Theorem resisoeq45d 43461

Description: Equality deduction for equally restricted isometries. (Contributed by RP, 14-Jan-2025.)

Hypotheses

Ref	Expression
resisoeq45.4	⊢ (𝜑 → 𝐴 = 𝐶)
resisoeq45.5	⊢ (𝜑 → 𝐵 = 𝐷)

Assertion

Ref	Expression
resisoeq45d	⊢ (𝜑 → ((𝐹 ↾ 𝐴) Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐹 ↾ 𝐶) Isom 𝑅, 𝑆 (𝐶, 𝐷)))

Proof of Theorem resisoeq45d

Step	Hyp	Ref	Expression
1		resisoeq45.4	. . 3 ⊢ (𝜑 → 𝐴 = 𝐶)
2	1	reseq2d 5927	. 2 ⊢ (𝜑 → (𝐹 ↾ 𝐴) = (𝐹 ↾ 𝐶))
3		resisoeq45.5	. 2 ⊢ (𝜑 → 𝐵 = 𝐷)
4	2, 1, 3	isoeq145d 43460	1 ⊢ (𝜑 → ((𝐹 ↾ 𝐴) Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐹 ↾ 𝐶) Isom 𝑅, 𝑆 (𝐶, 𝐷)))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1541   ↾ cres 5616   Isom wiso 6482

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-ext 2703

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-sb 2068  df-clab 2710  df-cleq 2723  df-clel 2806  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-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-isom 6490

This theorem is referenced by:  negslem1  43462