Theorem bnj1498 35258

Description: Technical lemma for bnj60 35259. 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 4928	. . . . . . 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 35036	. . . . . . . . . . . . . . 15 ⊢ (𝑓 ∈ 𝐶 → ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌)))
4	3	bnj1299 35015	. . . . . . . . . . . . . 14 ⊢ (𝑓 ∈ 𝐶 → ∃𝑑 ∈ 𝐵 𝑓 Fn 𝑑)
5		fndm 6592	. . . . . . . . . . . . . 14 ⊢ (𝑓 Fn 𝑑 → dom 𝑓 = 𝑑)
6	4, 5	bnj31 34917	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ 𝐶 → ∃𝑑 ∈ 𝐵 dom 𝑓 = 𝑑)
7	6	bnj1196 34991	. . . . . . . . . . . 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 35036	. . . . . . . . . . . . . 14 ⊢ (𝑑 ∈ 𝐵 → (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑))
10	9	simpld 496	. . . . . . . . . . . . 13 ⊢ (𝑑 ∈ 𝐵 → 𝑑 ⊆ 𝐴)
11	10	anim1i 622	. . . . . . . . . . . 12 ⊢ ((𝑑 ∈ 𝐵 ∧ dom 𝑓 = 𝑑) → (𝑑 ⊆ 𝐴 ∧ dom 𝑓 = 𝑑))
12	7, 11	bnj593 34943	. . . . . . . . . . 11 ⊢ (𝑓 ∈ 𝐶 → ∃𝑑(𝑑 ⊆ 𝐴 ∧ dom 𝑓 = 𝑑))
13		sseq1 3942	. . . . . . . . . . . 12 ⊢ (dom 𝑓 = 𝑑 → (dom 𝑓 ⊆ 𝐴 ↔ 𝑑 ⊆ 𝐴))
14	13	biimparc 481	. . . . . . . . . . 11 ⊢ ((𝑑 ⊆ 𝐴 ∧ dom 𝑓 = 𝑑) → dom 𝑓 ⊆ 𝐴)
15	12, 14	bnj593 34943	. . . . . . . . . 10 ⊢ (𝑓 ∈ 𝐶 → ∃𝑑dom 𝑓 ⊆ 𝐴)
16	15	bnj937 34969	. . . . . . . . 9 ⊢ (𝑓 ∈ 𝐶 → dom 𝑓 ⊆ 𝐴)
17	16	sselda 3917	. . . . . . . 8 ⊢ ((𝑓 ∈ 𝐶 ∧ 𝑧 ∈ dom 𝑓) → 𝑧 ∈ 𝐴)
18	17	rexlimiva 3134	. . . . . . 7 ⊢ (∃𝑓 ∈ 𝐶 𝑧 ∈ dom 𝑓 → 𝑧 ∈ 𝐴)
19	1, 18	sylbi 219	. . . . . 6 ⊢ (𝑧 ∈ ∪ 𝑓 ∈ 𝐶 dom 𝑓 → 𝑧 ∈ 𝐴)
20	2	bnj1317 35018	. . . . . . 7 ⊢ (𝑤 ∈ 𝐶 → ∀𝑓 𝑤 ∈ 𝐶)
21	20	bnj1400 35032	. . . . . 6 ⊢ dom ∪ 𝐶 = ∪ 𝑓 ∈ 𝐶 dom 𝑓
22	19, 21	eleq2s 2859	. . . . 5 ⊢ (𝑧 ∈ dom ∪ 𝐶 → 𝑧 ∈ 𝐴)
23		bnj1498.4	. . . . . 6 ⊢ 𝐹 = ∪ 𝐶
24	23	dmeqi 5853	. . . . 5 ⊢ dom 𝐹 = dom ∪ 𝐶
25	22, 24	eleq2s 2859	. . . 4 ⊢ (𝑧 ∈ dom 𝐹 → 𝑧 ∈ 𝐴)
26	25	ssriv 3921	. . 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 35256	. . . . . . 7 ⊢ (𝑅 FrSe 𝐴 → ∀𝑥 ∈ 𝐴 ∃𝑓 ∈ 𝐶 dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
30		vsnid 4598	. . . . . . . . . . 11 ⊢ 𝑥 ∈ {𝑥}
31		elun1 4114	. . . . . . . . . . 11 ⊢ (𝑥 ∈ {𝑥} → 𝑥 ∈ ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
32	30, 31	ax-mp 5	. . . . . . . . . 10 ⊢ 𝑥 ∈ ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))
33		eleq2 2830	. . . . . . . . . 10 ⊢ (dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)) → (𝑥 ∈ dom 𝑓 ↔ 𝑥 ∈ ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
34	32, 33	mpbiri 260	. . . . . . . . 9 ⊢ (dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)) → 𝑥 ∈ dom 𝑓)
35	34	reximi 3079	. . . . . . . 8 ⊢ (∃𝑓 ∈ 𝐶 dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)) → ∃𝑓 ∈ 𝐶 𝑥 ∈ dom 𝑓)
36	35	ralimi 3078	. . . . . . 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 4928	. . . . . . 7 ⊢ (𝑥 ∈ ∪ 𝑓 ∈ 𝐶 dom 𝑓 ↔ ∃𝑓 ∈ 𝐶 𝑥 ∈ dom 𝑓)
39	38	ralbii 3087	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ 𝑓 ∈ 𝐶 dom 𝑓 ↔ ∀𝑥 ∈ 𝐴 ∃𝑓 ∈ 𝐶 𝑥 ∈ dom 𝑓)
40	37, 39	sylibr 236	. . . . 5 ⊢ (𝑅 FrSe 𝐴 → ∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ 𝑓 ∈ 𝐶 dom 𝑓)
41		nfcv 2903	. . . . . 6 ⊢ Ⅎ𝑥𝐴
42	8	bnj1309 35219	. . . . . . . . 9 ⊢ (𝑡 ∈ 𝐵 → ∀𝑥 𝑡 ∈ 𝐵)
43	2, 42	bnj1307 35220	. . . . . . . 8 ⊢ (𝑡 ∈ 𝐶 → ∀𝑥 𝑡 ∈ 𝐶)
44	43	nfcii 2892	. . . . . . 7 ⊢ Ⅎ𝑥𝐶
45		nfcv 2903	. . . . . . 7 ⊢ Ⅎ𝑥dom 𝑓
46	44, 45	nfiun 4956	. . . . . 6 ⊢ Ⅎ𝑥∪ 𝑓 ∈ 𝐶 dom 𝑓
47	41, 46	dfss3f 3909	. . . . 5 ⊢ (𝐴 ⊆ ∪ 𝑓 ∈ 𝐶 dom 𝑓 ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ 𝑓 ∈ 𝐶 dom 𝑓)
48	40, 47	sylibr 236	. . . 4 ⊢ (𝑅 FrSe 𝐴 → 𝐴 ⊆ ∪ 𝑓 ∈ 𝐶 dom 𝑓)
49	48, 21	sseqtrrdi 3958	. . 3 ⊢ (𝑅 FrSe 𝐴 → 𝐴 ⊆ dom ∪ 𝐶)
50	49, 24	sseqtrrdi 3958	. 2 ⊢ (𝑅 FrSe 𝐴 → 𝐴 ⊆ dom 𝐹)
51	27, 50	eqssd 3934	1 ⊢ (𝑅 FrSe 𝐴 → dom 𝐹 = 𝐴)

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1548   ∈ wcel 2121  {cab 2719  ∀wral 3055  ∃wrex 3065   ∪ cun 3883   ⊆ wss 3885  {csn 4558  ⟨cop 4564  ∪ cuni 4841  ∪ ciun 4924  dom cdm 5621   ↾ cres 5623   Fn wfn 6484  ‘cfv 6489   predc-bnj14 34886   FrSe w-bnj15 34890   trClc-bnj18 34892

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-rep 5202  ax-sep 5221  ax-nul 5231  ax-pow 5297  ax-pr 5365  ax-un 7682  ax-reg 9501  ax-inf2 9557

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-reu 3347  df-rab 3394  df-v 3435  df-sbc 3726  df-csb 3834  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-pss 3905  df-nul 4265  df-if 4458  df-pw 4534  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-iun 4926  df-br 5076  df-opab 5138  df-mpt 5157  df-tr 5183  df-id 5516  df-eprel 5521  df-po 5529  df-so 5530  df-fr 5574  df-we 5576  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-ima 5634  df-ord 6317  df-on 6318  df-lim 6319  df-suc 6320  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-f1 6494  df-fo 6495  df-f1o 6496  df-fv 6497  df-om 7811  df-1o 8399  df-bnj17 34885  df-bnj14 34887  df-bnj13 34889  df-bnj15 34891  df-bnj18 34893  df-bnj19 34895

This theorem is referenced by:  bnj60  35259