Theorem resisoeq45d 43402

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 5952	. 2 ⊢ (𝜑 → (𝐹 ↾ 𝐴) = (𝐹 ↾ 𝐶))
3		resisoeq45.5	. 2 ⊢ (𝜑 → 𝐵 = 𝐷)
4	2, 1, 3	isoeq145d 43401	1 ⊢ (𝜑 → ((𝐹 ↾ 𝐴) Isom 𝑅, 𝑆 (𝐴, 𝐵) ↔ (𝐹 ↾ 𝐶) Isom 𝑅, 𝑆 (𝐶, 𝐷)))

Syntax hints:   → wi 4   ↔ wb 206   = wceq 1540   ↾ cres 5642   Isom wiso 6514

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-nul 4299  df-if 4491  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5110  df-opab 5172  df-xp 5646  df-rel 5647  df-cnv 5648  df-co 5649  df-dm 5650  df-rn 5651  df-res 5652  df-iota 6466  df-fun 6515  df-fn 6516  df-f 6517  df-f1 6518  df-fo 6519  df-f1o 6520  df-fv 6521  df-isom 6522

This theorem is referenced by:  negslem1  43403