Theorem tz6.12-2 6809

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 2144, ax-11 2160, ax-12 2180. (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 6489	. 2 ⊢ (𝐹‘𝐴) = (℩𝑦𝐴𝐹𝑦)
2		eu6im 2570	. . 3 ⊢ (∃𝑧∀𝑥(𝐴𝐹𝑥 ↔ 𝑥 = 𝑧) → ∃!𝑥 𝐴𝐹𝑥)
3		breq2 5093	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (𝐴𝐹𝑦 ↔ 𝐴𝐹𝑥))
4	3	eqabcbw 2805	. . . . . 6 ⊢ ({𝑦 ∣ 𝐴𝐹𝑦} = {𝑧} ↔ ∀𝑥(𝐴𝐹𝑥 ↔ 𝑥 ∈ {𝑧}))
5		velsn 4589	. . . . . . . 8 ⊢ (𝑥 ∈ {𝑧} ↔ 𝑥 = 𝑧)
6	5	bibi2i 337	. . . . . . 7 ⊢ ((𝐴𝐹𝑥 ↔ 𝑥 ∈ {𝑧}) ↔ (𝐴𝐹𝑥 ↔ 𝑥 = 𝑧))
7	6	albii 1820	. . . . . 6 ⊢ (∀𝑥(𝐴𝐹𝑥 ↔ 𝑥 ∈ {𝑧}) ↔ ∀𝑥(𝐴𝐹𝑥 ↔ 𝑥 = 𝑧))
8	4, 7	bitri 275	. . . . 5 ⊢ ({𝑦 ∣ 𝐴𝐹𝑦} = {𝑧} ↔ ∀𝑥(𝐴𝐹𝑥 ↔ 𝑥 = 𝑧))
9	8	exbii 1849	. . . 4 ⊢ (∃𝑧{𝑦 ∣ 𝐴𝐹𝑦} = {𝑧} ↔ ∃𝑧∀𝑥(𝐴𝐹𝑥 ↔ 𝑥 = 𝑧))
10		iotanul2 6454	. . . 4 ⊢ (¬ ∃𝑧{𝑦 ∣ 𝐴𝐹𝑦} = {𝑧} → (℩𝑦𝐴𝐹𝑦) = ∅)
11	9, 10	sylnbir 331	. . 3 ⊢ (¬ ∃𝑧∀𝑥(𝐴𝐹𝑥 ↔ 𝑥 = 𝑧) → (℩𝑦𝐴𝐹𝑦) = ∅)
12	2, 11	nsyl5 159	. 2 ⊢ (¬ ∃!𝑥 𝐴𝐹𝑥 → (℩𝑦𝐴𝐹𝑦) = ∅)
13	1, 12	eqtrid 2778	1 ⊢ (¬ ∃!𝑥 𝐴𝐹𝑥 → (𝐹‘𝐴) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206  ∀wal 1539   = wceq 1541  ∃wex 1780   ∈ wcel 2111  ∃!weu 2563  {cab 2709  ∅c0 4280  {csn 4573   class class class wbr 5089  ℩cio 6435  ‘cfv 6481

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-ext 2703

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-ne 2929  df-rab 3396  df-v 3438  df-dif 3900  df-un 3902  df-ss 3914  df-nul 4281  df-if 4473  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-iota 6437  df-fv 6489

This theorem is referenced by:  fvprc  6814  fvprcALT  6815  tz6.12i  6848  ndmfv  6854  nfunsn  6861  funpartfv  35989  setrec2lem1  49793