MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  eqfunressuc Structured version   Visualization version   GIF version

Theorem eqfunressuc 7307
Description: Law for equality of restriction to successors. This is primarily useful when 𝑋 is an ordinal, but it does not require that. (Contributed by Scott Fenton, 6-Dec-2021.)
Assertion
Ref Expression
eqfunressuc (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐹𝑋) = (𝐺𝑋) ∧ (𝑋 ∈ dom 𝐹𝑋 ∈ dom 𝐺 ∧ (𝐹𝑋) = (𝐺𝑋))) → (𝐹 ↾ suc 𝑋) = (𝐺 ↾ suc 𝑋))

Proof of Theorem eqfunressuc
StepHypRef Expression
1 eqfunresadj 7306 . 2 (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐹𝑋) = (𝐺𝑋) ∧ (𝑋 ∈ dom 𝐹𝑋 ∈ dom 𝐺 ∧ (𝐹𝑋) = (𝐺𝑋))) → (𝐹 ↾ (𝑋 ∪ {𝑋})) = (𝐺 ↾ (𝑋 ∪ {𝑋})))
2 df-suc 6324 . . 3 suc 𝑋 = (𝑋 ∪ {𝑋})
32reseq2i 5935 . 2 (𝐹 ↾ suc 𝑋) = (𝐹 ↾ (𝑋 ∪ {𝑋}))
42reseq2i 5935 . 2 (𝐺 ↾ suc 𝑋) = (𝐺 ↾ (𝑋 ∪ {𝑋}))
51, 3, 43eqtr4g 2802 1 (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐹𝑋) = (𝐺𝑋) ∧ (𝑋 ∈ dom 𝐹𝑋 ∈ dom 𝐺 ∧ (𝐹𝑋) = (𝐺𝑋))) → (𝐹 ↾ suc 𝑋) = (𝐺 ↾ suc 𝑋))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397  w3a 1088   = wceq 1542  wcel 2107  cun 3909  {csn 4587  dom cdm 5634  cres 5636  suc csuc 6320  Fun wfun 6491  cfv 6497
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-12 2172  ax-ext 2708  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-rex 3075  df-rab 3409  df-v 3448  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4867  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-res 5646  df-suc 6324  df-iota 6449  df-fun 6499  df-fn 6500  df-fv 6505
This theorem is referenced by:  nosupbnd1lem5  27063  noinfbnd1lem5  27078
  Copyright terms: Public domain W3C validator