Theorem resisoeq45d 43410

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 5977	. 2 ⊢ (𝜑 → (𝐹 ↾ 𝐴) = (𝐹 ↾ 𝐶))
3		resisoeq45.5	. 2 ⊢ (𝜑 → 𝐵 = 𝐷)
4	2, 1, 3	isoeq145d 43409	1 ⊢ (𝜑 → ((𝐹 ↾ 𝐴) Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐹 ↾ 𝐶) Isom 𝑅, 𝑆 (𝐶, 𝐷)))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1539   ↾ cres 5667   Isom wiso 6542

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2064  df-clab 2713  df-cleq 2726  df-clel 2808  df-ral 3051  df-rex 3060  df-rab 3420  df-v 3465  df-dif 3934  df-un 3936  df-in 3938  df-ss 3948  df-nul 4314  df-if 4506  df-sn 4607  df-pr 4609  df-op 4613  df-uni 4888  df-br 5124  df-opab 5186  df-xp 5671  df-rel 5672  df-cnv 5673  df-co 5674  df-dm 5675  df-rn 5676  df-res 5677  df-iota 6494  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-isom 6550

This theorem is referenced by:  negslem1  43411