Theorem bnj1498 35196

Description: Technical lemma for bnj60 35197. 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 4949	. . . . . . 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 34974	. . . . . . . . . . . . . . 15 ⊢ (𝑓 ∈ 𝐶 → ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌)))
4	3	bnj1299 34953	. . . . . . . . . . . . . 14 ⊢ (𝑓 ∈ 𝐶 → ∃𝑑 ∈ 𝐵 𝑓 Fn 𝑑)
5		fndm 6594	. . . . . . . . . . . . . 14 ⊢ (𝑓 Fn 𝑑 → dom 𝑓 = 𝑑)
6	4, 5	bnj31 34854	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ 𝐶 → ∃𝑑 ∈ 𝐵 dom 𝑓 = 𝑑)
7	6	bnj1196 34929	. . . . . . . . . . . 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 34974	. . . . . . . . . . . . . 14 ⊢ (𝑑 ∈ 𝐵 → (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑))
10	9	simpld 494	. . . . . . . . . . . . 13 ⊢ (𝑑 ∈ 𝐵 → 𝑑 ⊆ 𝐴)
11	10	anim1i 616	. . . . . . . . . . . 12 ⊢ ((𝑑 ∈ 𝐵 ∧ dom 𝑓 = 𝑑) → (𝑑 ⊆ 𝐴 ∧ dom 𝑓 = 𝑑))
12	7, 11	bnj593 34880	. . . . . . . . . . 11 ⊢ (𝑓 ∈ 𝐶 → ∃𝑑(𝑑 ⊆ 𝐴 ∧ dom 𝑓 = 𝑑))
13		sseq1 3958	. . . . . . . . . . . 12 ⊢ (dom 𝑓 = 𝑑 → (dom 𝑓 ⊆ 𝐴 ↔ 𝑑 ⊆ 𝐴))
14	13	biimparc 479	. . . . . . . . . . 11 ⊢ ((𝑑 ⊆ 𝐴 ∧ dom 𝑓 = 𝑑) → dom 𝑓 ⊆ 𝐴)
15	12, 14	bnj593 34880	. . . . . . . . . 10 ⊢ (𝑓 ∈ 𝐶 → ∃𝑑dom 𝑓 ⊆ 𝐴)
16	15	bnj937 34906	. . . . . . . . 9 ⊢ (𝑓 ∈ 𝐶 → dom 𝑓 ⊆ 𝐴)
17	16	sselda 3932	. . . . . . . 8 ⊢ ((𝑓 ∈ 𝐶 ∧ 𝑧 ∈ dom 𝑓) → 𝑧 ∈ 𝐴)
18	17	rexlimiva 3128	. . . . . . 7 ⊢ (∃𝑓 ∈ 𝐶 𝑧 ∈ dom 𝑓 → 𝑧 ∈ 𝐴)
19	1, 18	sylbi 217	. . . . . 6 ⊢ (𝑧 ∈ ∪ 𝑓 ∈ 𝐶 dom 𝑓 → 𝑧 ∈ 𝐴)
20	2	bnj1317 34956	. . . . . . 7 ⊢ (𝑤 ∈ 𝐶 → ∀𝑓 𝑤 ∈ 𝐶)
21	20	bnj1400 34970	. . . . . 6 ⊢ dom ∪ 𝐶 = ∪ 𝑓 ∈ 𝐶 dom 𝑓
22	19, 21	eleq2s 2853	. . . . 5 ⊢ (𝑧 ∈ dom ∪ 𝐶 → 𝑧 ∈ 𝐴)
23		bnj1498.4	. . . . . 6 ⊢ 𝐹 = ∪ 𝐶
24	23	dmeqi 5852	. . . . 5 ⊢ dom 𝐹 = dom ∪ 𝐶
25	22, 24	eleq2s 2853	. . . 4 ⊢ (𝑧 ∈ dom 𝐹 → 𝑧 ∈ 𝐴)
26	25	ssriv 3936	. . 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 35194	. . . . . . 7 ⊢ (𝑅 FrSe 𝐴 → ∀𝑥 ∈ 𝐴 ∃𝑓 ∈ 𝐶 dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
30		vsnid 4619	. . . . . . . . . . 11 ⊢ 𝑥 ∈ {𝑥}
31		elun1 4133	. . . . . . . . . . 11 ⊢ (𝑥 ∈ {𝑥} → 𝑥 ∈ ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
32	30, 31	ax-mp 5	. . . . . . . . . 10 ⊢ 𝑥 ∈ ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))
33		eleq2 2824	. . . . . . . . . 10 ⊢ (dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)) → (𝑥 ∈ dom 𝑓 ↔ 𝑥 ∈ ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
34	32, 33	mpbiri 258	. . . . . . . . 9 ⊢ (dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)) → 𝑥 ∈ dom 𝑓)
35	34	reximi 3073	. . . . . . . 8 ⊢ (∃𝑓 ∈ 𝐶 dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)) → ∃𝑓 ∈ 𝐶 𝑥 ∈ dom 𝑓)
36	35	ralimi 3072	. . . . . . 7 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑓 ∈ 𝐶 dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)) → ∀𝑥 ∈ 𝐴 ∃𝑓 ∈ 𝐶 𝑥 ∈ dom 𝑓)
37	29, 36	syl 17	. . . . . 6 ⊢ (𝑅 FrSe 𝐴 → ∀𝑥 ∈ 𝐴 ∃𝑓 ∈ 𝐶 𝑥 ∈ dom 𝑓)
38		eliun 4949	. . . . . . 7 ⊢ (𝑥 ∈ ∪ 𝑓 ∈ 𝐶 dom 𝑓 ↔ ∃𝑓 ∈ 𝐶 𝑥 ∈ dom 𝑓)
39	38	ralbii 3081	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ 𝑓 ∈ 𝐶 dom 𝑓 ↔ ∀𝑥 ∈ 𝐴 ∃𝑓 ∈ 𝐶 𝑥 ∈ dom 𝑓)
40	37, 39	sylibr 234	. . . . 5 ⊢ (𝑅 FrSe 𝐴 → ∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ 𝑓 ∈ 𝐶 dom 𝑓)
41		nfcv 2897	. . . . . 6 ⊢ Ⅎ𝑥𝐴
42	8	bnj1309 35157	. . . . . . . . 9 ⊢ (𝑡 ∈ 𝐵 → ∀𝑥 𝑡 ∈ 𝐵)
43	2, 42	bnj1307 35158	. . . . . . . 8 ⊢ (𝑡 ∈ 𝐶 → ∀𝑥 𝑡 ∈ 𝐶)
44	43	nfcii 2886	. . . . . . 7 ⊢ Ⅎ𝑥𝐶
45		nfcv 2897	. . . . . . 7 ⊢ Ⅎ𝑥dom 𝑓
46	44, 45	nfiun 4977	. . . . . 6 ⊢ Ⅎ𝑥∪ 𝑓 ∈ 𝐶 dom 𝑓
47	41, 46	dfss3f 3924	. . . . 5 ⊢ (𝐴 ⊆ ∪ 𝑓 ∈ 𝐶 dom 𝑓 ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ 𝑓 ∈ 𝐶 dom 𝑓)
48	40, 47	sylibr 234	. . . 4 ⊢ (𝑅 FrSe 𝐴 → 𝐴 ⊆ ∪ 𝑓 ∈ 𝐶 dom 𝑓)
49	48, 21	sseqtrrdi 3974	. . 3 ⊢ (𝑅 FrSe 𝐴 → 𝐴 ⊆ dom ∪ 𝐶)
50	49, 24	sseqtrrdi 3974	. 2 ⊢ (𝑅 FrSe 𝐴 → 𝐴 ⊆ dom 𝐹)
51	27, 50	eqssd 3950	1 ⊢ (𝑅 FrSe 𝐴 → dom 𝐹 = 𝐴)

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {cab 2713  ∀wral 3050  ∃wrex 3059   ∪ cun 3898   ⊆ wss 3900  {csn 4579  ⟨cop 4585  ∪ cuni 4862  ∪ ciun 4945  dom cdm 5623   ↾ cres 5625   Fn wfn 6486  ‘cfv 6491   predc-bnj14 34823   FrSe w-bnj15 34827   trClc-bnj18 34829

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2183  ax-ext 2707  ax-rep 5223  ax-sep 5240  ax-nul 5250  ax-pow 5309  ax-pr 5376  ax-un 7680  ax-reg 9499  ax-inf2 9552

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2538  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2810  df-nfc 2884  df-ne 2932  df-ral 3051  df-rex 3060  df-reu 3350  df-rab 3399  df-v 3441  df-sbc 3740  df-csb 3849  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-pss 3920  df-nul 4285  df-if 4479  df-pw 4555  df-sn 4580  df-pr 4582  df-op 4586  df-uni 4863  df-iun 4947  df-br 5098  df-opab 5160  df-mpt 5179  df-tr 5205  df-id 5518  df-eprel 5523  df-po 5531  df-so 5532  df-fr 5576  df-we 5578  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636  df-ord 6319  df-on 6320  df-lim 6321  df-suc 6322  df-iota 6447  df-fun 6493  df-fn 6494  df-f 6495  df-f1 6496  df-fo 6497  df-f1o 6498  df-fv 6499  df-om 7809  df-1o 8397  df-bnj17 34822  df-bnj14 34824  df-bnj13 34826  df-bnj15 34828  df-bnj18 34830  df-bnj19 34832

This theorem is referenced by:  bnj60  35197