Theorem fmptapd 5840

Description: Append an additional value to a function. (Contributed by Thierry Arnoux, 3-Jan-2017.)

Hypotheses

Ref	Expression
fmptapd.0a	⊢ (𝜑 → 𝐴 ∈ V)
fmptapd.0b	⊢ (𝜑 → 𝐵 ∈ V)
fmptapd.1	⊢ (𝜑 → (𝑅 ∪ {𝐴}) = 𝑆)
fmptapd.2	⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐶 = 𝐵)

Assertion

Ref	Expression
fmptapd	⊢ (𝜑 → ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ {⟨𝐴, 𝐵⟩}) = (𝑥 ∈ 𝑆 ↦ 𝐶))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝑅   𝑥,𝑆   𝜑,𝑥

Allowed substitution hint:   𝐶(𝑥)

Proof of Theorem fmptapd

Step	Hyp	Ref	Expression
1		fmptapd.0a	. . . . 5 ⊢ (𝜑 → 𝐴 ∈ V)
2		fmptapd.0b	. . . . 5 ⊢ (𝜑 → 𝐵 ∈ V)
3		fmptsn 5838	. . . . 5 ⊢ ((𝐴 ∈ V ∧ 𝐵 ∈ V) → {⟨𝐴, 𝐵⟩} = (𝑥 ∈ {𝐴} ↦ 𝐵))
4	1, 2, 3	syl2anc 411	. . . 4 ⊢ (𝜑 → {⟨𝐴, 𝐵⟩} = (𝑥 ∈ {𝐴} ↦ 𝐵))
5		elsni 3685	. . . . . 6 ⊢ (𝑥 ∈ {𝐴} → 𝑥 = 𝐴)
6		fmptapd.2	. . . . . 6 ⊢ ((𝜑 ∧ 𝑥 = 𝐴) → 𝐶 = 𝐵)
7	5, 6	sylan2 286	. . . . 5 ⊢ ((𝜑 ∧ 𝑥 ∈ {𝐴}) → 𝐶 = 𝐵)
8	7	mpteq2dva 4177	. . . 4 ⊢ (𝜑 → (𝑥 ∈ {𝐴} ↦ 𝐶) = (𝑥 ∈ {𝐴} ↦ 𝐵))
9	4, 8	eqtr4d 2265	. . 3 ⊢ (𝜑 → {⟨𝐴, 𝐵⟩} = (𝑥 ∈ {𝐴} ↦ 𝐶))
10	9	uneq2d 3359	. 2 ⊢ (𝜑 → ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ {⟨𝐴, 𝐵⟩}) = ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ (𝑥 ∈ {𝐴} ↦ 𝐶)))
11		mptun 5461	. . 3 ⊢ (𝑥 ∈ (𝑅 ∪ {𝐴}) ↦ 𝐶) = ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ (𝑥 ∈ {𝐴} ↦ 𝐶))
12	11	a1i 9	. 2 ⊢ (𝜑 → (𝑥 ∈ (𝑅 ∪ {𝐴}) ↦ 𝐶) = ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ (𝑥 ∈ {𝐴} ↦ 𝐶)))
13		fmptapd.1	. . 3 ⊢ (𝜑 → (𝑅 ∪ {𝐴}) = 𝑆)
14	13	mpteq1d 4172	. 2 ⊢ (𝜑 → (𝑥 ∈ (𝑅 ∪ {𝐴}) ↦ 𝐶) = (𝑥 ∈ 𝑆 ↦ 𝐶))
15	10, 12, 14	3eqtr2d 2268	1 ⊢ (𝜑 → ((𝑥 ∈ 𝑅 ↦ 𝐶) ∪ {⟨𝐴, 𝐵⟩}) = (𝑥 ∈ 𝑆 ↦ 𝐶))

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1395   ∈ wcel 2200  Vcvv 2800   ∪ cun 3196  {csn 3667  ⟨cop 3670   ↦ cmpt 4148

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-14 2203  ax-ext 2211  ax-sep 4205  ax-pow 4262  ax-pr 4297

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-reu 2515  df-v 2802  df-un 3202  df-in 3204  df-ss 3211  df-pw 3652  df-sn 3673  df-pr 3674  df-op 3676  df-br 4087  df-opab 4149  df-mpt 4150  df-id 4388  df-xp 4729  df-rel 4730  df-cnv 4731  df-co 4732  df-dm 4733  df-rn 4734  df-fun 5326  df-fn 5327  df-f 5328  df-f1 5329  df-fo 5330  df-f1o 5331

This theorem is referenced by:  fmptpr  5841