MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  tz7.44-1 Structured version   Visualization version   GIF version

Theorem tz7.44-1 7655
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 6332 . . . 4 (𝑦 = ∅ → (𝐹𝑦) = (𝐹‘∅))
2 reseq2 5529 . . . . . 6 (𝑦 = ∅ → (𝐹𝑦) = (𝐹 ↾ ∅))
3 res0 5538 . . . . . 6 (𝐹 ↾ ∅) = ∅
42, 3syl6eq 2821 . . . . 5 (𝑦 = ∅ → (𝐹𝑦) = ∅)
54fveq2d 6336 . . . 4 (𝑦 = ∅ → (𝐺‘(𝐹𝑦)) = (𝐺‘∅))
61, 5eqeq12d 2786 . . 3 (𝑦 = ∅ → ((𝐹𝑦) = (𝐺‘(𝐹𝑦)) ↔ (𝐹‘∅) = (𝐺‘∅)))
7 tz7.44.2 . . 3 (𝑦𝑋 → (𝐹𝑦) = (𝐺‘(𝐹𝑦)))
86, 7vtoclga 3423 . 2 (∅ ∈ 𝑋 → (𝐹‘∅) = (𝐺‘∅))
9 0ex 4924 . . 3 ∅ ∈ V
10 iftrue 4231 . . . 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 6424 . . 3 (∅ ∈ V → (𝐺‘∅) = 𝐴)
149, 13ax-mp 5 . 2 (𝐺‘∅) = 𝐴
158, 14syl6eq 2821 1 (∅ ∈ 𝑋 → (𝐹‘∅) = 𝐴)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1631  wcel 2145  Vcvv 3351  c0 4063  ifcif 4225   cuni 4574  cmpt 4863  dom cdm 5249  ran crn 5250  cres 5251  Lim wlim 5867  cfv 6031
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-uni 4575  df-br 4787  df-opab 4847  df-mpt 4864  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-res 5261  df-iota 5994  df-fun 6033  df-fv 6039
This theorem is referenced by:  rdg0  7670
  Copyright terms: Public domain W3C validator