Theorem fmptunsnop 32796

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 6635	. 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 4744	. . . 4 ⊢ (𝑋 ∈ 𝐴 → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) = 𝐴)
4	2, 3	syl 17	. . 3 ⊢ (𝜑 → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) = 𝐴)
5	4	mpteq1d 5165	. 2 ⊢ (𝜑 → (𝑥 ∈ ((𝐴 ∖ {𝑋}) ∪ {𝑋}) ↦ if(𝑥 = 𝑋, 𝑌, (𝐹‘𝑥))) = (𝑥 ∈ 𝐴 ↦ if(𝑥 = 𝑋, 𝑌, (𝐹‘𝑥))))
6		eldifsni 4726	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐴 ∖ {𝑋}) → 𝑥 ≠ 𝑋)
7	6	adantl 483	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝑋})) → 𝑥 ≠ 𝑋)
8	7	neneqd 2941	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝑋})) → ¬ 𝑥 = 𝑋)
9	8	iffalsed 4468	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝑋})) → if(𝑥 = 𝑋, 𝑌, (𝐹‘𝑥)) = (𝐹‘𝑥))
10	9	mpteq2dva 5168	. . . 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 6671	. . . . . 6 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟶ran 𝐹)
13	11, 12	sylib 220	. . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶ran 𝐹)
14		difssd 4070	. . . . 5 ⊢ (𝜑 → (𝐴 ∖ {𝑋}) ⊆ 𝐴)
15	13, 14	feqresmpt 6900	. . . 4 ⊢ (𝜑 → (𝐹 ↾ (𝐴 ∖ {𝑋})) = (𝑥 ∈ (𝐴 ∖ {𝑋}) ↦ (𝐹‘𝑥)))
16	10, 15	eqtr4d 2779	. . 3 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∖ {𝑋}) ↦ if(𝑥 = 𝑋, 𝑌, (𝐹‘𝑥))) = (𝐹 ↾ (𝐴 ∖ {𝑋})))
17		iftrue 4463	. . . . . 6 ⊢ (𝑥 = 𝑋 → if(𝑥 = 𝑋, 𝑌, (𝐹‘𝑥)) = 𝑌)
18	17	adantl 483	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → if(𝑥 = 𝑋, 𝑌, (𝐹‘𝑥)) = 𝑌)
19		fmptunsnop.3	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
20	18, 2, 19	fmptsnd 7117	. . . 4 ⊢ (𝜑 → {⟨𝑋, 𝑌⟩} = (𝑥 ∈ {𝑋} ↦ if(𝑥 = 𝑋, 𝑌, (𝐹‘𝑥))))
21	20	eqcomd 2747	. . 3 ⊢ (𝜑 → (𝑥 ∈ {𝑋} ↦ if(𝑥 = 𝑋, 𝑌, (𝐹‘𝑥))) = {⟨𝑋, 𝑌⟩})
22	16, 21	uneq12d 4102	. 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 2800	1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(𝑥 = 𝑋, 𝑌, (𝐹‘𝑥))) = ((𝐹 ↾ (𝐴 ∖ {𝑋})) ∪ {⟨𝑋, 𝑌⟩}))

Syntax hints:   → wi 4   ∧ wa 397   = wceq 1548   ∈ wcel 2121   ≠ wne 2936   ∖ cdif 3882   ∪ cun 3883  ifcif 4457  {csn 4558  ⟨cop 4564   ↦ cmpt 5156  ran crn 5622   ↾ cres 5623   Fn wfn 6484  ⟶wf 6485  ‘cfv 6489

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-10 2154  ax-11 2170  ax-12 2191  ax-ext 2713  ax-sep 5221  ax-nul 5231  ax-pr 5365

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-nf 1792  df-sb 2075  df-mo 2545  df-eu 2575  df-clab 2720  df-cleq 2733  df-clel 2816  df-nfc 2890  df-ne 2937  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-sbc 3726  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-sn 4559  df-pr 4561  df-op 4565  df-uni 4842  df-br 5076  df-opab 5138  df-mpt 5157  df-id 5516  df-xp 5627  df-rel 5628  df-cnv 5629  df-co 5630  df-dm 5631  df-rn 5632  df-res 5633  df-iota 6445  df-fun 6491  df-fn 6492  df-f 6493  df-fv 6497

This theorem is referenced by:  elrgspnlem4  33330