Theorem fmptunsnop 32904

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 6669	. 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 4770	. . . 4 ⊢ (𝑋 ∈ 𝐴 → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) = 𝐴)
4	2, 3	syl 17	. . 3 ⊢ (𝜑 → ((𝐴 ∖ {𝑋}) ∪ {𝑋}) = 𝐴)
5	4	mpteq1d 5192	. 2 ⊢ (𝜑 → (𝑥 ∈ ((𝐴 ∖ {𝑋}) ∪ {𝑋}) ↦ if(𝑥 = 𝑋, 𝑌, (𝐹‘𝑥))) = (𝑥 ∈ 𝐴 ↦ if(𝑥 = 𝑋, 𝑌, (𝐹‘𝑥))))
6		eldifsni 4752	. . . . . . . 8 ⊢ (𝑥 ∈ (𝐴 ∖ {𝑋}) → 𝑥 ≠ 𝑋)
7	6	adantl 485	. . . . . . 7 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝑋})) → 𝑥 ≠ 𝑋)
8	7	neneqd 2964	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝑋})) → ¬ 𝑥 = 𝑋)
9	8	iffalsed 4493	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ (𝐴 ∖ {𝑋})) → if(𝑥 = 𝑋, 𝑌, (𝐹‘𝑥)) = (𝐹‘𝑥))
10	9	mpteq2dva 5195	. . . 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 6706	. . . . . 6 ⊢ (𝐹 Fn 𝐴 ↔ 𝐹:𝐴⟶ran 𝐹)
13	11, 12	sylib 220	. . . . 5 ⊢ (𝜑 → 𝐹:𝐴⟶ran 𝐹)
14		difssd 4092	. . . . 5 ⊢ (𝜑 → (𝐴 ∖ {𝑋}) ⊆ 𝐴)
15	13, 14	feqresmpt 6938	. . . 4 ⊢ (𝜑 → (𝐹 ↾ (𝐴 ∖ {𝑋})) = (𝑥 ∈ (𝐴 ∖ {𝑋}) ↦ (𝐹‘𝑥)))
16	10, 15	eqtr4d 2802	. . 3 ⊢ (𝜑 → (𝑥 ∈ (𝐴 ∖ {𝑋}) ↦ if(𝑥 = 𝑋, 𝑌, (𝐹‘𝑥))) = (𝐹 ↾ (𝐴 ∖ {𝑋})))
17		iftrue 4488	. . . . . 6 ⊢ (𝑥 = 𝑋 → if(𝑥 = 𝑋, 𝑌, (𝐹‘𝑥)) = 𝑌)
18	17	adantl 485	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 = 𝑋) → if(𝑥 = 𝑋, 𝑌, (𝐹‘𝑥)) = 𝑌)
19		fmptunsnop.3	. . . . 5 ⊢ (𝜑 → 𝑌 ∈ 𝐵)
20	18, 2, 19	fmptsnd 7155	. . . 4 ⊢ (𝜑 → {⟨𝑋, 𝑌⟩} = (𝑥 ∈ {𝑋} ↦ if(𝑥 = 𝑋, 𝑌, (𝐹‘𝑥))))
21	20	eqcomd 2770	. . 3 ⊢ (𝜑 → (𝑥 ∈ {𝑋} ↦ if(𝑥 = 𝑋, 𝑌, (𝐹‘𝑥))) = {⟨𝑋, 𝑌⟩})
22	16, 21	uneq12d 4124	. 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 2823	1 ⊢ (𝜑 → (𝑥 ∈ 𝐴 ↦ if(𝑥 = 𝑋, 𝑌, (𝐹‘𝑥))) = ((𝐹 ↾ (𝐴 ∖ {𝑋})) ∪ {⟨𝑋, 𝑌⟩}))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1562   ∈ wcel 2144   ≠ wne 2959   ∖ cdif 3903   ∪ cun 3904  ifcif 4482  {csn 4584  ⟨cop 4590   ↦ cmpt 5183  ran crn 5650   ↾ cres 5651   Fn wfn 6518  ⟶wf 6519  ‘cfv 6523

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-sep 5248  ax-nul 5258  ax-pr 5392

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-sbc 3747  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5544  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-fv 6531

This theorem is referenced by:  elrgspnlem4  33428