Theorem tz6.12-2 2845

Description: Theorem 6.12(2) of [TakeutiZaring] p. 27.

Assertion

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

Distinct variable group(s):   y,A   y,F

Proof of Theorem tz6.12-2

Step	Hyp	Ref	Expression
1		ax-17 925	. . . . . . 7 ⊢ (¬ ∃!y AFy → ∀z ¬ ∃!y AFy)
2		eq0 1719	. . . . . . . 8 ⊢ ({x∣∃!y AFy} = ∅ ↔ ∀z ¬ z ∈ {x∣∃!y AFy})
3		visset 1350	. . . . . . . . . . 11 ⊢ z ∈ V
4		pm4.2i 149	. . . . . . . . . . 11 ⊢ (x = z → (∃!y AFy ↔ ∃!y AFy))
5	3, 4	elab 1415	. . . . . . . . . 10 ⊢ (z ∈ {x∣∃!y AFy} ↔ ∃!y AFy)
6	5	negbii 162	. . . . . . . . 9 ⊢ (¬ z ∈ {x∣∃!y AFy} ↔ ¬ ∃!y AFy)
7	6	bial 695	. . . . . . . 8 ⊢ (∀z ¬ z ∈ {x∣∃!y AFy} ↔ ∀z ¬ ∃!y AFy)
8	2, 7	bitr2 152	. . . . . . 7 ⊢ (∀z ¬ ∃!y AFy ↔ {x∣∃!y AFy} = ∅)
9	1, 8	sylib 173	. . . . . 6 ⊢ (¬ ∃!y AFy → {x∣∃!y AFy} = ∅)
10	9	sseq2d 1528	. . . . 5 ⊢ (¬ ∃!y AFy → ((F ‘A) ⊆ {x∣∃!y AFy} ↔ (F ‘A) ⊆ ∅))
11		fveq2 2832	. . . . . . 7 ⊢ (z = A → (F ‘z) = (F ‘A))
12		breq1 2065	. . . . . . . . 9 ⊢ (z = A → (zFy ↔ AFy))
13	12	bieudv 1013	. . . . . . . 8 ⊢ (z = A → (∃!y zFy ↔ ∃!y AFy))
14	13	biabdv 1183	. . . . . . 7 ⊢ (z = A → {x∣∃!y zFy} = {x∣∃!y AFy})
15	11, 14	sseq12d 1529	. . . . . 6 ⊢ (z = A → ((F ‘z) ⊆ {x∣∃!y zFy} ↔ (F ‘A) ⊆ {x∣∃!y AFy}))
16	3	fv3 2839	. . . . . . 7 ⊢ (F ‘z) = {x∣(∃y(x ∈ y ∧ zFy) ∧ ∃!y zFy)}
17		pm3.27 260	. . . . . . . 8 ⊢ ((∃y(x ∈ y ∧ zFy) ∧ ∃!y zFy) → ∃!y zFy)
18	17	ss2abi 1552	. . . . . . 7 ⊢ {x∣(∃y(x ∈ y ∧ zFy) ∧ ∃!y zFy)} ⊆ {x∣∃!y zFy}
19	16, 18	eqsstr 1530	. . . . . 6 ⊢ (F ‘z) ⊆ {x∣∃!y zFy}
20	15, 19	vtoclg 1383	. . . . 5 ⊢ (A ∈ V → (F ‘A) ⊆ {x∣∃!y AFy})
21	10, 20	syl5bi 183	. . . 4 ⊢ (¬ ∃!y AFy → (A ∈ V → (F ‘A) ⊆ ∅))
22	21	com12 13	. . 3 ⊢ (A ∈ V → (¬ ∃!y AFy → (F ‘A) ⊆ ∅))
23		ss0 1727	. . 3 ⊢ ((F ‘A) ⊆ ∅ → (F ‘A) = ∅)
24	22, 23	syl6 23	. 2 ⊢ (A ∈ V → (¬ ∃!y AFy → (F ‘A) = ∅))
25		fvprc 2829	. . 3 ⊢ (¬ A ∈ V → (F ‘A) = ∅)
26	25	a1d 14	. 2 ⊢ (¬ A ∈ V → (¬ ∃!y AFy → (F ‘A) = ∅))
27	24, 26	pm2.61i 110	1 ⊢ (¬ ∃!y AFy → (F ‘A) = ∅)

Syntax hints:  ¬ wn 1   → wi 2   ∧ wa 196  ∀wal 672  ∃wex 678   = weq 797   ∈ wel 803  ∃!weu 1007  {cab 1090   = wceq 1091   ∈ wcel 1092  Vcvv 1348   ⊆ wss 1487  ∅c0 1707   class class class wbr 2054   ‘cfv 2422

This theorem is referenced by:  tz6.12i 2847  ndmfv 2848

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-eu 1009  df-clab 1093  df-cleq 1097  df-clel 1099  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-br 2063  df-opab 2098  df-xp 2424  df-cnv 2426  df-dm 2428  df-rn 2429  df-res 2430  df-ima 2431  df-fv 2438

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