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Theorem bj-fvsnun1 34550
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 4695 . . . 4 (𝐷 ∈ (𝐶 ∖ {𝐴}) → ¬ 𝐷 = 𝐴)
42, 3syl 17 . . 3 (𝜑 → ¬ 𝐷 = 𝐴)
51, 4bj-fununsn1 34548 . 2 (𝜑 → (𝐺𝐷) = ((𝐹 ↾ (𝐶 ∖ {𝐴}))‘𝐷))
62fvresd 6662 . 2 (𝜑 → ((𝐹 ↾ (𝐶 ∖ {𝐴}))‘𝐷) = (𝐹𝐷))
75, 6eqtrd 2855 1 (𝜑 → (𝐺𝐷) = (𝐹𝐷))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4   = wceq 1537  wcel 2114  cdif 3906  cun 3907  {csn 4539  cop 4545  cres 5529  cfv 6327
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2792  ax-sep 5175  ax-nul 5182  ax-pow 5238  ax-pr 5302
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2799  df-cleq 2813  df-clel 2891  df-nfc 2959  df-ne 3007  df-ral 3130  df-rex 3131  df-rab 3134  df-v 3472  df-sbc 3749  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-uni 4811  df-br 5039  df-opab 5101  df-xp 5533  df-cnv 5535  df-dm 5537  df-rn 5538  df-res 5539  df-ima 5540  df-iota 6286  df-fv 6335
This theorem is referenced by: (None)
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