Theorem bnj1491 35754

Description: Technical lemma for bnj60 35759. 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
bnj1491.1	⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1491.2	⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1491.3	⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
bnj1491.4	⊢ (𝜏 ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
bnj1491.5	⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏}
bnj1491.6	⊢ (𝜓 ↔ (𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅))
bnj1491.7	⊢ (𝜒 ↔ (𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥))
bnj1491.8	⊢ (𝜏′ ↔ [𝑦 / 𝑥]𝜏)
bnj1491.9	⊢ 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
bnj1491.10	⊢ 𝑃 = ∪ 𝐻
bnj1491.11	⊢ 𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1491.12	⊢ 𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺‘𝑍)⟩})
bnj1491.13	⊢ (𝜒 → (𝑄 ∈ 𝐶 ∧ dom 𝑄 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))

Assertion

Ref	Expression
bnj1491	⊢ ((𝜒 ∧ 𝑄 ∈ V) → ∃𝑓(𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))

Distinct variable groups:   𝐴,𝑓   𝑓,𝐺   𝑅,𝑓   𝑥,𝑓

Allowed substitution hints:   𝜓(𝑥, 𝑦, 𝑓, 𝑑)   𝜒(𝑥, 𝑦, 𝑓, 𝑑)   𝜏(𝑥, 𝑦, 𝑓, 𝑑)   𝐴(𝑥, 𝑦, 𝑑)   𝐵(𝑥, 𝑦, 𝑓, 𝑑)   𝐶(𝑥, 𝑦, 𝑓, 𝑑)   𝐷(𝑥, 𝑦, 𝑓, 𝑑)   𝑃(𝑥, 𝑦, 𝑓, 𝑑)   𝑄(𝑥, 𝑦, 𝑓, 𝑑)   𝑅(𝑥, 𝑦, 𝑑)   𝐺(𝑥, 𝑦, 𝑑)   𝐻(𝑥, 𝑦, 𝑓, 𝑑)   𝑌(𝑥, 𝑦, 𝑓, 𝑑)   𝑍(𝑥, 𝑦, 𝑓, 𝑑)   𝜏′(𝑥, 𝑦, 𝑓, 𝑑)

Proof of Theorem bnj1491

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		bnj1491.13	. 2 ⊢ (𝜒 → (𝑄 ∈ 𝐶 ∧ dom 𝑄 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
2		bnj1491.1	. . . . 5 ⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
3		bnj1491.2	. . . . 5 ⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
4		bnj1491.3	. . . . 5 ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
5		bnj1491.4	. . . . 5 ⊢ (𝜏 ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
6		bnj1491.5	. . . . 5 ⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏}
7		bnj1491.6	. . . . 5 ⊢ (𝜓 ↔ (𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅))
8		bnj1491.7	. . . . 5 ⊢ (𝜒 ↔ (𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥))
9		bnj1491.8	. . . . 5 ⊢ (𝜏′ ↔ [𝑦 / 𝑥]𝜏)
10		bnj1491.9	. . . . 5 ⊢ 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
11		bnj1491.10	. . . . 5 ⊢ 𝑃 = ∪ 𝐻
12		bnj1491.11	. . . . 5 ⊢ 𝑍 = ⟨𝑥, (𝑃 ↾ pred(𝑥, 𝐴, 𝑅))⟩
13		bnj1491.12	. . . . 5 ⊢ 𝑄 = (𝑃 ∪ {⟨𝑥, (𝐺‘𝑍)⟩})
14	2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13	bnj1466 35750	. . . 4 ⊢ (𝑤 ∈ 𝑄 → ∀𝑓 𝑤 ∈ 𝑄)
15	14	nfcii 2915	. . 3 ⊢ Ⅎ𝑓𝑄
16	4	bnj1317 35518	. . . . . 6 ⊢ (𝑤 ∈ 𝐶 → ∀𝑓 𝑤 ∈ 𝐶)
17	16	nfcii 2915	. . . . 5 ⊢ Ⅎ𝑓𝐶
18	15, 17	nfel 2940	. . . 4 ⊢ Ⅎ𝑓 𝑄 ∈ 𝐶
19	15	nfdm 5939	. . . . 5 ⊢ Ⅎ𝑓dom 𝑄
20	19	nfeq1 2941	. . . 4 ⊢ Ⅎ𝑓dom 𝑄 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))
21	18, 20	nfan 1934	. . 3 ⊢ Ⅎ𝑓(𝑄 ∈ 𝐶 ∧ dom 𝑄 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
22		eleq1 2852	. . . 4 ⊢ (𝑓 = 𝑄 → (𝑓 ∈ 𝐶 ↔ 𝑄 ∈ 𝐶))
23		dmeq 5891	. . . . 5 ⊢ (𝑓 = 𝑄 → dom 𝑓 = dom 𝑄)
24	23	eqeq1d 2765	. . . 4 ⊢ (𝑓 = 𝑄 → (dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)) ↔ dom 𝑄 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
25	22, 24	anbi12d 644	. . 3 ⊢ (𝑓 = 𝑄 → ((𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))) ↔ (𝑄 ∈ 𝐶 ∧ dom 𝑄 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))))
26	15, 21, 25	spcegf 3550	. 2 ⊢ (𝑄 ∈ V → ((𝑄 ∈ 𝐶 ∧ dom 𝑄 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))) → ∃𝑓(𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))))
27	1, 26	mpan9 516	1 ⊢ ((𝜒 ∧ 𝑄 ∈ V) → ∃𝑓(𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∧ w3a 1103   = wceq 1570  ∃wex 1812   ∈ wcel 2147  {cab 2741   ≠ wne 2959  ∀wral 3080  ∃wrex 3090  {crab 3416  Vcvv 3454  [wsbc 3742   ⊆ wss 3900   ∪ cun 4152  ∅c0 4281  {csn 4588  ⟨cop 4838  ∪ cuni 4870   class class class wbr 5106  dom cdm 5655   ↾ cres 5657   Fn wfn 6550  ‘cfv 6555   predc-bnj14 35386   FrSe w-bnj15 35390   trClc-bnj18 35392

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-10 2180  ax-11 2197  ax-12 2216  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-nfc 2913  df-ral 3081  df-rex 3091  df-rab 3417  df-v 3456  df-ss 3901  df-dif 4050  df-in 4094  df-un 4154  df-nul 4282  df-if 4486  df-sn 4589  df-pr 4591  df-op 4839  df-uni 4871  df-br 5107  df-opab 5171  df-xp 5661  df-dm 5665  df-res 5667  df-iota 6509  df-fv 6563

This theorem is used by:  bnj1312  35755