Theorem fnunsn 5295

Description: Extension of a function with a new ordered pair. (Contributed by NM, 28-Sep-2013.) (Revised by Mario Carneiro, 30-Apr-2015.)

Hypotheses

Ref	Expression
fnunop.x	⊢ (𝜑 → 𝑋 ∈ V)
fnunop.y	⊢ (𝜑 → 𝑌 ∈ V)
fnunop.f	⊢ (𝜑 → 𝐹 Fn 𝐷)
fnunop.g	⊢ 𝐺 = (𝐹 ∪ {⟨𝑋, 𝑌⟩})
fnunop.e	⊢ 𝐸 = (𝐷 ∪ {𝑋})
fnunop.d	⊢ (𝜑 → ¬ 𝑋 ∈ 𝐷)

Assertion

Ref	Expression
fnunsn	⊢ (𝜑 → 𝐺 Fn 𝐸)

Proof of Theorem fnunsn

Step	Hyp	Ref	Expression
1		fnunop.f	. . 3 ⊢ (𝜑 → 𝐹 Fn 𝐷)
2		fnunop.x	. . . 4 ⊢ (𝜑 → 𝑋 ∈ V)
3		fnunop.y	. . . 4 ⊢ (𝜑 → 𝑌 ∈ V)
4		fnsng 5235	. . . 4 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → {⟨𝑋, 𝑌⟩} Fn {𝑋})
5	2, 3, 4	syl2anc 409	. . 3 ⊢ (𝜑 → {⟨𝑋, 𝑌⟩} Fn {𝑋})
6		fnunop.d	. . . 4 ⊢ (𝜑 → ¬ 𝑋 ∈ 𝐷)
7		disjsn 3638	. . . 4 ⊢ ((𝐷 ∩ {𝑋}) = ∅ ↔ ¬ 𝑋 ∈ 𝐷)
8	6, 7	sylibr 133	. . 3 ⊢ (𝜑 → (𝐷 ∩ {𝑋}) = ∅)
9		fnun 5294	. . 3 ⊢ (((𝐹 Fn 𝐷 ∧ {⟨𝑋, 𝑌⟩} Fn {𝑋}) ∧ (𝐷 ∩ {𝑋}) = ∅) → (𝐹 ∪ {⟨𝑋, 𝑌⟩}) Fn (𝐷 ∪ {𝑋}))
10	1, 5, 8, 9	syl21anc 1227	. 2 ⊢ (𝜑 → (𝐹 ∪ {⟨𝑋, 𝑌⟩}) Fn (𝐷 ∪ {𝑋}))
11		fnunop.g	. . . 4 ⊢ 𝐺 = (𝐹 ∪ {⟨𝑋, 𝑌⟩})
12	11	fneq1i 5282	. . 3 ⊢ (𝐺 Fn 𝐸 ↔ (𝐹 ∪ {⟨𝑋, 𝑌⟩}) Fn 𝐸)
13		fnunop.e	. . . 4 ⊢ 𝐸 = (𝐷 ∪ {𝑋})
14	13	fneq2i 5283	. . 3 ⊢ ((𝐹 ∪ {⟨𝑋, 𝑌⟩}) Fn 𝐸 ↔ (𝐹 ∪ {⟨𝑋, 𝑌⟩}) Fn (𝐷 ∪ {𝑋}))
15	12, 14	bitri 183	. 2 ⊢ (𝐺 Fn 𝐸 ↔ (𝐹 ∪ {⟨𝑋, 𝑌⟩}) Fn (𝐷 ∪ {𝑋}))
16	10, 15	sylibr 133	1 ⊢ (𝜑 → 𝐺 Fn 𝐸)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1343   ∈ wcel 2136  Vcvv 2726   ∪ cun 3114   ∩ cin 3115  ∅c0 3409  {csn 3576  ⟨cop 3579   Fn wfn 5183

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187

This theorem depends on definitions:  df-bi 116  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ral 2449  df-rex 2450  df-v 2728  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-br 3983  df-opab 4044  df-id 4271  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-fun 5190  df-fn 5191

This theorem is referenced by:  tfrlemisucfn  6292  tfr1onlemsucfn  6308