Theorem bnj1416 35054

Description: Technical lemma for bnj60 35077. 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
bnj1416.1	⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1416.2	⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1416.3	⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
bnj1416.4	⊢ (𝜏 ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
bnj1416.5	⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏}
bnj1416.6	⊢ (𝜓 ↔ (𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅))
bnj1416.7	⊢ (𝜒 ↔ (𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥))
bnj1416.8	⊢ (𝜏′ ↔ [𝑦 / 𝑥]𝜏)
bnj1416.9	⊢ 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
bnj1416.10	⊢ 𝑃 = ∪ 𝐻
bnj1416.11	⊢ 𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1416.12	⊢ 𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺‘𝑍)⟩})
bnj1416.28	⊢ (𝜒 → dom 𝑃 = trCl(𝑥, 𝐴, 𝑅))

Assertion

Ref	Expression
bnj1416	⊢ (𝜒 → dom 𝑄 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))

Proof of Theorem bnj1416

Step	Hyp	Ref	Expression
1		bnj1416.12	. . . 4 ⊢ 𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺‘𝑍)⟩})
2	1	dmeqi 5914	. . 3 ⊢ dom 𝑄 = dom (𝑃 ∪ {⟨𝑥, (𝐺‘𝑍)⟩})
3		dmun 5920	. . 3 ⊢ dom (𝑃 ∪ {⟨𝑥, (𝐺‘𝑍)⟩}) = (dom 𝑃 ∪ dom {⟨𝑥, (𝐺‘𝑍)⟩})
4		fvex 6918	. . . . 5 ⊢ (𝐺‘𝑍) ∈ V
5	4	dmsnop 6235	. . . 4 ⊢ dom {⟨𝑥, (𝐺‘𝑍)⟩} = {𝑥}
6	5	uneq2i 4164	. . 3 ⊢ (dom 𝑃 ∪ dom {⟨𝑥, (𝐺‘𝑍)⟩}) = (dom 𝑃 ∪ {𝑥})
7	2, 3, 6	3eqtri 2768	. 2 ⊢ dom 𝑄 = (dom 𝑃 ∪ {𝑥})
8		bnj1416.28	. . . 4 ⊢ (𝜒 → dom 𝑃 = trCl(𝑥, 𝐴, 𝑅))
9	8	uneq1d 4166	. . 3 ⊢ (𝜒 → (dom 𝑃 ∪ {𝑥}) = ( trCl(𝑥, 𝐴, 𝑅) ∪ {𝑥}))
10		uncom 4157	. . 3 ⊢ ( trCl(𝑥, 𝐴, 𝑅) ∪ {𝑥}) = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))
11	9, 10	eqtrdi 2792	. 2 ⊢ (𝜒 → (dom 𝑃 ∪ {𝑥}) = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
12	7, 11	eqtrid 2788	1 ⊢ (𝜒 → dom 𝑄 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∧ w3a 1086   = wceq 1539  ∃wex 1778   ∈ wcel 2107  {cab 2713   ≠ wne 2939  ∀wral 3060  ∃wrex 3069  {crab 3435  [wsbc 3787   ∪ cun 3948   ⊆ wss 3950  ∅c0 4332  {csn 4625  ⟨cop 4631  ∪ cuni 4906   class class class wbr 5142  dom cdm 5684   ↾ cres 5686   Fn wfn 6555  ‘cfv 6560   predc-bnj14 34703   FrSe w-bnj15 34707   trClc-bnj18 34709

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-10 2140  ax-12 2176  ax-ext 2707  ax-sep 5295  ax-nul 5305  ax-pr 5431

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-nf 1783  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4626  df-pr 4628  df-op 4632  df-uni 4907  df-br 5143  df-dm 5694  df-iota 6513  df-fv 6568

This theorem is referenced by:  bnj1312  35073