Theorem bj-fvsnun1 36643

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 4789	. . . 4 ⊢ (𝐷 ∈ (𝐶 ∖ {𝐴}) → ¬ 𝐷 = 𝐴)
4	2, 3	syl 17	. . 3 ⊢ (𝜑 → ¬ 𝐷 = 𝐴)
5	1, 4	bj-fununsn1 36641	. 2 ⊢ (𝜑 → (𝐺‘𝐷) = ((𝐹 ↾ (𝐶 ∖ {𝐴}))‘𝐷))
6	2	fvresd 6905	. 2 ⊢ (𝜑 → ((𝐹 ↾ (𝐶 ∖ {𝐴}))‘𝐷) = (𝐹‘𝐷))
7	5, 6	eqtrd 2766	1 ⊢ (𝜑 → (𝐺‘𝐷) = (𝐹‘𝐷))

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1533   ∈ wcel 2098   ∖ cdif 3940   ∪ cun 3941  {csn 4623  ⟨cop 4629   ↾ cres 5671  ‘cfv 6537

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-ne 2935  df-ral 3056  df-rex 3065  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-xp 5675  df-cnv 5677  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-iota 6489  df-fv 6545

This theorem is referenced by: (None)