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Theorem fmptunsnop 32714
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 4814 . . . 4 (𝑋𝐴 → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) = 𝐴)
42, 3syl 17 . . 3 (𝜑 → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) = 𝐴)
54mpteq1d 5242 . 2 (𝜑 → (𝑥 ∈ ((𝐴 ∖ {𝑋}) ∪ {𝑋}) ↦ if(𝑥 = 𝑋, 𝑌, (𝐹𝑥))) = (𝑥𝐴 ↦ if(𝑥 = 𝑋, 𝑌, (𝐹𝑥))))
6 eldifsni 4794 . . . . . . . 8 (𝑥 ∈ (𝐴 ∖ {𝑋}) → 𝑥𝑋)
76adantl 481 . . . . . . 7 ((𝜑𝑥 ∈ (𝐴 ∖ {𝑋})) → 𝑥𝑋)
87neneqd 2942 . . . . . 6 ((𝜑𝑥 ∈ (𝐴 ∖ {𝑋})) → ¬ 𝑥 = 𝑋)
98iffalsed 4541 . . . . 5 ((𝜑𝑥 ∈ (𝐴 ∖ {𝑋})) → if(𝑥 = 𝑋, 𝑌, (𝐹𝑥)) = (𝐹𝑥))
109mpteq2dva 5247 . . . 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 4146 . . . . 5 (𝜑 → (𝐴 ∖ {𝑋}) ⊆ 𝐴)
1513, 14feqresmpt 6977 . . . 4 (𝜑 → (𝐹 ↾ (𝐴 ∖ {𝑋})) = (𝑥 ∈ (𝐴 ∖ {𝑋}) ↦ (𝐹𝑥)))
1610, 15eqtr4d 2777 . . 3 (𝜑 → (𝑥 ∈ (𝐴 ∖ {𝑋}) ↦ if(𝑥 = 𝑋, 𝑌, (𝐹𝑥))) = (𝐹 ↾ (𝐴 ∖ {𝑋})))
17 iftrue 4536 . . . . . 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 7188 . . . 4 (𝜑 → {⟨𝑋, 𝑌⟩} = (𝑥 ∈ {𝑋} ↦ if(𝑥 = 𝑋, 𝑌, (𝐹𝑥))))
2120eqcomd 2740 . . 3 (𝜑 → (𝑥 ∈ {𝑋} ↦ if(𝑥 = 𝑋, 𝑌, (𝐹𝑥))) = {⟨𝑋, 𝑌⟩})
2216, 21uneq12d 4178 . 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 2798 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 1536  wcel 2105  wne 2937  cdif 3959  cun 3960  ifcif 4530  {csn 4630  cop 4636  cmpt 5230  ran crn 5689  cres 5690   Fn wfn 6557  wf 6558  cfv 6562
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-10 2138  ax-11 2154  ax-12 2174  ax-ext 2705  ax-sep 5301  ax-nul 5311  ax-pr 5437
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-nf 1780  df-sb 2062  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2726  df-clel 2813  df-nfc 2889  df-ne 2938  df-ral 3059  df-rex 3068  df-rab 3433  df-v 3479  df-sbc 3791  df-dif 3965  df-un 3967  df-in 3969  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-uni 4912  df-br 5148  df-opab 5210  df-mpt 5231  df-id 5582  df-xp 5694  df-rel 5695  df-cnv 5696  df-co 5697  df-dm 5698  df-rn 5699  df-res 5700  df-iota 6515  df-fun 6564  df-fn 6565  df-f 6566  df-fv 6570
This theorem is referenced by:  elrgspnlem4  33234
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