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Theorem eqfunressuc 31988
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 31987 . 2 (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐹𝑋) = (𝐺𝑋) ∧ (𝑋 ∈ dom 𝐹𝑋 ∈ dom 𝐺 ∧ (𝐹𝑋) = (𝐺𝑋))) → (𝐹 ↾ (𝑋 ∪ {𝑋})) = (𝐺 ↾ (𝑋 ∪ {𝑋})))
2 df-suc 5890 . . 3 suc 𝑋 = (𝑋 ∪ {𝑋})
32reseq2i 5548 . 2 (𝐹 ↾ suc 𝑋) = (𝐹 ↾ (𝑋 ∪ {𝑋}))
42reseq2i 5548 . 2 (𝐺 ↾ suc 𝑋) = (𝐺 ↾ (𝑋 ∪ {𝑋}))
51, 3, 43eqtr4g 2819 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 383  w3a 1072   = wceq 1632  wcel 2139  cun 3713  {csn 4321  dom cdm 5266  cres 5268  suc csuc 5886  Fun wfun 6043  cfv 6049
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-res 5278  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-fv 6057
This theorem is referenced by:  nosupbnd1lem5  32185
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