Theorem bnj1371 31325

Description: Technical lemma for bnj60 31358. 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
bnj1371.1	⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1371.2	⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1371.3	⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
bnj1371.4	⊢ (𝜏 ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑥} ∪ trCl(𝑥, 𝐴, 𝑅))))
bnj1371.5	⊢ 𝐷 = {𝑥 ∈ 𝐴 ∣ ¬ ∃𝑓𝜏}
bnj1371.6	⊢ (𝜓 ↔ (𝑅 FrSe 𝐴 ∧ 𝐷 ≠ ∅))
bnj1371.7	⊢ (𝜒 ↔ (𝜓 ∧ 𝑥 ∈ 𝐷 ∧ ∀𝑦 ∈ 𝐷 ¬ 𝑦𝑅𝑥))
bnj1371.8	⊢ (𝜏′ ↔ [𝑦 / 𝑥]𝜏)
bnj1371.9	⊢ 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
bnj1371.10	⊢ 𝑃 = ∪ 𝐻
bnj1371.11	⊢ (𝜏′ ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))

Assertion

Ref	Expression
bnj1371	⊢ (𝑓 ∈ 𝐻 → Fun 𝑓)

Distinct variable groups:   𝑓,𝑑   𝑦,𝑓

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

Proof of Theorem bnj1371

Step	Hyp	Ref	Expression
1		bnj1371.9	. . . . . . 7 ⊢ 𝐻 = {𝑓 ∣ ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′}
2	1	bnj1436 31138	. . . . . 6 ⊢ (𝑓 ∈ 𝐻 → ∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′)
3		rexex 3104	. . . . . 6 ⊢ (∃𝑦 ∈ pred (𝑥, 𝐴, 𝑅)𝜏′ → ∃𝑦𝜏′)
4	2, 3	syl 17	. . . . 5 ⊢ (𝑓 ∈ 𝐻 → ∃𝑦𝜏′)
5		bnj1371.11	. . . . . 6 ⊢ (𝜏′ ↔ (𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
6	5	exbii 1887	. . . . 5 ⊢ (∃𝑦𝜏′ ↔ ∃𝑦(𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
7	4, 6	sylib 208	. . . 4 ⊢ (𝑓 ∈ 𝐻 → ∃𝑦(𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))))
8		exsimpl 1908	. . . 4 ⊢ (∃𝑦(𝑓 ∈ 𝐶 ∧ dom 𝑓 = ({𝑦} ∪ trCl(𝑦, 𝐴, 𝑅))) → ∃𝑦 𝑓 ∈ 𝐶)
9	7, 8	syl 17	. . 3 ⊢ (𝑓 ∈ 𝐻 → ∃𝑦 𝑓 ∈ 𝐶)
10		bnj1371.3	. . . . . . 7 ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
11	10	abeq2i 2837	. . . . . 6 ⊢ (𝑓 ∈ 𝐶 ↔ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌)))
12	11	bnj1238 31105	. . . . 5 ⊢ (𝑓 ∈ 𝐶 → ∃𝑑 ∈ 𝐵 𝑓 Fn 𝑑)
13		fnfun 6101	. . . . 5 ⊢ (𝑓 Fn 𝑑 → Fun 𝑓)
14	12, 13	bnj31 31015	. . . 4 ⊢ (𝑓 ∈ 𝐶 → ∃𝑑 ∈ 𝐵 Fun 𝑓)
15	14	bnj1265 31111	. . 3 ⊢ (𝑓 ∈ 𝐶 → Fun 𝑓)
16	9, 15	bnj593 31043	. 2 ⊢ (𝑓 ∈ 𝐻 → ∃𝑦Fun 𝑓)
17	16	bnj937 31070	1 ⊢ (𝑓 ∈ 𝐻 → Fun 𝑓)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∧ wa 383   ∧ w3a 1072   = wceq 1596  ∃wex 1817   ∈ wcel 2103  {cab 2710   ≠ wne 2896  ∀wral 3014  ∃wrex 3015  {crab 3018  [wsbc 3541   ∪ cun 3678   ⊆ wss 3680  ∅c0 4023  {csn 4285  ⟨cop 4291  ∪ cuni 4544   class class class wbr 4760  dom cdm 5218   ↾ cres 5220  Fun wfun 5995   Fn wfn 5996  ‘cfv 6001   predc-bnj14 30984   FrSe w-bnj15 30988   trClc-bnj18 30990

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1835  ax-4 1850  ax-5 1952  ax-6 2018  ax-7 2054  ax-9 2112  ax-12 2160  ax-ext 2704

This theorem depends on definitions:  df-bi 197  df-an 385  df-tru 1599  df-ex 1818  df-sb 2011  df-clab 2711  df-cleq 2717  df-clel 2720  df-ral 3019  df-rex 3020  df-fn 6004

This theorem is referenced by:  bnj1384  31328