Theorem tz7.44-1 8392

Description: The value of 𝐹 at ∅. Part 1 of Theorem 7.44 of [TakeutiZaring] p. 49. (Contributed by NM, 23-Apr-1995.) (Revised by Mario Carneiro, 14-Nov-2014.)

Hypotheses

Ref	Expression
tz7.44.1	⊢ 𝐺 = (𝑥 ∈ V ↦ if(𝑥 = ∅, 𝐴, if(Lim dom 𝑥, ∪ ran 𝑥, (𝐻‘(𝑥‘∪ dom 𝑥)))))
tz7.44.2	⊢ (𝑦 ∈ 𝑋 → (𝐹‘𝑦) = (𝐺‘(𝐹 ↾ 𝑦)))
tz7.44-1.3	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
tz7.44-1	⊢ (∅ ∈ 𝑋 → (𝐹‘∅) = 𝐴)

Distinct variable groups:   𝑥,𝐴   𝑥,𝑦,𝐹   𝑦,𝐺   𝑥,𝐻   𝑦,𝑋

Allowed substitution hints:   𝐴(𝑦)   𝐺(𝑥)   𝐻(𝑦)   𝑋(𝑥)

Proof of Theorem tz7.44-1

Step	Hyp	Ref	Expression
1		fveq2 6873	. . . 4 ⊢ (𝑦 = ∅ → (𝐹‘𝑦) = (𝐹‘∅))
2		reseq2 5961	. . . . . 6 ⊢ (𝑦 = ∅ → (𝐹 ↾ 𝑦) = (𝐹 ↾ ∅))
3		res0 5970	. . . . . 6 ⊢ (𝐹 ↾ ∅) = ∅
4	2, 3	eqtrdi 2811	. . . . 5 ⊢ (𝑦 = ∅ → (𝐹 ↾ 𝑦) = ∅)
5	4	fveq2d 6877	. . . 4 ⊢ (𝑦 = ∅ → (𝐺‘(𝐹 ↾ 𝑦)) = (𝐺‘∅))
6	1, 5	eqeq12d 2776	. . 3 ⊢ (𝑦 = ∅ → ((𝐹‘𝑦) = (𝐺‘(𝐹 ↾ 𝑦)) ↔ (𝐹‘∅) = (𝐺‘∅)))
7		tz7.44.2	. . 3 ⊢ (𝑦 ∈ 𝑋 → (𝐹‘𝑦) = (𝐺‘(𝐹 ↾ 𝑦)))
8	6, 7	vtoclga 3536	. 2 ⊢ (∅ ∈ 𝑋 → (𝐹‘∅) = (𝐺‘∅))
9		0ex 5260	. . 3 ⊢ ∅ ∈ V
10		iftrue 4487	. . . 4 ⊢ (𝑥 = ∅ → if(𝑥 = ∅, 𝐴, if(Lim dom 𝑥, ∪ ran 𝑥, (𝐻‘(𝑥‘∪ dom 𝑥)))) = 𝐴)
11		tz7.44.1	. . . 4 ⊢ 𝐺 = (𝑥 ∈ V ↦ if(𝑥 = ∅, 𝐴, if(Lim dom 𝑥, ∪ ran 𝑥, (𝐻‘(𝑥‘∪ dom 𝑥)))))
12		tz7.44-1.3	. . . 4 ⊢ 𝐴 ∈ V
13	10, 11, 12	fvmpt 6981	. . 3 ⊢ (∅ ∈ V → (𝐺‘∅) = 𝐴)
14	9, 13	ax-mp 5	. 2 ⊢ (𝐺‘∅) = 𝐴
15	8, 14	eqtrdi 2811	1 ⊢ (∅ ∈ 𝑋 → (𝐹‘∅) = 𝐴)

This proof depends on syntax axioms:   → wi 4   = wceq 1570   ∈ wcel 2145  Vcvv 3450  ∅c0 4278  ifcif 4481  ∪ cuni 4866   ↦ cmpt 5185  dom cdm 5647  ran crn 5648   ↾ cres 5649  Lim wlim 6352  ‘cfv 6527

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 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2213  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390

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 2100  df-mo 2564  df-eu 2594  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-dif 3901  df-un 3903  df-in 3905  df-ss 3915  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-mpt 5186  df-id 5542  df-xp 5653  df-rel 5654  df-cnv 5655  df-co 5656  df-dm 5657  df-res 5659  df-iota 6483  df-fun 6529  df-fv 6535

This theorem is used by:  rdg0  8407