Theorem tz7.44-1 8340

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 6836	. . . 4 ⊢ (𝑦 = ∅ → (𝐹‘𝑦) = (𝐹‘∅))
2		reseq2 5935	. . . . . 6 ⊢ (𝑦 = ∅ → (𝐹 ↾ 𝑦) = (𝐹 ↾ ∅))
3		res0 5944	. . . . . 6 ⊢ (𝐹 ↾ ∅) = ∅
4	2, 3	eqtrdi 2788	. . . . 5 ⊢ (𝑦 = ∅ → (𝐹 ↾ 𝑦) = ∅)
5	4	fveq2d 6840	. . . 4 ⊢ (𝑦 = ∅ → (𝐺‘(𝐹 ↾ 𝑦)) = (𝐺‘∅))
6	1, 5	eqeq12d 2753	. . 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 3521	. 2 ⊢ (∅ ∈ 𝑋 → (𝐹‘∅) = (𝐺‘∅))
9		0ex 5243	. . 3 ⊢ ∅ ∈ V
10		iftrue 4473	. . . 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 6943	. . 3 ⊢ (∅ ∈ V → (𝐺‘∅) = 𝐴)
14	9, 13	ax-mp 5	. 2 ⊢ (𝐺‘∅) = 𝐴
15	8, 14	eqtrdi 2788	1 ⊢ (∅ ∈ 𝑋 → (𝐹‘∅) = 𝐴)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  Vcvv 3430  ∅c0 4274  ifcif 4467  ∪ cuni 4851   ↦ cmpt 5167  dom cdm 5626  ran crn 5627   ↾ cres 5628  Lim wlim 6320  ‘cfv 6494

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-sep 5232  ax-nul 5242  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-uni 4852  df-br 5087  df-opab 5149  df-mpt 5168  df-id 5521  df-xp 5632  df-rel 5633  df-cnv 5634  df-co 5635  df-dm 5636  df-res 5638  df-iota 6450  df-fun 6496  df-fv 6502

This theorem is referenced by:  rdg0  8355