Theorem idfudiag1lem 49805

Description: Lemma for idfudiag1bas 49806 and idfudiag1 49807. (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 6033	. . 3 ⊢ ran ( I ↾ 𝐴) = 𝐴
2		idfudiag1lem.1	. . . 4 ⊢ (𝜑 → ( I ↾ 𝐴) = (𝐴 × {𝐵}))
3	2	rneqd 5886	. . 3 ⊢ (𝜑 → ran ( I ↾ 𝐴) = ran (𝐴 × {𝐵}))
4	1, 3	eqtr3id 2784	. 2 ⊢ (𝜑 → 𝐴 = ran (𝐴 × {𝐵}))
5		idfudiag1lem.2	. . 3 ⊢ (𝜑 → 𝐴 ≠ ∅)
6		rnxp 6127	. . 3 ⊢ (𝐴 ≠ ∅ → ran (𝐴 × {𝐵}) = {𝐵})
7	5, 6	syl 17	. 2 ⊢ (𝜑 → ran (𝐴 × {𝐵}) = {𝐵})
8	4, 7	eqtrd 2770	1 ⊢ (𝜑 → 𝐴 = {𝐵})

Syntax hints:   → wi 4   = wceq 1542   ≠ wne 2931  ∅c0 4284  {csn 4579   I cid 5517   × cxp 5621  ran crn 5624   ↾ cres 5625

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-11 2163  ax-ext 2707  ax-sep 5240  ax-nul 5250  ax-pr 5376

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2714  df-cleq 2727  df-clel 2810  df-ne 2932  df-ral 3051  df-rex 3060  df-rab 3399  df-v 3441  df-dif 3903  df-un 3905  df-in 3907  df-ss 3917  df-nul 4285  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-br 5098  df-opab 5160  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-dm 5633  df-rn 5634  df-res 5635  df-ima 5636

This theorem is referenced by:  idfudiag1bas  49806  idfudiag1  49807