Theorem eqfunressuc 7295

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) ∧ (𝐹 ↾ 𝑋) = (𝐺 ↾ 𝑋) ∧ (𝑋 ∈ dom 𝐹 ∧ 𝑋 ∈ dom 𝐺 ∧ (𝐹‘𝑋) = (𝐺‘𝑋))) → (𝐹 ↾ suc 𝑋) = (𝐺 ↾ suc 𝑋))

Proof of Theorem eqfunressuc

Step	Hyp	Ref	Expression
1		eqfunresadj 7294	. 2 ⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐹 ↾ 𝑋) = (𝐺 ↾ 𝑋) ∧ (𝑋 ∈ dom 𝐹 ∧ 𝑋 ∈ dom 𝐺 ∧ (𝐹‘𝑋) = (𝐺‘𝑋))) → (𝐹 ↾ (𝑋 ∪ {𝑋})) = (𝐺 ↾ (𝑋 ∪ {𝑋})))
2		df-suc 6312	. . 3 ⊢ suc 𝑋 = (𝑋 ∪ {𝑋})
3	2	reseq2i 5924	. 2 ⊢ (𝐹 ↾ suc 𝑋) = (𝐹 ↾ (𝑋 ∪ {𝑋}))
4	2	reseq2i 5924	. 2 ⊢ (𝐺 ↾ suc 𝑋) = (𝐺 ↾ (𝑋 ∪ {𝑋}))
5	1, 3, 4	3eqtr4g 2791	1 ⊢ (((Fun 𝐹 ∧ Fun 𝐺) ∧ (𝐹 ↾ 𝑋) = (𝐺 ↾ 𝑋) ∧ (𝑋 ∈ dom 𝐹 ∧ 𝑋 ∈ dom 𝐺 ∧ (𝐹‘𝑋) = (𝐺‘𝑋))) → (𝐹 ↾ suc 𝑋) = (𝐺 ↾ suc 𝑋))

Syntax hints:   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111   ∪ cun 3895  {csn 4573  dom cdm 5614   ↾ cres 5616  suc csuc 6308  Fun wfun 6475  ‘cfv 6481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pr 5368

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-res 5626  df-suc 6312  df-iota 6437  df-fun 6483  df-fn 6484  df-fv 6489

This theorem is referenced by:  nosupbnd1lem5  27651  noinfbnd1lem5  27666