Theorem bj-fvsnun1 37880

Description: The value of a function with one of its ordered pairs replaced, at arguments other than the replaced one. (Contributed by NM, 23-Sep-2007.) Put in deduction form and remove two sethood hypotheses. (Revised by BJ, 18-Mar-2023.)

Hypotheses

Ref	Expression
bj-fvsnun.un	⊢ (𝜑 → 𝐺 = ((𝐹 ↾ (𝐶 ∖ {𝐴})) ∪ {⟨𝐴, 𝐵⟩}))
bj-fvsnun1.eldif	⊢ (𝜑 → 𝐷 ∈ (𝐶 ∖ {𝐴}))

Assertion

Ref	Expression
bj-fvsnun1	⊢ (𝜑 → (𝐺‘𝐷) = (𝐹‘𝐷))

Proof of Theorem bj-fvsnun1

Step	Hyp	Ref	Expression
1		bj-fvsnun.un	. . 3 ⊢ (𝜑 → 𝐺 = ((𝐹 ↾ (𝐶 ∖ {𝐴})) ∪ {⟨𝐴, 𝐵⟩}))
2		bj-fvsnun1.eldif	. . . 4 ⊢ (𝜑 → 𝐷 ∈ (𝐶 ∖ {𝐴}))
3		eldifsnneq 4760	. . . 4 ⊢ (𝐷 ∈ (𝐶 ∖ {𝐴}) → ¬ 𝐷 = 𝐴)
4	2, 3	syl 18	. . 3 ⊢ (𝜑 → ¬ 𝐷 = 𝐴)
5	1, 4	bj-fununsn1 37878	. 2 ⊢ (𝜑 → (𝐺‘𝐷) = ((𝐹 ↾ (𝐶 ∖ {𝐴}))‘𝐷))
6	2	fvresd 6903	. 2 ⊢ (𝜑 → ((𝐹 ↾ (𝐶 ∖ {𝐴}))‘𝐷) = (𝐹‘𝐷))
7	5, 6	eqtrd 2798	1 ⊢ (𝜑 → (𝐺‘𝐷) = (𝐹‘𝐷))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1570   ∈ wcel 2143   ∖ cdif 3903   ∪ cun 3904  {csn 4590  ⟨cop 4596   ↾ cres 5665  ‘cfv 6538

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-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pr 5406

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-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  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 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-xp 5669  df-cnv 5671  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-iota 6494  df-fv 6546

This theorem is referenced by: (None)