Theorem bnj1498 30383

Description: Technical lemma for bnj60 30384. 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 4460	. . . . . . 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 30164	. . . . . . . . . . . . . . 15 ⊢ (𝑓 ∈ 𝐶 → ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌)))
4	3	bnj1299 30143	. . . . . . . . . . . . . 14 ⊢ (𝑓 ∈ 𝐶 → ∃𝑑 ∈ 𝐵 𝑓 Fn 𝑑)
5		fndm 5904	. . . . . . . . . . . . . 14 ⊢ (𝑓 Fn 𝑑 → dom 𝑓 = 𝑑)
6	4, 5	bnj31 30039	. . . . . . . . . . . . 13 ⊢ (𝑓 ∈ 𝐶 → ∃𝑑 ∈ 𝐵 dom 𝑓 = 𝑑)
7	6	bnj1196 30119	. . . . . . . . . . . 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 30164	. . . . . . . . . . . . . 14 ⊢ (𝑑 ∈ 𝐵 → (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑))
10	9	simpld 474	. . . . . . . . . . . . 13 ⊢ (𝑑 ∈ 𝐵 → 𝑑 ⊆ 𝐴)
11	10	anim1i 590	. . . . . . . . . . . 12 ⊢ ((𝑑 ∈ 𝐵 ∧ dom 𝑓 = 𝑑) → (𝑑 ⊆ 𝐴 ∧ dom 𝑓 = 𝑑))
12	7, 11	bnj593 30069	. . . . . . . . . . 11 ⊢ (𝑓 ∈ 𝐶 → ∃𝑑(𝑑 ⊆ 𝐴 ∧ dom 𝑓 = 𝑑))
13		sseq1 3589	. . . . . . . . . . . 12 ⊢ (dom 𝑓 = 𝑑 → (dom 𝑓 ⊆ 𝐴 ↔ 𝑑 ⊆ 𝐴))
14	13	biimparc 503	. . . . . . . . . . 11 ⊢ ((𝑑 ⊆ 𝐴 ∧ dom 𝑓 = 𝑑) → dom 𝑓 ⊆ 𝐴)
15	12, 14	bnj593 30069	. . . . . . . . . 10 ⊢ (𝑓 ∈ 𝐶 → ∃𝑑dom 𝑓 ⊆ 𝐴)
16	15	bnj937 30096	. . . . . . . . 9 ⊢ (𝑓 ∈ 𝐶 → dom 𝑓 ⊆ 𝐴)
17	16	sselda 3568	. . . . . . . 8 ⊢ ((𝑓 ∈ 𝐶 ∧ 𝑧 ∈ dom 𝑓) → 𝑧 ∈ 𝐴)
18	17	rexlimiva 3010	. . . . . . 7 ⊢ (∃𝑓 ∈ 𝐶 𝑧 ∈ dom 𝑓 → 𝑧 ∈ 𝐴)
19	1, 18	sylbi 206	. . . . . 6 ⊢ (𝑧 ∈ ∪ 𝑓 ∈ 𝐶 dom 𝑓 → 𝑧 ∈ 𝐴)
20	2	bnj1317 30146	. . . . . . 7 ⊢ (𝑤 ∈ 𝐶 → ∀𝑓 𝑤 ∈ 𝐶)
21	20	bnj1400 30160	. . . . . 6 ⊢ dom ∪ 𝐶 = ∪ 𝑓 ∈ 𝐶 dom 𝑓
22	19, 21	eleq2s 2706	. . . . 5 ⊢ (𝑧 ∈ dom ∪ 𝐶 → 𝑧 ∈ 𝐴)
23		bnj1498.4	. . . . . 6 ⊢ 𝐹 = ∪ 𝐶
24	23	dmeqi 5247	. . . . 5 ⊢ dom 𝐹 = dom ∪ 𝐶
25	22, 24	eleq2s 2706	. . . 4 ⊢ (𝑧 ∈ dom 𝐹 → 𝑧 ∈ 𝐴)
26	25	ssriv 3572	. . 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 30381	. . . . . . 7 ⊢ (𝑅 FrSe 𝐴 → ∀𝑥 ∈ 𝐴 ∃𝑓 ∈ 𝐶 dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
30		vsnid 4156	. . . . . . . . . . 11 ⊢ 𝑥 ∈ {𝑥}
31		elun1 3742	. . . . . . . . . . 11 ⊢ (𝑥 ∈ {𝑥} → 𝑥 ∈ ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
32	30, 31	ax-mp 5	. . . . . . . . . 10 ⊢ 𝑥 ∈ ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))
33		eleq2 2677	. . . . . . . . . 10 ⊢ (dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)) → (𝑥 ∈ dom 𝑓 ↔ 𝑥 ∈ ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
34	32, 33	mpbiri 247	. . . . . . . . 9 ⊢ (dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)) → 𝑥 ∈ dom 𝑓)
35	34	reximi 2994	. . . . . . . 8 ⊢ (∃𝑓 ∈ 𝐶 dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)) → ∃𝑓 ∈ 𝐶 𝑥 ∈ dom 𝑓)
36	35	ralimi 2936	. . . . . . 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 4460	. . . . . . 7 ⊢ (𝑥 ∈ ∪ 𝑓 ∈ 𝐶 dom 𝑓 ↔ ∃𝑓 ∈ 𝐶 𝑥 ∈ dom 𝑓)
39	38	ralbii 2963	. . . . . 6 ⊢ (∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ 𝑓 ∈ 𝐶 dom 𝑓 ↔ ∀𝑥 ∈ 𝐴 ∃𝑓 ∈ 𝐶 𝑥 ∈ dom 𝑓)
40	37, 39	sylibr 223	. . . . 5 ⊢ (𝑅 FrSe 𝐴 → ∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ 𝑓 ∈ 𝐶 dom 𝑓)
41		nfcv 2751	. . . . . 6 ⊢ Ⅎ𝑥𝐴
42	8	bnj1309 30344	. . . . . . . . 9 ⊢ (𝑡 ∈ 𝐵 → ∀𝑥 𝑡 ∈ 𝐵)
43	2, 42	bnj1307 30345	. . . . . . . 8 ⊢ (𝑡 ∈ 𝐶 → ∀𝑥 𝑡 ∈ 𝐶)
44	43	nfcii 2742	. . . . . . 7 ⊢ Ⅎ𝑥𝐶
45		nfcv 2751	. . . . . . 7 ⊢ Ⅎ𝑥dom 𝑓
46	44, 45	nfiun 4484	. . . . . 6 ⊢ Ⅎ𝑥∪ 𝑓 ∈ 𝐶 dom 𝑓
47	41, 46	dfss3f 3560	. . . . 5 ⊢ (𝐴 ⊆ ∪ 𝑓 ∈ 𝐶 dom 𝑓 ↔ ∀𝑥 ∈ 𝐴 𝑥 ∈ ∪ 𝑓 ∈ 𝐶 dom 𝑓)
48	40, 47	sylibr 223	. . . 4 ⊢ (𝑅 FrSe 𝐴 → 𝐴 ⊆ ∪ 𝑓 ∈ 𝐶 dom 𝑓)
49	48, 21	syl6sseqr 3615	. . 3 ⊢ (𝑅 FrSe 𝐴 → 𝐴 ⊆ dom ∪ 𝐶)
50	49, 24	syl6sseqr 3615	. 2 ⊢ (𝑅 FrSe 𝐴 → 𝐴 ⊆ dom 𝐹)
51	27, 50	eqssd 3585	1 ⊢ (𝑅 FrSe 𝐴 → dom 𝐹 = 𝐴)

Syntax hints:   → wi 4   ∧ wa 383   = wceq 1475   ∈ wcel 1977  {cab 2596  ∀wral 2896  ∃wrex 2897   ∪ cun 3538   ⊆ wss 3540  {csn 4125  ⟨cop 4131  ∪ cuni 4372  ∪ ciun 4455  dom cdm 5038   ↾ cres 5040   Fn wfn 5799  ‘cfv 5804   predc-bnj14 30007   FrSe w-bnj15 30011   trClc-bnj18 30013

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-rep 4699  ax-sep 4709  ax-nul 4717  ax-pow 4769  ax-pr 4833  ax-un 6847  ax-reg 8380  ax-inf2 8421

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3or 1032  df-3an 1033  df-tru 1478  df-fal 1481  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-reu 2903  df-rab 2905  df-v 3175  df-sbc 3403  df-csb 3500  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-pss 3556  df-nul 3875  df-if 4037  df-pw 4110  df-sn 4126  df-pr 4128  df-tp 4130  df-op 4132  df-uni 4373  df-iun 4457  df-br 4584  df-opab 4644  df-mpt 4645  df-tr 4681  df-eprel 4949  df-id 4953  df-po 4959  df-so 4960  df-fr 4997  df-we 4999  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051  df-ord 5643  df-on 5644  df-lim 5645  df-suc 5646  df-iota 5768  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-fv 5812  df-om 6958  df-1o 7447  df-bnj17 30006  df-bnj14 30008  df-bnj13 30010  df-bnj15 30012  df-bnj18 30014  df-bnj19 30016

This theorem is referenced by:  bnj60  30384