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Theorem eqfunressuc 7345
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 7344 . 2 (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐹𝑋) = (𝐺𝑋) ∧ (𝑋 ∈ dom 𝐹𝑋 ∈ dom 𝐺 ∧ (𝐹𝑋) = (𝐺𝑋))) → (𝐹 ↾ (𝑋 ∪ {𝑋})) = (𝐺 ↾ (𝑋 ∪ {𝑋})))
2 df-suc 6362 . . 3 suc 𝑋 = (𝑋 ∪ {𝑋})
32reseq2i 5973 . 2 (𝐹 ↾ suc 𝑋) = (𝐹 ↾ (𝑋 ∪ {𝑋}))
42reseq2i 5973 . 2 (𝐺 ↾ suc 𝑋) = (𝐺 ↾ (𝑋 ∪ {𝑋}))
51, 3, 43eqtr4g 2798 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 3944  {csn 4624  dom cdm 5672  cres 5674  suc csuc 6358  Fun wfun 6529  cfv 6535
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 2704  ax-sep 5295  ax-nul 5302  ax-pr 5423
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 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3949  df-un 3951  df-in 3953  df-ss 3963  df-nul 4321  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4905  df-br 5145  df-opab 5207  df-id 5570  df-xp 5678  df-rel 5679  df-cnv 5680  df-co 5681  df-dm 5682  df-res 5684  df-suc 6362  df-iota 6487  df-fun 6537  df-fn 6538  df-fv 6543
This theorem is referenced by:  nosupbnd1lem5  27182  noinfbnd1lem5  27197
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