Theorem fmptunsnop 32623

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 6683	. 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 4786	. . . 4 ⊢ (𝑋 ∈ 𝐴 → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) = 𝐴)
4	2, 3	syl 17	. . 3 ⊢ (𝜑 → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) = 𝐴)
5	4	mpteq1d 5210	. 2 ⊢ (𝜑 → (𝑥 ∈ ((𝐴 ∖ {𝑋}) ∪ {𝑋}) ↦ if(𝑥 = 𝑋, 𝑌, (𝐹‘𝑥))) = (𝑥 ∈ 𝐴 ↦ if(𝑥 = 𝑋, 𝑌, (𝐹‘𝑥))))
6		eldifsni 4766	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐴 ∖ {𝑋}) → 𝑥 ≠ 𝑋)
7	6	adantl 481	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝑋})) → 𝑥 ≠ 𝑋)
8	7	neneqd 2937	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝑋})) → ¬ 𝑥 = 𝑋)
9	8	iffalsed 4511	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝑋})) → if(𝑥 = 𝑋, 𝑌, (𝐹‘𝑥)) = (𝐹‘𝑥))
10	9	mpteq2dva 5214	. . . 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 6717	. . . . . 6 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟶ran 𝐹)
13	11, 12	sylib 218	. . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶ran 𝐹)
14		difssd 4112	. . . . 5 ⊢ (𝜑 → (𝐴 ∖ {𝑋}) ⊆ 𝐴)
15	13, 14	feqresmpt 6947	. . . 4 ⊢ (𝜑 → (𝐹 ↾ (𝐴 ∖ {𝑋})) = (𝑥 ∈ (𝐴 ∖ {𝑋}) ↦ (𝐹‘𝑥)))
16	10, 15	eqtr4d 2773	. . 3 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∖ {𝑋}) ↦ if(𝑥 = 𝑋, 𝑌, (𝐹‘𝑥))) = (𝐹 ↾ (𝐴 ∖ {𝑋})))
17		iftrue 4506	. . . . . 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 7160	. . . 4 ⊢ (𝜑 → {⟨𝑋, 𝑌⟩} = (𝑥 ∈ {𝑋} ↦ if(𝑥 = 𝑋, 𝑌, (𝐹‘𝑥))))
21	20	eqcomd 2741	. . 3 ⊢ (𝜑 → (𝑥 ∈ {𝑋} ↦ if(𝑥 = 𝑋, 𝑌, (𝐹‘𝑥))) = {⟨𝑋, 𝑌⟩})
22	16, 21	uneq12d 4144	. 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 2794	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 2932   ∖ cdif 3923   ∪ cun 3924  ifcif 4500  {csn 4601  ⟨cop 4607   ↦ cmpt 5201  ran crn 5655   ↾ cres 5656   Fn wfn 6525  ⟶wf 6526  ‘cfv 6530

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 2707  ax-sep 5266  ax-nul 5276  ax-pr 5402

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2539  df-eu 2568  df-clab 2714  df-cleq 2727  df-clel 2809  df-nfc 2885  df-ne 2933  df-ral 3052  df-rex 3061  df-rab 3416  df-v 3461  df-sbc 3766  df-dif 3929  df-un 3931  df-in 3933  df-ss 3943  df-nul 4309  df-if 4501  df-sn 4602  df-pr 4604  df-op 4608  df-uni 4884  df-br 5120  df-opab 5182  df-mpt 5202  df-id 5548  df-xp 5660  df-rel 5661  df-cnv 5662  df-co 5663  df-dm 5664  df-rn 5665  df-res 5666  df-iota 6483  df-fun 6532  df-fn 6533  df-f 6534  df-fv 6538

This theorem is referenced by:  elrgspnlem4  33186