Theorem tz7.44-1 8462

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 6920	. . . 4 ⊢ (𝑦 = ∅ → (𝐹‘𝑦) = (𝐹‘∅))
2		reseq2 6004	. . . . . 6 ⊢ (𝑦 = ∅ → (𝐹 ↾ 𝑦) = (𝐹 ↾ ∅))
3		res0 6013	. . . . . 6 ⊢ (𝐹 ↾ ∅) = ∅
4	2, 3	eqtrdi 2796	. . . . 5 ⊢ (𝑦 = ∅ → (𝐹 ↾ 𝑦) = ∅)
5	4	fveq2d 6924	. . . 4 ⊢ (𝑦 = ∅ → (𝐺‘(𝐹 ↾ 𝑦)) = (𝐺‘∅))
6	1, 5	eqeq12d 2756	. . 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 3589	. 2 ⊢ (∅ ∈ 𝑋 → (𝐹‘∅) = (𝐺‘∅))
9		0ex 5325	. . 3 ⊢ ∅ ∈ V
10		iftrue 4554	. . . 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 7029	. . 3 ⊢ (∅ ∈ V → (𝐺‘∅) = 𝐴)
14	9, 13	ax-mp 5	. 2 ⊢ (𝐺‘∅) = 𝐴
15	8, 14	eqtrdi 2796	1 ⊢ (∅ ∈ 𝑋 → (𝐹‘∅) = 𝐴)

Syntax hints:   → wi 4   = wceq 1537   ∈ wcel 2108  Vcvv 3488  ∅c0 4352  ifcif 4548  ∪ cuni 4931   ↦ cmpt 5249  dom cdm 5700  ran crn 5701   ↾ cres 5702  Lim wlim 6396  ‘cfv 6573

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ral 3068  df-rex 3077  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-mpt 5250  df-id 5593  df-xp 5706  df-rel 5707  df-cnv 5708  df-co 5709  df-dm 5710  df-res 5712  df-iota 6525  df-fun 6575  df-fv 6581

This theorem is referenced by:  rdg0  8477