Theorem bnj1491 35687

Description: Technical lemma for bnj60 35692. 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 35683	. . . 4 ⊢ (𝑤 ∈ 𝑄 → ∀𝑓 𝑤 ∈ 𝑄)
15	14	nfcii 2911	. . 3 ⊢ Ⅎ𝑓𝑄
16	4	bnj1317 35451	. . . . . 6 ⊢ (𝑤 ∈ 𝐶 → ∀𝑓 𝑤 ∈ 𝐶)
17	16	nfcii 2911	. . . . 5 ⊢ Ⅎ𝑓𝐶
18	15, 17	nfel 2936	. . . 4 ⊢ Ⅎ𝑓 𝑄 ∈ 𝐶
19	15	nfdm 5932	. . . . 5 ⊢ Ⅎ𝑓dom 𝑄
20	19	nfeq1 2937	. . . 4 ⊢ Ⅎ𝑓dom 𝑄 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))
21	18, 20	nfan 1931	. . 3 ⊢ Ⅎ𝑓(𝑄 ∈ 𝐶 ∧ dom 𝑄 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅)))
22		eleq1 2848	. . . 4 ⊢ (𝑓 = 𝑄 → (𝑓 ∈ 𝐶 ↔ 𝑄 ∈ 𝐶))
23		dmeq 5884	. . . . 5 ⊢ (𝑓 = 𝑄 → dom 𝑓 = dom 𝑄)
24	23	eqeq1d 2762	. . . 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 3546	. 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 1569  ∃wex 1811   ∈ wcel 2144  {cab 2738   ≠ wne 2955  ∀wral 3076  ∃wrex 3086  {crab 3412  Vcvv 3450  [wsbc 3738   ⊆ wss 3895   ∪ cun 4149  ∅c0 4278  {csn 4583  ⟨cop 4589  ∪ cuni 4866   class class class wbr 5102  dom cdm 5650   ↾ cres 5652   Fn wfn 6533  ‘cfv 6538   predc-bnj14 35319   FrSe w-bnj15 35323   trClc-bnj18 35325

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-br 5103  df-opab 5167  df-xp 5656  df-dm 5660  df-res 5662  df-iota 6494  df-fv 6546

This theorem is used by:  bnj1312  35688