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	⊢ (𝜑 → 𝐹 Fn 𝐴)
fmptunsnop.2	⊢ (𝜑 → 𝑋 ∈ 𝐴)
fmptunsnop.3	⊢ (𝜑 → 𝑌 ∈ 𝐵)

Assertion

Ref	Expression
fmptunsnop	⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(𝑥 = 𝑋, 𝑌, (𝐹‘𝑥))) = ((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

Step	Hyp	Ref	Expression
1		mptun 6714	. 2 ⊢ (𝑥 ∈ ((𝐴 ∖ {𝑋}) ∪ {𝑋}) ↦ if(𝑥 = 𝑋, 𝑌, (𝐹‘𝑥))) = ((𝑥 ∈ (𝐴 ∖ {𝑋}) ↦ if(𝑥 = 𝑋, 𝑌, (𝐹‘𝑥))) ∪ (𝑥 ∈ {𝑋} ↦ if(𝑥 = 𝑋, 𝑌, (𝐹‘𝑥))))
2		fmptunsnop.2	. . . 4 ⊢ (𝜑 → 𝑋 ∈ 𝐴)
3		difsnid 4810	. . . 4 ⊢ (𝑋 ∈ 𝐴 → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) = 𝐴)
4	2, 3	syl 17	. . 3 ⊢ (𝜑 → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) = 𝐴)
5	4	mpteq1d 5237	. 2 ⊢ (𝜑 → (𝑥 ∈ ((𝐴 ∖ {𝑋}) ∪ {𝑋}) ↦ if(𝑥 = 𝑋, 𝑌, (𝐹‘𝑥))) = (𝑥 ∈ 𝐴 ↦ if(𝑥 = 𝑋, 𝑌, (𝐹‘𝑥))))
6		eldifsni 4790	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐴 ∖ {𝑋}) → 𝑥 ≠ 𝑋)
7	6	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝑋})) → 𝑥 ≠ 𝑋)
8	7	neneqd 2945	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝑋})) → ¬ 𝑥 = 𝑋)
9	8	iffalsed 4536	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝑋})) → if(𝑥 = 𝑋, 𝑌, (𝐹‘𝑥)) = (𝐹‘𝑥))
10	9	mpteq2dva 5242	. . . 4 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∖ {𝑋}) ↦ if(𝑥 = 𝑋, 𝑌, (𝐹‘𝑥))) = (𝑥 ∈ (𝐴 ∖ {𝑋}) ↦ (𝐹‘𝑥)))
11		fmptunsnop.1	. . . . . 6 ⊢ (𝜑 → 𝐹 Fn 𝐴)
12		dffn3 6748	. . . . . 6 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟶ran 𝐹)
13	11, 12	sylib 218	. . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶ran 𝐹)
14		difssd 4137	. . . . 5 ⊢ (𝜑 → (𝐴 ∖ {𝑋}) ⊆ 𝐴)
15	13, 14	feqresmpt 6978	. . . 4 ⊢ (𝜑 → (𝐹 ↾ (𝐴 ∖ {𝑋})) = (𝑥 ∈ (𝐴 ∖ {𝑋}) ↦ (𝐹‘𝑥)))
16	10, 15	eqtr4d 2780	. . 3 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∖ {𝑋}) ↦ if(𝑥 = 𝑋, 𝑌, (𝐹‘𝑥))) = (𝐹 ↾ (𝐴 ∖ {𝑋})))
17		iftrue 4531	. . . . . 6 ⊢ (𝑥 = 𝑋 → if(𝑥 = 𝑋, 𝑌, (𝐹‘𝑥)) = 𝑌)
18	17	adantl 481	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → if(𝑥 = 𝑋, 𝑌, (𝐹‘𝑥)) = 𝑌)
19		fmptunsnop.3	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
20	18, 2, 19	fmptsnd 7189	. . . 4 ⊢ (𝜑 → {⟨𝑋, 𝑌⟩} = (𝑥 ∈ {𝑋} ↦ if(𝑥 = 𝑋, 𝑌, (𝐹‘𝑥))))
21	20	eqcomd 2743	. . 3 ⊢ (𝜑 → (𝑥 ∈ {𝑋} ↦ if(𝑥 = 𝑋, 𝑌, (𝐹‘𝑥))) = {⟨𝑋, 𝑌⟩})
22	16, 21	uneq12d 4169	. 2 ⊢ (𝜑 → ((𝑥 ∈ (𝐴 ∖ {𝑋}) ↦ if(𝑥 = 𝑋, 𝑌, (𝐹‘𝑥))) ∪ (𝑥 ∈ {𝑋} ↦ if(𝑥 = 𝑋, 𝑌, (𝐹‘𝑥)))) = ((𝐹 ↾ (𝐴 ∖ {𝑋})) ∪ {⟨𝑋, 𝑌⟩}))
23	1, 5, 22	3eqtr3a 2801	1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(𝑥 = 𝑋, 𝑌, (𝐹‘𝑥))) = ((𝐹 ↾ (𝐴 ∖ {𝑋})) ∪ {⟨𝑋, 𝑌⟩}))

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