Theorem bnj1416 35408

Description: Technical lemma for bnj60 35431. 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 5896	. . 3 ⊢ dom 𝑄 = dom (𝑃 ∪ {⟨𝑥, (𝐺‘𝑍)⟩})
3		dmun 5902	. . 3 ⊢ dom (𝑃 ∪ {⟨𝑥, (𝐺‘𝑍)⟩}) = (dom 𝑃 ∪ dom {⟨𝑥, (𝐺‘𝑍)⟩})
4		fvex 6896	. . . . 5 ⊢ (𝐺‘𝑍) ∈ V
5	4	dmsnop 6219	. . . 4 ⊢ dom {⟨𝑥, (𝐺‘𝑍)⟩} = {𝑥}
6	5	uneq2i 4120	. . 3 ⊢ (dom 𝑃 ∪ dom {⟨𝑥, (𝐺‘𝑍)⟩}) = (dom 𝑃 ∪ {𝑥})
7	2, 3, 6	3eqtri 2790	. 2 ⊢ dom 𝑄 = (dom 𝑃 ∪ {𝑥})
8		bnj1416.28	. . . 4 ⊢ (𝜒 → dom 𝑃 = trCl(𝑥, 𝐴, 𝑅))
9	8	uneq1d 4122	. . 3 ⊢ (𝜒 → (dom 𝑃 ∪ {𝑥}) = ( trCl(𝑥, 𝐴, 𝑅) ∪ {𝑥}))
10		uncom 4113	. . 3 ⊢ ( trCl(𝑥, 𝐴, 𝑅) ∪ {𝑥}) = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))
11	9, 10	eqtrdi 2814	. 2 ⊢ (𝜒 → (dom 𝑃 ∪ {𝑥}) = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
12	7, 11	eqtrid 2810	1 ⊢ (𝜒 → dom 𝑄 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 400   ∧ w3a 1103   = wceq 1570  ∃wex 1809   ∈ wcel 2143  {cab 2741   ≠ wne 2958  ∀wral 3079  ∃wrex 3089  {crab 3416  [wsbc 3745   ∪ cun 3904   ⊆ wss 3906  ∅c0 4287  {csn 4590  ⟨cop 4596  ∪ cuni 4873   class class class wbr 5110  dom cdm 5663   ↾ cres 5665   Fn wfn 6533  ‘cfv 6538   predc-bnj14 35058   FrSe w-bnj15 35062   trClc-bnj18 35064

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-dm 5673  df-iota 6494  df-fv 6546

This theorem is referenced by:  bnj1312  35427