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Theorem tz7.44-1 7389
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 6103 . . . 4 (𝑦 = ∅ → (𝐹𝑦) = (𝐹‘∅))
2 reseq2 5312 . . . . . 6 (𝑦 = ∅ → (𝐹𝑦) = (𝐹 ↾ ∅))
3 res0 5321 . . . . . 6 (𝐹 ↾ ∅) = ∅
42, 3syl6eq 2660 . . . . 5 (𝑦 = ∅ → (𝐹𝑦) = ∅)
54fveq2d 6107 . . . 4 (𝑦 = ∅ → (𝐺‘(𝐹𝑦)) = (𝐺‘∅))
61, 5eqeq12d 2625 . . 3 (𝑦 = ∅ → ((𝐹𝑦) = (𝐺‘(𝐹𝑦)) ↔ (𝐹‘∅) = (𝐺‘∅)))
7 tz7.44.2 . . 3 (𝑦𝑋 → (𝐹𝑦) = (𝐺‘(𝐹𝑦)))
86, 7vtoclga 3245 . 2 (∅ ∈ 𝑋 → (𝐹‘∅) = (𝐺‘∅))
9 0ex 4718 . . 3 ∅ ∈ V
10 iftrue 4042 . . . 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 6191 . . 3 (∅ ∈ V → (𝐺‘∅) = 𝐴)
149, 13ax-mp 5 . 2 (𝐺‘∅) = 𝐴
158, 14syl6eq 2660 1 (∅ ∈ 𝑋 → (𝐹‘∅) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1475  wcel 1977  Vcvv 3173  c0 3874  ifcif 4036   cuni 4372  cmpt 4643  dom cdm 5038  ran crn 5039  cres 5040  Lim wlim 5641  cfv 5804
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833
This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-sbc 3403  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-mpt 4645  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-res 5050  df-iota 5768  df-fun 5806  df-fv 5812
This theorem is referenced by:  rdg0  7404
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