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Theorem bj-fvsnun1 37234
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 (𝜑𝐺 = ((𝐹 ↾ (𝐶 ∖ {𝐴})) ∪ {⟨𝐴, 𝐵⟩}))
bj-fvsnun1.eldif (𝜑𝐷 ∈ (𝐶 ∖ {𝐴}))
Assertion
Ref Expression
bj-fvsnun1 (𝜑 → (𝐺𝐷) = (𝐹𝐷))

Proof of Theorem bj-fvsnun1
StepHypRef Expression
1 bj-fvsnun.un . . 3 (𝜑𝐺 = ((𝐹 ↾ (𝐶 ∖ {𝐴})) ∪ {⟨𝐴, 𝐵⟩}))
2 bj-fvsnun1.eldif . . . 4 (𝜑𝐷 ∈ (𝐶 ∖ {𝐴}))
3 eldifsnneq 4789 . . . 4 (𝐷 ∈ (𝐶 ∖ {𝐴}) → ¬ 𝐷 = 𝐴)
42, 3syl 17 . . 3 (𝜑 → ¬ 𝐷 = 𝐴)
51, 4bj-fununsn1 37232 . 2 (𝜑 → (𝐺𝐷) = ((𝐹 ↾ (𝐶 ∖ {𝐴}))‘𝐷))
62fvresd 6924 . 2 (𝜑 → ((𝐹 ↾ (𝐶 ∖ {𝐴}))‘𝐷) = (𝐹𝐷))
75, 6eqtrd 2776 1 (𝜑 → (𝐺𝐷) = (𝐹𝐷))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1540  wcel 2108  cdif 3947  cun 3948  {csn 4624  cop 4630  cres 5685  cfv 6559
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 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2707  ax-sep 5294  ax-nul 5304  ax-pr 5430
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2728  df-clel 2815  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3436  df-v 3481  df-dif 3953  df-un 3955  df-in 3957  df-ss 3967  df-nul 4333  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4906  df-br 5142  df-opab 5204  df-xp 5689  df-cnv 5691  df-dm 5693  df-rn 5694  df-res 5695  df-ima 5696  df-iota 6512  df-fv 6567
This theorem is referenced by: (None)
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