Theorem bnj1514 32943

Description: Technical lemma for bnj1500 32948. 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
bnj1514.1	⊢ 𝐵 = {𝑑 ∣ (𝑑 ⊆ 𝐴 ∧ ∀𝑥 ∈ 𝑑 pred(𝑥, 𝐴, 𝑅) ⊆ 𝑑)}
bnj1514.2	⊢ 𝑌 = ⟨𝑥, (𝑓 ↾ pred(𝑥, 𝐴, 𝑅))⟩
bnj1514.3	⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}

Assertion

Ref	Expression
bnj1514	⊢ (𝑓 ∈ 𝐶 → ∀𝑥 ∈ dom 𝑓(𝑓‘𝑥) = (𝐺‘𝑌))

Distinct variable groups:   𝑥,𝐴   𝐺,𝑑   𝑌,𝑑   𝑓,𝑑,𝑥

Allowed substitution hints:   𝐴(𝑓,𝑑)   𝐵(𝑥,𝑓,𝑑)   𝐶(𝑥,𝑓,𝑑)   𝑅(𝑥,𝑓,𝑑)   𝐺(𝑥,𝑓)   𝑌(𝑥,𝑓)

Proof of Theorem bnj1514

Step	Hyp	Ref	Expression
1		bnj1514.3	. . . . 5 ⊢ 𝐶 = {𝑓 ∣ ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))}
2	1	bnj1436 32719	. . . 4 ⊢ (𝑓 ∈ 𝐶 → ∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌)))
3		df-rex 3069	. . . . 5 ⊢ (∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌)) ↔ ∃𝑑(𝑑 ∈ 𝐵 ∧ (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))))
4		3anass 1093	. . . . 5 ⊢ ((𝑑 ∈ 𝐵 ∧ 𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌)) ↔ (𝑑 ∈ 𝐵 ∧ (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))))
5	3, 4	bnj133 32606	. . . 4 ⊢ (∃𝑑 ∈ 𝐵 (𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌)) ↔ ∃𝑑(𝑑 ∈ 𝐵 ∧ 𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌)))
6	2, 5	sylib 217	. . 3 ⊢ (𝑓 ∈ 𝐶 → ∃𝑑(𝑑 ∈ 𝐵 ∧ 𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌)))
7		simp3 1136	. . . 4 ⊢ ((𝑑 ∈ 𝐵 ∧ 𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌)) → ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌))
8		fndm 6520	. . . . . 6 ⊢ (𝑓 Fn 𝑑 → dom 𝑓 = 𝑑)
9	8	3ad2ant2 1132	. . . . 5 ⊢ ((𝑑 ∈ 𝐵 ∧ 𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌)) → dom 𝑓 = 𝑑)
10	9	raleqdv 3339	. . . 4 ⊢ ((𝑑 ∈ 𝐵 ∧ 𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌)) → (∀𝑥 ∈ dom 𝑓(𝑓‘𝑥) = (𝐺‘𝑌) ↔ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌)))
11	7, 10	mpbird 256	. . 3 ⊢ ((𝑑 ∈ 𝐵 ∧ 𝑓 Fn 𝑑 ∧ ∀𝑥 ∈ 𝑑 (𝑓‘𝑥) = (𝐺‘𝑌)) → ∀𝑥 ∈ dom 𝑓(𝑓‘𝑥) = (𝐺‘𝑌))
12	6, 11	bnj593 32625	. 2 ⊢ (𝑓 ∈ 𝐶 → ∃𝑑∀𝑥 ∈ dom 𝑓(𝑓‘𝑥) = (𝐺‘𝑌))
13	12	bnj937 32651	1 ⊢ (𝑓 ∈ 𝐶 → ∀𝑥 ∈ dom 𝑓(𝑓‘𝑥) = (𝐺‘𝑌))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1085   = wceq 1539  ∃wex 1783   ∈ wcel 2108  {cab 2715  ∀wral 3063  ∃wrex 3064   ⊆ wss 3883  ⟨cop 4564  dom cdm 5580   ↾ cres 5582   Fn wfn 6413  ‘cfv 6418   predc-bnj14 32567

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-12 2173  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-3an 1087  df-tru 1542  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-ral 3068  df-rex 3069  df-fn 6421

This theorem is referenced by:  bnj1501  32947