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Theorem fmptunsnop 32709
Description: Two ways to express a function with a value replaced. (Contributed by Thierry Arnoux, 5-Oct-2025.)
Hypotheses
Ref Expression
fmptunsnop.1 (𝜑𝐹 Fn 𝐴)
fmptunsnop.2 (𝜑𝑋𝐴)
fmptunsnop.3 (𝜑𝑌𝐵)
Assertion
Ref Expression
fmptunsnop (𝜑 → (𝑥𝐴 ↦ if(𝑥 = 𝑋, 𝑌, (𝐹𝑥))) = ((𝐹 ↾ (𝐴 ∖ {𝑋})) ∪ {⟨𝑋, 𝑌⟩}))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐹   𝑥,𝑋   𝑥,𝑌   𝜑,𝑥
Allowed substitution hint:   𝐵(𝑥)

Proof of Theorem fmptunsnop
StepHypRef Expression
1 mptun 6714 . 2 (𝑥 ∈ ((𝐴 ∖ {𝑋}) ∪ {𝑋}) ↦ if(𝑥 = 𝑋, 𝑌, (𝐹𝑥))) = ((𝑥 ∈ (𝐴 ∖ {𝑋}) ↦ if(𝑥 = 𝑋, 𝑌, (𝐹𝑥))) ∪ (𝑥 ∈ {𝑋} ↦ if(𝑥 = 𝑋, 𝑌, (𝐹𝑥))))
2 fmptunsnop.2 . . . 4 (𝜑𝑋𝐴)
3 difsnid 4810 . . . 4 (𝑋𝐴 → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) = 𝐴)
42, 3syl 17 . . 3 (𝜑 → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) = 𝐴)
54mpteq1d 5237 . 2 (𝜑 → (𝑥 ∈ ((𝐴 ∖ {𝑋}) ∪ {𝑋}) ↦ if(𝑥 = 𝑋, 𝑌, (𝐹𝑥))) = (𝑥𝐴 ↦ if(𝑥 = 𝑋, 𝑌, (𝐹𝑥))))
6 eldifsni 4790 . . . . . . . 8 (𝑥 ∈ (𝐴 ∖ {𝑋}) → 𝑥𝑋)
76adantl 481 . . . . . . 7 ((𝜑𝑥 ∈ (𝐴 ∖ {𝑋})) → 𝑥𝑋)
87neneqd 2945 . . . . . 6 ((𝜑𝑥 ∈ (𝐴 ∖ {𝑋})) → ¬ 𝑥 = 𝑋)
98iffalsed 4536 . . . . 5 ((𝜑𝑥 ∈ (𝐴 ∖ {𝑋})) → if(𝑥 = 𝑋, 𝑌, (𝐹𝑥)) = (𝐹𝑥))
109mpteq2dva 5242 . . . 4 (𝜑 → (𝑥 ∈ (𝐴 ∖ {𝑋}) ↦ if(𝑥 = 𝑋, 𝑌, (𝐹𝑥))) = (𝑥 ∈ (𝐴 ∖ {𝑋}) ↦ (𝐹𝑥)))
11 fmptunsnop.1 . . . . . 6 (𝜑𝐹 Fn 𝐴)
12 dffn3 6748 . . . . . 6 (𝐹 Fn 𝐴𝐹:𝐴⟶ran 𝐹)
1311, 12sylib 218 . . . . 5 (𝜑𝐹:𝐴⟶ran 𝐹)
14 difssd 4137 . . . . 5 (𝜑 → (𝐴 ∖ {𝑋}) ⊆ 𝐴)
1513, 14feqresmpt 6978 . . . 4 (𝜑 → (𝐹 ↾ (𝐴 ∖ {𝑋})) = (𝑥 ∈ (𝐴 ∖ {𝑋}) ↦ (𝐹𝑥)))
1610, 15eqtr4d 2780 . . 3 (𝜑 → (𝑥 ∈ (𝐴 ∖ {𝑋}) ↦ if(𝑥 = 𝑋, 𝑌, (𝐹𝑥))) = (𝐹 ↾ (𝐴 ∖ {𝑋})))
17 iftrue 4531 . . . . . 6 (𝑥 = 𝑋 → if(𝑥 = 𝑋, 𝑌, (𝐹𝑥)) = 𝑌)
1817adantl 481 . . . . 5 ((𝜑𝑥 = 𝑋) → if(𝑥 = 𝑋, 𝑌, (𝐹𝑥)) = 𝑌)
19 fmptunsnop.3 . . . . 5 (𝜑𝑌𝐵)
2018, 2, 19fmptsnd 7189 . . . 4 (𝜑 → {⟨𝑋, 𝑌⟩} = (𝑥 ∈ {𝑋} ↦ if(𝑥 = 𝑋, 𝑌, (𝐹𝑥))))
2120eqcomd 2743 . . 3 (𝜑 → (𝑥 ∈ {𝑋} ↦ if(𝑥 = 𝑋, 𝑌, (𝐹𝑥))) = {⟨𝑋, 𝑌⟩})
2216, 21uneq12d 4169 . 2 (𝜑 → ((𝑥 ∈ (𝐴 ∖ {𝑋}) ↦ if(𝑥 = 𝑋, 𝑌, (𝐹𝑥))) ∪ (𝑥 ∈ {𝑋} ↦ if(𝑥 = 𝑋, 𝑌, (𝐹𝑥)))) = ((𝐹 ↾ (𝐴 ∖ {𝑋})) ∪ {⟨𝑋, 𝑌⟩}))
231, 5, 223eqtr3a 2801 1 (𝜑 → (𝑥𝐴 ↦ if(𝑥 = 𝑋, 𝑌, (𝐹𝑥))) = ((𝐹 ↾ (𝐴 ∖ {𝑋})) ∪ {⟨𝑋, 𝑌⟩}))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 395   = wceq 1540  wcel 2108  wne 2940  cdif 3948  cun 3949  ifcif 4525  {csn 4626  cop 4632  cmpt 5225  ran crn 5686  cres 5687   Fn wfn 6556  wf 6557  cfv 6561
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 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
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 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-nfc 2892  df-ne 2941  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-sbc 3789  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-mpt 5226  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-fv 6569
This theorem is referenced by:  elrgspnlem4  33249
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