Theorem tz6.12-2 3730

Description: Function value when F is not a function. Theorem 6.12(2) of [TakeutiZaring] p. 27.

Assertion

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

Distinct variable groups:   y,A   y,F

Proof of Theorem tz6.12-2

Step	Hyp	Ref	Expression
1		ax-17 969	. . . . . 6 ⊢ (¬ ∃!y AFy → ∀z ¬ ∃!y AFy)
2		eq0 2290	. . . . . . 7 ⊢ ({x∣∃!y AFy} = ∅ ↔ ∀z ¬ z ∈ {x∣∃!y AFy})
3		visset 1809	. . . . . . . . . 10 ⊢ z ∈ V
4		pm4.2i 171	. . . . . . . . . 10 ⊢ (x = z → (∃!y AFy ↔ ∃!y AFy))
5	3, 4	elab 1893	. . . . . . . . 9 ⊢ (z ∈ {x∣∃!y AFy} ↔ ∃!y AFy)
6	5	negbii 187	. . . . . . . 8 ⊢ (¬ z ∈ {x∣∃!y AFy} ↔ ¬ ∃!y AFy)
7	6	albii 997	. . . . . . 7 ⊢ (∀z ¬ z ∈ {x∣∃!y AFy} ↔ ∀z ¬ ∃!y AFy)
8	2, 7	bitr2 174	. . . . . 6 ⊢ (∀z ¬ ∃!y AFy ↔ {x∣∃!y AFy} = ∅)
9	1, 8	sylib 198	. . . . 5 ⊢ (¬ ∃!y AFy → {x∣∃!y AFy} = ∅)
10	9	sseq2d 2085	. . . 4 ⊢ (¬ ∃!y AFy → ((F ‘A) ⊆ {x∣∃!y AFy} ↔ (F ‘A) ⊆ ∅))
11		fveq2 3715	. . . . . 6 ⊢ (z = A → (F ‘z) = (F ‘A))
12		breq1 2617	. . . . . . . 8 ⊢ (z = A → (zFy ↔ AFy))
13	12	eubidv 1384	. . . . . . 7 ⊢ (z = A → (∃!y zFy ↔ ∃!y AFy))
14	13	abbidv 1574	. . . . . 6 ⊢ (z = A → {x∣∃!y zFy} = {x∣∃!y AFy})
15	11, 14	sseq12d 2086	. . . . 5 ⊢ (z = A → ((F ‘z) ⊆ {x∣∃!y zFy} ↔ (F ‘A) ⊆ {x∣∃!y AFy}))
16	3	fv3 3724	. . . . . 6 ⊢ (F ‘z) = {x∣(∃y(x ∈ y ⋀ zFy) ⋀ ∃!y zFy)}
17		pm3.27 323	. . . . . . 7 ⊢ ((∃y(x ∈ y ⋀ zFy) ⋀ ∃!y zFy) → ∃!y zFy)
18	17	ss2abi 2116	. . . . . 6 ⊢ {x∣(∃y(x ∈ y ⋀ zFy) ⋀ ∃!y zFy)} ⊆ {x∣∃!y zFy}
19	16, 18	eqsstr 2087	. . . . 5 ⊢ (F ‘z) ⊆ {x∣∃!y zFy}
20	15, 19	vtoclg 1843	. . . 4 ⊢ (A ∈ V → (F ‘A) ⊆ {x∣∃!y AFy})
21	10, 20	syl5bi 208	. . 3 ⊢ (¬ ∃!y AFy → (A ∈ V → (F ‘A) ⊆ ∅))
22		ss0 2299	. . 3 ⊢ ((F ‘A) ⊆ ∅ → (F ‘A) = ∅)
23	21, 22	syl6com 53	. 2 ⊢ (A ∈ V → (¬ ∃!y AFy → (F ‘A) = ∅))
24		fvprc 3712	. . 3 ⊢ (¬ A ∈ V → (F ‘A) = ∅)
25	24	a1d 12	. 2 ⊢ (¬ A ∈ V → (¬ ∃!y AFy → (F ‘A) = ∅))
26	23, 25	pm2.61i 126	1 ⊢ (¬ ∃!y AFy → (F ‘A) = ∅)

Syntax hints:  ¬ wn 2   → wi 3   ⋀ wa 223  ∀wal 952   = wceq 954   ∈ wcel 956  ∃wex 978  ∃!weu 1378  {cab 1461  Vcvv 1807   ⊆ wss 2043  ∅c0 2276   class class class wbr 2614   ‘cfv 3177

This theorem is referenced by:  tz6.12i 3732  ndmfv 3736

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-11 965  ax-12 966  ax-13 967  ax-14 968  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457  ax-sep 2698  ax-pow 2737  ax-pr 2774

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-eu 1380  df-mo 1381  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-ral 1646  df-rex 1647  df-v 1808  df-dif 2045  df-un 2046  df-in 2047  df-ss 2049  df-nul 2277  df-pw 2398  df-sn 2408  df-pr 2409  df-op 2412  df-uni 2499  df-br 2615  df-opab 2662  df-xp 3179  df-cnv 3181  df-dm 3183  df-rn 3184  df-res 3185  df-ima 3186  df-fv 3193

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