Theorem idfudiag1lem 50301

Description: Lemma for idfudiag1bas 50302 and idfudiag1 50303. (Contributed by Zhi Wang, 19-Oct-2025.)

Hypotheses

Ref	Expression
idfudiag1lem.1	⊢ (𝜑 → ( I ↾ 𝐴) = (𝐴 × {𝐵}))
idfudiag1lem.2	⊢ (𝜑 → 𝐴 ≠ ∅)

Assertion

Ref	Expression
idfudiag1lem	⊢ (𝜑 → 𝐴 = {𝐵})

Proof of Theorem idfudiag1lem

Step	Hyp	Ref	Expression
1		rnresi 6077	. . 3 ⊢ ran ( I ↾ 𝐴) = 𝐴
2		idfudiag1lem.1	. . . 4 ⊢ (𝜑 → ( I ↾ 𝐴) = (𝐴 × {𝐵}))
3	2	rneqd 5928	. . 3 ⊢ (𝜑 → ran ( I ↾ 𝐴) = ran (𝐴 × {𝐵}))
4	1, 3	eqtr3id 2812	. 2 ⊢ (𝜑 → 𝐴 = ran (𝐴 × {𝐵}))
5		idfudiag1lem.2	. . 3 ⊢ (𝜑 → 𝐴 ≠ ∅)
6		rnxp 6168	. . 3 ⊢ (𝐴 ≠ ∅ → ran (𝐴 × {𝐵}) = {𝐵})
7	5, 6	syl 18	. 2 ⊢ (𝜑 → ran (𝐴 × {𝐵}) = {𝐵})
8	4, 7	eqtrd 2798	1 ⊢ (𝜑 → 𝐴 = {𝐵})

Syntax hints:   → wi 4   = wceq 1570   ≠ wne 2958  ∅c0 4286  {csn 4589   I cid 5555   × cxp 5659  ran crn 5662   ↾ cres 5663

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-11 2192  ax-ext 2735  ax-sep 5257  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-br 5110  df-opab 5174  df-id 5556  df-xp 5667  df-rel 5668  df-cnv 5669  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674

This theorem is referenced by:  idfudiag1bas  50302  idfudiag1  50303