Theorem bj-fvsnun1 37231

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 4771	. . . 4 ⊢ (𝐷 ∈ (𝐶 ∖ {𝐴}) → ¬ 𝐷 = 𝐴)
4	2, 3	syl 17	. . 3 ⊢ (𝜑 → ¬ 𝐷 = 𝐴)
5	1, 4	bj-fununsn1 37229	. 2 ⊢ (𝜑 → (𝐺‘𝐷) = ((𝐹 ↾ (𝐶 ∖ {𝐴}))‘𝐷))
6	2	fvresd 6906	. 2 ⊢ (𝜑 → ((𝐹 ↾ (𝐶 ∖ {𝐴}))‘𝐷) = (𝐹‘𝐷))
7	5, 6	eqtrd 2769	1 ⊢ (𝜑 → (𝐺‘𝐷) = (𝐹‘𝐷))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1539   ∈ wcel 2107   ∖ cdif 3928   ∪ cun 3929  {csn 4606  ⟨cop 4612   ↾ cres 5667  ‘cfv 6541

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-10 2140  ax-11 2156  ax-12 2176  ax-ext 2706  ax-sep 5276  ax-nul 5286  ax-pr 5412

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-nf 1783  df-sb 2064  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2726  df-clel 2808  df-ne 2932  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-cnv 5673  df-dm 5675  df-rn 5676  df-res 5677  df-ima 5678  df-iota 6494  df-fv 6549

This theorem is referenced by: (None)