 𝐴 ∧ dom 𝑓 = 𝑑))
13		sseq1 3933	. . . . . . . . . . . 12 ⊢ (dom 𝑓 = 𝑑 → (dom 𝑓 ⊆ 𝐴 ↔ 𝑑 ⊆ 𝐴))
14	13	biimparc 485	. . . . . . . . . . 11 ⊢ ((𝑑 ⊆ 𝐴 ∧ dom 𝑓 = 𝑑) → dom 𝑓 ⊆ 𝐴)
15	12, 14	bnj593 35376	. . . . . . . . . 10 ⊢ (𝑓 ∈ 𝐶 → ∃𝑑dom 𝑓 ⊆ 𝐴)
16	15	bnj937 35402	. . . . . . . . 9 ⊢ (𝑓 ∈ 𝐶 → dom 𝑓 ⊆ 𝐴)
17	16	sselda 3908	. . . . . . . 8 ⊢ ((𝑓 ∈ 𝐶 ∧ 𝑧 ∈ dom 𝑓) → 𝑧 ∈ 𝐴)
18	17	rexlimiva 3155	. . . . . . 7 ⊢ (∃𝑓 ∈ 𝐶 𝑧 ∈ dom 𝑓 → 𝑧 ∈ 𝐴)
19	1, 18	sylbi 220	. . . . . 6 ⊢ (𝑧 ∈ ∪ 𝑓 ∈ 𝐶 dom 𝑓 → 𝑧 ∈ 𝐴)
20	2	bnj1317 35451	. . . . . . 7 ⊢ (𝑤 ∈ 𝐶 → ∀𝑓 𝑤 ∈ 𝐶)
21	20	bnj1400 35465	. . . . . 6 ⊢ dom ∪ 𝐶 = ∪ 𝑓 ∈ 𝐶 dom 𝑓
22	19, 21	eleq2s 2878	. . . . 5 ⊢ (𝑧 ∈ dom ∪ 𝐶 → 𝑧 ∈ 𝐴)
23		bnj1498.4	. . . . . 6 ⊢ 𝐹 = ∪ 𝐶
24	23	dmeqi 5885	. . . . 5 ⊢ dom 𝐹 = dom ∪ 𝐶
25	22, 24	eleq2s 2878	. . . 4 ⊢ (𝑧 ∈ dom 𝐹 → 𝑧 ∈ 𝐴)
26	25	ssriv 3912	. . 3 ⊢ dom 𝐹 ⊆ 𝐴
27	26	a1i 11	. 2 ⊢ (𝑅 FrSe 𝐴 → dom 𝐹 ⊆ 𝐴)
28		bnj1498.2	. . . . . . . 8 ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
29	8, 28, 2	bnj1493 35689	. . . . . . 7 ⊢ (𝑅 FrSe 𝐴 → ∀𝑥 ∈ 𝐴 ∃𝑓 ∈ 𝐶 dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
30		vsnid 4623	. . . . . . . . . . 11 ⊢ 𝑥 ∈ {𝑥}
31		elun1 4180	. . . . . . . . . . 11 ⊢ (𝑥 ∈ {𝑥} → 𝑥 ∈ ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
32	30, 31	ax-mp 5	. . . . . . . . . 10 ⊢ 𝑥 ∈ ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))
33		eleq2 2849	. . . . . . . . . 10 ⊢ (dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)) → (𝑥 ∈ dom 𝑓 ↔ 𝑥 ∈ ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
34	32, 33	mpbiri 261	. . . . . . . . 9 ⊢ (dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)) → 𝑥 ∈ dom 𝑓)
35	34	reximi 3100	. . . . . . . 8 ⊢ (∃𝑓 ∈ 𝐶 dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)) → ∃𝑓 ∈ 𝐶 𝑥 ∈ dom 𝑓)
36	35	ralimi 3099	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑓 ∈ 𝐶 dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)) → ∀𝑥 ∈ 𝐴 ∃𝑓 ∈ 𝐶 𝑥 ∈ dom 𝑓)
37	29, 36	syl 18	. . . . . 6 ⊢ (𝑅 FrSe 𝐴 → ∀𝑥 ∈ 𝐴 ∃𝑓 ∈ 𝐶 𝑥 ∈ dom 𝑓)
38		eliun 4954	. . . . . . 7 ⊢ (𝑥 ∈ ∪ 𝑓 ∈ 𝐶 dom 𝑓 ↔ ∃𝑓 ∈ 𝐶 𝑥 ∈ dom 𝑓)
39	38	ralbii 3108	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ 𝑓 ∈ 𝐶 dom 𝑓 ↔ ∀𝑥 ∈ 𝐴 ∃𝑓 ∈ 𝐶 𝑥 ∈ dom 𝑓)
40	37, 39	sylibr 237	. . . . 5 ⊢ (𝑅 FrSe 𝐴 → ∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ 𝑓 ∈ 𝐶 dom 𝑓)
41		nfcv 2922	. . . . . 6 ⊢ Ⅎ𝑥𝐴
42	8	bnj1309 35652	. . . . . . . . 9 ⊢ (𝑡 ∈ 𝐵 → ∀𝑥 𝑡 ∈ 𝐵)
43	2, 42	bnj1307 35653	. . . . . . . 8 ⊢ (𝑡 ∈ 𝐶 → ∀𝑥 𝑡 ∈ 𝐶)
44	43	nfcii 2911	. . . . . . 7 ⊢ Ⅎ𝑥𝐶
45		nfcv 2922	. . . . . . 7 ⊢ Ⅎ𝑥dom 𝑓
46	44, 45	nfiun 4981	. . . . . 6 ⊢ Ⅎ𝑥∪ 𝑓 ∈ 𝐶 dom 𝑓
47	41, 46	dfss3f 3900	. . . . 5 ⊢ (𝐴 ⊆ ∪ 𝑓 ∈ 𝐶 dom 𝑓 ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ 𝑓 ∈ 𝐶 dom 𝑓)
48	40, 47	sylibr 237	. . . 4 ⊢ (𝑅 FrSe 𝐴 → 𝐴 ⊆ ∪ 𝑓 ∈ 𝐶 dom 𝑓)
49	48, 21	sseqtrrdi 3949	. . 3 ⊢ (𝑅 FrSe 𝐴 → 𝐴 ⊆ dom ∪ 𝐶)
50	49, 24	sseqtrrdi 3949	. 2 ⊢ (𝑅 FrSe 𝐴 → 𝐴 ⊆ dom 𝐹)
51	27, 50	eqssd 3925	1 ⊢ (𝑅 FrSe 𝐴 → dom 𝐹 = 𝐴)

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  {cab 2738  ∀wral 3076  ∃wrex 3086   ⊆ wss 3895   ∪ cun 4149  {csn 4583  ⟨cop 4589  ∪ cuni 4866  ∪ ciun 4950  dom cdm 5650   ↾ cres 5652   Fn wfn 6533  ‘cfv 6538   predc-bnj14 35319   FrSe w-bnj15 35323   trClc-bnj18 35325

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751  ax-reg 9586  ax-inf2 9642

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-reu 3366  df-rab 3413  df-v 3452  df-sbc 3739  df-csb 3847  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-om 7878  df-1o 8476  df-bnj17 35318  df-bnj14 35320  df-bnj13 35322  df-bnj15 35324  df-bnj18 35326  df-bnj19 35328

This theorem is used by:  bnj60  35692