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Theorem bj-fvsnun1 36790
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 4790 . . . 4 (𝐷 ∈ (𝐶 ∖ {𝐴}) → ¬ 𝐷 = 𝐴)
42, 3syl 17 . . 3 (𝜑 → ¬ 𝐷 = 𝐴)
51, 4bj-fununsn1 36788 . 2 (𝜑 → (𝐺𝐷) = ((𝐹 ↾ (𝐶 ∖ {𝐴}))‘𝐷))
62fvresd 6911 . 2 (𝜑 → ((𝐹 ↾ (𝐶 ∖ {𝐴}))‘𝐷) = (𝐹𝐷))
75, 6eqtrd 2765 1 (𝜑 → (𝐺𝐷) = (𝐹𝐷))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1533  wcel 2098  cdif 3937  cun 3938  {csn 4624  cop 4630  cres 5674  cfv 6542
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 2166  ax-ext 2696  ax-sep 5294  ax-nul 5301  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  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 2703  df-cleq 2717  df-clel 2802  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3420  df-v 3465  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-uni 4904  df-br 5144  df-opab 5206  df-xp 5678  df-cnv 5680  df-dm 5682  df-rn 5683  df-res 5684  df-ima 5685  df-iota 6494  df-fv 6550
This theorem is referenced by: (None)
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