Theorem resisoeq45d 44558

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 5971	. 2 ⊢ (𝜑 → (𝐹 ↾ 𝐴) = (𝐹 ↾ 𝐶))
3		resisoeq45.5	. 2 ⊢ (𝜑 → 𝐵 = 𝐷)
4	2, 1, 3	isoeq145d 44557	1 ⊢ (𝜑 → ((𝐹 ↾ 𝐴) Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐹 ↾ 𝐶) Isom 𝑅, 𝑆 (𝐶, 𝐷)))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1569   ↾ cres 5653   Isom wiso 6544

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-iota 6497  df-fun 6545  df-fn 6546  df-f 6547  df-f1 6548  df-fo 6549  df-f1o 6550  df-fv 6551  df-isom 6552

This theorem is used by:  negslem1  44559