Theorem fnunsn 5446

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 5384	. . . 4 ⊢ ((𝑋 ∈ V ∧ 𝑌 ∈ V) → {⟨𝑋, 𝑌⟩} Fn {𝑋})
5	2, 3, 4	syl2anc 411	. . 3 ⊢ (𝜑 → {⟨𝑋, 𝑌⟩} Fn {𝑋})
6		fnunop.d	. . . 4 ⊢ (𝜑 → ¬ 𝑋 ∈ 𝐷)
7		disjsn 3735	. . . 4 ⊢ ((𝐷 ∩ {𝑋}) = ∅ ↔ ¬ 𝑋 ∈ 𝐷)
8	6, 7	sylibr 134	. . 3 ⊢ (𝜑 → (𝐷 ∩ {𝑋}) = ∅)
9		fnun 5445	. . 3 ⊢ (((𝐹 Fn 𝐷 ∧ {⟨𝑋, 𝑌⟩} Fn {𝑋}) ∧ (𝐷 ∩ {𝑋}) = ∅) → (𝐹 ∪ {⟨𝑋, 𝑌⟩}) Fn (𝐷 ∪ {𝑋}))
10	1, 5, 8, 9	syl21anc 1273	. 2 ⊢ (𝜑 → (𝐹 ∪ {⟨𝑋, 𝑌⟩}) Fn (𝐷 ∪ {𝑋}))
11		fnunop.g	. . . 4 ⊢ 𝐺 = (𝐹 ∪ {⟨𝑋, 𝑌⟩})
12	11	fneq1i 5431	. . 3 ⊢ (𝐺 Fn 𝐸 ↔ (𝐹 ∪ {⟨𝑋, 𝑌⟩}) Fn 𝐸)
13		fnunop.e	. . . 4 ⊢ 𝐸 = (𝐷 ∪ {𝑋})
14	13	fneq2i 5432	. . 3 ⊢ ((𝐹 ∪ {⟨𝑋, 𝑌⟩}) Fn 𝐸 ↔ (𝐹 ∪ {⟨𝑋, 𝑌⟩}) Fn (𝐷 ∪ {𝑋}))
15	12, 14	bitri 184	. 2 ⊢ (𝐺 Fn 𝐸 ↔ (𝐹 ∪ {⟨𝑋, 𝑌⟩}) Fn (𝐷 ∪ {𝑋}))
16	10, 15	sylibr 134	1 ⊢ (𝜑 → 𝐺 Fn 𝐸)

Syntax hints:  ¬ wn 3   → wi 4   = wceq 1398   ∈ wcel 2202  Vcvv 2803   ∪ cun 3199   ∩ cin 3200  ∅c0 3496  {csn 3673  ⟨cop 3676   Fn wfn 5328

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-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2205  ax-ext 2213  ax-sep 4212  ax-pow 4270  ax-pr 4305

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-v 2805  df-dif 3203  df-un 3205  df-in 3207  df-ss 3214  df-nul 3497  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-br 4094  df-opab 4156  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-fun 5335  df-fn 5336

This theorem is referenced by:  tfrlemisucfn  6533  tfr1onlemsucfn  6549