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Theorem tz7.44-1 7547
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 (𝑦𝑋 → (𝐹𝑦) = (𝐺‘(𝐹𝑦)))
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
StepHypRef Expression
1 fveq2 6229 . . . 4 (𝑦 = ∅ → (𝐹𝑦) = (𝐹‘∅))
2 reseq2 5423 . . . . . 6 (𝑦 = ∅ → (𝐹𝑦) = (𝐹 ↾ ∅))
3 res0 5432 . . . . . 6 (𝐹 ↾ ∅) = ∅
42, 3syl6eq 2701 . . . . 5 (𝑦 = ∅ → (𝐹𝑦) = ∅)
54fveq2d 6233 . . . 4 (𝑦 = ∅ → (𝐺‘(𝐹𝑦)) = (𝐺‘∅))
61, 5eqeq12d 2666 . . 3 (𝑦 = ∅ → ((𝐹𝑦) = (𝐺‘(𝐹𝑦)) ↔ (𝐹‘∅) = (𝐺‘∅)))
7 tz7.44.2 . . 3 (𝑦𝑋 → (𝐹𝑦) = (𝐺‘(𝐹𝑦)))
86, 7vtoclga 3303 . 2 (∅ ∈ 𝑋 → (𝐹‘∅) = (𝐺‘∅))
9 0ex 4823 . . 3 ∅ ∈ V
10 iftrue 4125 . . . 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
1310, 11, 12fvmpt 6321 . . 3 (∅ ∈ V → (𝐺‘∅) = 𝐴)
149, 13ax-mp 5 . 2 (𝐺‘∅) = 𝐴
158, 14syl6eq 2701 1 (∅ ∈ 𝑋 → (𝐹‘∅) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523  wcel 2030  Vcvv 3231  c0 3948  ifcif 4119   cuni 4468  cmpt 4762  dom cdm 5143  ran crn 5144  cres 5145  Lim wlim 5762  cfv 5926
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-mpt 4763  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-res 5155  df-iota 5889  df-fun 5928  df-fv 5934
This theorem is referenced by:  rdg0  7562
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