Theorem tz7.44-1 8351

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 6840	. . . 4 ⊢ (𝑦 = ∅ → (𝐹‘𝑦) = (𝐹‘∅))
2		reseq2 5934	. . . . . 6 ⊢ (𝑦 = ∅ → (𝐹 ↾ 𝑦) = (𝐹 ↾ ∅))
3		res0 5943	. . . . . 6 ⊢ (𝐹 ↾ ∅) = ∅
4	2, 3	eqtrdi 2780	. . . . 5 ⊢ (𝑦 = ∅ → (𝐹 ↾ 𝑦) = ∅)
5	4	fveq2d 6844	. . . 4 ⊢ (𝑦 = ∅ → (𝐺‘(𝐹 ↾ 𝑦)) = (𝐺‘∅))
6	1, 5	eqeq12d 2745	. . 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 3540	. 2 ⊢ (∅ ∈ 𝑋 → (𝐹‘∅) = (𝐺‘∅))
9		0ex 5257	. . 3 ⊢ ∅ ∈ V
10		iftrue 4490	. . . 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 6950	. . 3 ⊢ (∅ ∈ V → (𝐺‘∅) = 𝐴)
14	9, 13	ax-mp 5	. 2 ⊢ (𝐺‘∅) = 𝐴
15	8, 14	eqtrdi 2780	1 ⊢ (∅ ∈ 𝑋 → (𝐹‘∅) = 𝐴)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  Vcvv 3444  ∅c0 4292  ifcif 4484  ∪ cuni 4867   ↦ cmpt 5183  dom cdm 5631  ran crn 5632   ↾ cres 5633  Lim wlim 6321  ‘cfv 6499

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pr 5382

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-res 5643  df-iota 6452  df-fun 6501  df-fv 6507

This theorem is referenced by:  rdg0  8366