Theorem fsnunf 7162

Description: Adjoining a point to a function gives a function. (Contributed by Stefan O'Rear, 28-Feb-2015.)

Assertion

Ref	Expression
fsnunf	⊢ ((𝐹:𝑆⟶𝑇 ∧ (𝑋 ∈ 𝑉 ∧ ¬ 𝑋 ∈ 𝑆) ∧ 𝑌 ∈ 𝑇) → (𝐹 ∪ {⟨𝑋, 𝑌⟩}):(𝑆 ∪ {𝑋})⟶𝑇)

Proof of Theorem fsnunf

Step	Hyp	Ref	Expression
1		simp1 1136	. . 3 ⊢ ((𝐹:𝑆⟶𝑇 ∧ (𝑋 ∈ 𝑉 ∧ ¬ 𝑋 ∈ 𝑆) ∧ 𝑌 ∈ 𝑇) → 𝐹:𝑆⟶𝑇)
2		simp2l 1200	. . . . 5 ⊢ ((𝐹:𝑆⟶𝑇 ∧ (𝑋 ∈ 𝑉 ∧ ¬ 𝑋 ∈ 𝑆) ∧ 𝑌 ∈ 𝑇) → 𝑋 ∈ 𝑉)
3		simp3 1138	. . . . 5 ⊢ ((𝐹:𝑆⟶𝑇 ∧ (𝑋 ∈ 𝑉 ∧ ¬ 𝑋 ∈ 𝑆) ∧ 𝑌 ∈ 𝑇) → 𝑌 ∈ 𝑇)
4		f1osng 6844	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑇) → {⟨𝑋, 𝑌⟩}:{𝑋}–1-1-onto→{𝑌})
5	2, 3, 4	syl2anc 584	. . . 4 ⊢ ((𝐹:𝑆⟶𝑇 ∧ (𝑋 ∈ 𝑉 ∧ ¬ 𝑋 ∈ 𝑆) ∧ 𝑌 ∈ 𝑇) → {⟨𝑋, 𝑌⟩}:{𝑋}–1-1-onto→{𝑌})
6		f1of 6803	. . . 4 ⊢ ({⟨𝑋, 𝑌⟩}:{𝑋}–1-1-onto→{𝑌} → {⟨𝑋, 𝑌⟩}:{𝑋}⟶{𝑌})
7	5, 6	syl 17	. . 3 ⊢ ((𝐹:𝑆⟶𝑇 ∧ (𝑋 ∈ 𝑉 ∧ ¬ 𝑋 ∈ 𝑆) ∧ 𝑌 ∈ 𝑇) → {⟨𝑋, 𝑌⟩}:{𝑋}⟶{𝑌})
8		simp2r 1201	. . . 4 ⊢ ((𝐹:𝑆⟶𝑇 ∧ (𝑋 ∈ 𝑉 ∧ ¬ 𝑋 ∈ 𝑆) ∧ 𝑌 ∈ 𝑇) → ¬ 𝑋 ∈ 𝑆)
9		disjsn 4678	. . . 4 ⊢ ((𝑆 ∩ {𝑋}) = ∅ ↔ ¬ 𝑋 ∈ 𝑆)
10	8, 9	sylibr 234	. . 3 ⊢ ((𝐹:𝑆⟶𝑇 ∧ (𝑋 ∈ 𝑉 ∧ ¬ 𝑋 ∈ 𝑆) ∧ 𝑌 ∈ 𝑇) → (𝑆 ∩ {𝑋}) = ∅)
11		fun 6725	. . 3 ⊢ (((𝐹:𝑆⟶𝑇 ∧ {⟨𝑋, 𝑌⟩}:{𝑋}⟶{𝑌}) ∧ (𝑆 ∩ {𝑋}) = ∅) → (𝐹 ∪ {⟨𝑋, 𝑌⟩}):(𝑆 ∪ {𝑋})⟶(𝑇 ∪ {𝑌}))
12	1, 7, 10, 11	syl21anc 837	. 2 ⊢ ((𝐹:𝑆⟶𝑇 ∧ (𝑋 ∈ 𝑉 ∧ ¬ 𝑋 ∈ 𝑆) ∧ 𝑌 ∈ 𝑇) → (𝐹 ∪ {⟨𝑋, 𝑌⟩}):(𝑆 ∪ {𝑋})⟶(𝑇 ∪ {𝑌}))
13		snssi 4775	. . . . 5 ⊢ (𝑌 ∈ 𝑇 → {𝑌} ⊆ 𝑇)
14	13	3ad2ant3 1135	. . . 4 ⊢ ((𝐹:𝑆⟶𝑇 ∧ (𝑋 ∈ 𝑉 ∧ ¬ 𝑋 ∈ 𝑆) ∧ 𝑌 ∈ 𝑇) → {𝑌} ⊆ 𝑇)
15		ssequn2 4155	. . . 4 ⊢ ({𝑌} ⊆ 𝑇 ↔ (𝑇 ∪ {𝑌}) = 𝑇)
16	14, 15	sylib 218	. . 3 ⊢ ((𝐹:𝑆⟶𝑇 ∧ (𝑋 ∈ 𝑉 ∧ ¬ 𝑋 ∈ 𝑆) ∧ 𝑌 ∈ 𝑇) → (𝑇 ∪ {𝑌}) = 𝑇)
17	16	feq3d 6676	. 2 ⊢ ((𝐹:𝑆⟶𝑇 ∧ (𝑋 ∈ 𝑉 ∧ ¬ 𝑋 ∈ 𝑆) ∧ 𝑌 ∈ 𝑇) → ((𝐹 ∪ {⟨𝑋, 𝑌⟩}):(𝑆 ∪ {𝑋})⟶(𝑇 ∪ {𝑌}) ↔ (𝐹 ∪ {⟨𝑋, 𝑌⟩}):(𝑆 ∪ {𝑋})⟶𝑇))
18	12, 17	mpbid 232	1 ⊢ ((𝐹:𝑆⟶𝑇 ∧ (𝑋 ∈ 𝑉 ∧ ¬ 𝑋 ∈ 𝑆) ∧ 𝑌 ∈ 𝑇) → (𝐹 ∪ {⟨𝑋, 𝑌⟩}):(𝑆 ∪ {𝑋})⟶𝑇)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∧ w3a 1086   = wceq 1540   ∈ wcel 2109   ∪ cun 3915   ∩ cin 3916   ⊆ wss 3917  ∅c0 4299  {csn 4592  ⟨cop 4598  ⟶wf 6510  –1-1-onto→wf1o 6513

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-12 2178  ax-ext 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

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 2066  df-mo 2534  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-in 3924  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-br 5111  df-opab 5173  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521

This theorem is referenced by:  fsnunf2  7163  f1ounsn  7250  fnchoice  45030  nnsum4primeseven  47805  nnsum4primesevenALTV  47806