Theorem bj-fvsnun1 37249

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 4742	. . . 4 ⊢ (𝐷 ∈ (𝐶 ∖ {𝐴}) → ¬ 𝐷 = 𝐴)
4	2, 3	syl 17	. . 3 ⊢ (𝜑 → ¬ 𝐷 = 𝐴)
5	1, 4	bj-fununsn1 37247	. 2 ⊢ (𝜑 → (𝐺‘𝐷) = ((𝐹 ↾ (𝐶 ∖ {𝐴}))‘𝐷))
6	2	fvresd 6842	. 2 ⊢ (𝜑 → ((𝐹 ↾ (𝐶 ∖ {𝐴}))‘𝐷) = (𝐹‘𝐷))
7	5, 6	eqtrd 2764	1 ⊢ (𝜑 → (𝐺‘𝐷) = (𝐹‘𝐷))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1540   ∈ wcel 2109   ∖ cdif 3900   ∪ cun 3901  {csn 4577  ⟨cop 4583   ↾ cres 5621  ‘cfv 6482

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-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5235  ax-nul 5245  ax-pr 5371

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-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-ne 2926  df-ral 3045  df-rex 3054  df-rab 3395  df-v 3438  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-nul 4285  df-if 4477  df-sn 4578  df-pr 4580  df-op 4584  df-uni 4859  df-br 5093  df-opab 5155  df-xp 5625  df-cnv 5627  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6438  df-fv 6490

This theorem is referenced by: (None)