Theorem fsnunf 7169

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 1150	. . 3 ⊢ ((𝐹:𝑆⟶𝑇 ∧ (𝑋 ∈ 𝑉 ∧ ¬ 𝑋 ∈ 𝑆) ∧ 𝑌 ∈ 𝑇) → 𝐹:𝑆⟶𝑇)
2		simp2l 1214	. . . . 5 ⊢ ((𝐹:𝑆⟶𝑇 ∧ (𝑋 ∈ 𝑉 ∧ ¬ 𝑋 ∈ 𝑆) ∧ 𝑌 ∈ 𝑇) → 𝑋 ∈ 𝑉)
3		simp3 1152	. . . . 5 ⊢ ((𝐹:𝑆⟶𝑇 ∧ (𝑋 ∈ 𝑉 ∧ ¬ 𝑋 ∈ 𝑆) ∧ 𝑌 ∈ 𝑇) → 𝑌 ∈ 𝑇)
4		f1osng 6849	. . . . 5 ⊢ ((𝑋 ∈ 𝑉 ∧ 𝑌 ∈ 𝑇) → {⟨𝑋, 𝑌⟩}:{𝑋}–1-1-onto→{𝑌})
5	2, 3, 4	syl2anc 593	. . . 4 ⊢ ((𝐹:𝑆⟶𝑇 ∧ (𝑋 ∈ 𝑉 ∧ ¬ 𝑋 ∈ 𝑆) ∧ 𝑌 ∈ 𝑇) → {⟨𝑋, 𝑌⟩}:{𝑋}–1-1-onto→{𝑌})
6		f1of 6806	. . . 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 1215	. . . 4 ⊢ ((𝐹:𝑆⟶𝑇 ∧ (𝑋 ∈ 𝑉 ∧ ¬ 𝑋 ∈ 𝑆) ∧ 𝑌 ∈ 𝑇) → ¬ 𝑋 ∈ 𝑆)
9		disjsn 4671	. . . 4 ⊢ ((𝑆 ∩ {𝑋}) = ∅ ↔ ¬ 𝑋 ∈ 𝑆)
10	8, 9	sylibr 236	. . 3 ⊢ ((𝐹:𝑆⟶𝑇 ∧ (𝑋 ∈ 𝑉 ∧ ¬ 𝑋 ∈ 𝑆) ∧ 𝑌 ∈ 𝑇) → (𝑆 ∩ {𝑋}) = ∅)
11		fun 6726	. . 3 ⊢ (((𝐹:𝑆⟶𝑇 ∧ {⟨𝑋, 𝑌⟩}:{𝑋}⟶{𝑌}) ∧ (𝑆 ∩ {𝑋}) = ∅) → (𝐹 ∪ {⟨𝑋, 𝑌⟩}):(𝑆 ∪ {𝑋})⟶(𝑇 ∪ {𝑌}))
12	1, 7, 10, 11	syl21anc 848	. 2 ⊢ ((𝐹:𝑆⟶𝑇 ∧ (𝑋 ∈ 𝑉 ∧ ¬ 𝑋 ∈ 𝑆) ∧ 𝑌 ∈ 𝑇) → (𝐹 ∪ {⟨𝑋, 𝑌⟩}):(𝑆 ∪ {𝑋})⟶(𝑇 ∪ {𝑌}))
13		snssi 4745	. . . . 5 ⊢ (𝑌 ∈ 𝑇 → {𝑌} ⊆ 𝑇)
14	13	3ad2ant3 1149	. . . 4 ⊢ ((𝐹:𝑆⟶𝑇 ∧ (𝑋 ∈ 𝑉 ∧ ¬ 𝑋 ∈ 𝑆) ∧ 𝑌 ∈ 𝑇) → {𝑌} ⊆ 𝑇)
15		ssequn2 4142	. . . 4 ⊢ ({𝑌} ⊆ 𝑇 ↔ (𝑇 ∪ {𝑌}) = 𝑇)
16	14, 15	sylib 220	. . 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 234	1 ⊢ ((𝐹:𝑆⟶𝑇 ∧ (𝑋 ∈ 𝑉 ∧ ¬ 𝑋 ∈ 𝑆) ∧ 𝑌 ∈ 𝑇) → (𝐹 ∪ {⟨𝑋, 𝑌⟩}):(𝑆 ∪ {𝑋})⟶𝑇)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∧ w3a 1099   = wceq 1561   ∈ wcel 2143   ∪ cun 3903   ∩ cin 3904   ⊆ wss 3905  ∅c0 4286  {csn 4583  ⟨cop 4589  ⟶wf 6517  –1-1-onto→wf1o 6520

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1816  ax-4 1830  ax-5 1931  ax-6 1988  ax-7 2029  ax-8 2145  ax-9 2153  ax-12 2213  ax-ext 2735  ax-sep 5247  ax-pr 5391

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3an 1101  df-tru 1564  df-fal 1574  df-ex 1801  df-sb 2092  df-mo 2567  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3078  df-rex 3088  df-rab 3416  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-br 5102  df-opab 5164  df-id 5543  df-xp 5654  df-rel 5655  df-cnv 5656  df-co 5657  df-dm 5658  df-rn 5659  df-fun 6523  df-fn 6524  df-f 6525  df-f1 6526  df-fo 6527  df-f1o 6528

This theorem is referenced by:  fsnunf2  7170  f1ounsn  7256  fnchoice  45600  nnsum4primeseven  48413  nnsum4primesevenALTV  48414