Theorem tz6.12-2 6869

Description: Function value when 𝐹 is not a function. Theorem 6.12(2) of [TakeutiZaring] p. 27. (Contributed by NM, 30-Apr-2004.) (Proof shortened by Mario Carneiro, 31-Aug-2015.) Avoid ax-10 2178, ax-11 2194, ax-12 2215. (Revised by TM, 25-Jan-2026.)

Assertion

Ref	Expression
tz6.12-2	⊢ (¬ ∃!𝑥 𝐴𝐹𝑥 → (𝐹‘𝐴) = ∅)

Distinct variable groups:   𝑥,𝐴   𝑥,𝐹

Proof of Theorem tz6.12-2

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-fv 6545	. 2 ⊢ (𝐹‘𝐴) = (℩𝑦𝐴𝐹𝑦)
2		eu6im 2602	. . 3 ⊢ (∃𝑧∀𝑥(𝐴𝐹𝑥 ↔ 𝑥 = 𝑧) → ∃!𝑥 𝐴𝐹𝑥)
3		breq2 5111	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (𝐴𝐹𝑦 ↔ 𝐴𝐹𝑥))
4	3	eqabcbw 2836	. . . . . 6 ⊢ ({𝑦 ∣ 𝐴𝐹𝑦} = {𝑧} ↔ ∀𝑥(𝐴𝐹𝑥 ↔ 𝑥 ∈ {𝑧}))
5		velsn 4603	. . . . . . . 8 ⊢ (𝑥 ∈ {𝑧} ↔ 𝑥 = 𝑧)
6	5	bibi2i 340	. . . . . . 7 ⊢ ((𝐴𝐹𝑥 ↔ 𝑥 ∈ {𝑧}) ↔ (𝐴𝐹𝑥 ↔ 𝑥 = 𝑧))
7	6	albii 1852	. . . . . 6 ⊢ (∀𝑥(𝐴𝐹𝑥 ↔ 𝑥 ∈ {𝑧}) ↔ ∀𝑥(𝐴𝐹𝑥 ↔ 𝑥 = 𝑧))
8	4, 7	bitri 278	. . . . 5 ⊢ ({𝑦 ∣ 𝐴𝐹𝑦} = {𝑧} ↔ ∀𝑥(𝐴𝐹𝑥 ↔ 𝑥 = 𝑧))
9	8	exbii 1881	. . . 4 ⊢ (∃𝑧{𝑦 ∣ 𝐴𝐹𝑦} = {𝑧} ↔ ∃𝑧∀𝑥(𝐴𝐹𝑥 ↔ 𝑥 = 𝑧))
10		iotanul2 6510	. . . 4 ⊢ (¬ ∃𝑧{𝑦 ∣ 𝐴𝐹𝑦} = {𝑧} → (℩𝑦𝐴𝐹𝑦) = ∅)
11	9, 10	sylnbir 334	. . 3 ⊢ (¬ ∃𝑧∀𝑥(𝐴𝐹𝑥 ↔ 𝑥 = 𝑧) → (℩𝑦𝐴𝐹𝑦) = ∅)
12	2, 11	nsyl5 160	. 2 ⊢ (¬ ∃!𝑥 𝐴𝐹𝑥 → (℩𝑦𝐴𝐹𝑦) = ∅)
13	1, 12	eqtrid 2809	1 ⊢ (¬ ∃!𝑥 𝐴𝐹𝑥 → (𝐹‘𝐴) = ∅)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209  ∀wal 1568   = wceq 1570  ∃wex 1812   ∈ wcel 2145  ∃!weu 2595  {cab 2740  ∅c0 4282  {csn 4587   class class class wbr 5107  ℩cio 6491  ‘cfv 6537

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-ne 2958  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-iota 6493  df-fv 6545

This theorem is used by:  fvprc  6874  fvprcALT  6875  tz6.12i  6908  ndmfv  6914  nfunsn  6921  noinfepregs  35661  funpartfv  36526  setrec2lem1  50621