Theorem tz6.12f 2844

Description: Function value requiring only that y not be 'free' in F (but not necessarily absent from it).

Hypothesis

Ref	Expression
tz6.12f.1	⊢ (w ∈ F → ∀y w ∈ F)

Assertion

Ref	Expression
tz6.12f	⊢ ((⟨x, y⟩ ∈ F ∧ ∃!y⟨x, y⟩ ∈ F) → (F ‘x) = y)

Distinct variable group(s):   x,y,w   w,F

Proof of Theorem tz6.12f

Step	Hyp	Ref	Expression
1		ax-17 925	. 2 ⊢ (((⟨x, y⟩ ∈ F ∧ ∃!y⟨x, y⟩ ∈ F) → (F ‘x) = y) → ∀z((⟨x, y⟩ ∈ F ∧ ∃!y⟨x, y⟩ ∈ F) → (F ‘x) = y))
2		opeq2 1877	. . . . 5 ⊢ (z = y → ⟨x, z⟩ = ⟨x, y⟩)
3	2	eleq1d 1155	. . . 4 ⊢ (z = y → (⟨x, z⟩ ∈ F ↔ ⟨x, y⟩ ∈ F))
4		ax-17 925	. . . . . . 7 ⊢ (w ∈ ⟨x, z⟩ → ∀y w ∈ ⟨x, z⟩)
5		tz6.12f.1	. . . . . . 7 ⊢ (w ∈ F → ∀y w ∈ F)
6	4, 5	hbel 1172	. . . . . 6 ⊢ (⟨x, z⟩ ∈ F → ∀y⟨x, z⟩ ∈ F)
7		ax-17 925	. . . . . 6 ⊢ (⟨x, y⟩ ∈ F → ∀z⟨x, y⟩ ∈ F)
8	6, 7, 3	cbveu 1018	. . . . 5 ⊢ (∃!z⟨x, z⟩ ∈ F ↔ ∃!y⟨x, y⟩ ∈ F)
9	8	a1i 7	. . . 4 ⊢ (z = y → (∃!z⟨x, z⟩ ∈ F ↔ ∃!y⟨x, y⟩ ∈ F))
10	3, 9	anbi12d 476	. . 3 ⊢ (z = y → ((⟨x, z⟩ ∈ F ∧ ∃!z⟨x, z⟩ ∈ F) ↔ (⟨x, y⟩ ∈ F ∧ ∃!y⟨x, y⟩ ∈ F)))
11		cleq2 1110	. . 3 ⊢ (z = y → ((F ‘x) = z ↔ (F ‘x) = y))
12	10, 11	imbi12d 474	. 2 ⊢ (z = y → (((⟨x, z⟩ ∈ F ∧ ∃!z⟨x, z⟩ ∈ F) → (F ‘x) = z) ↔ ((⟨x, y⟩ ∈ F ∧ ∃!y⟨x, y⟩ ∈ F) → (F ‘x) = y)))
13		visset 1350	. . 3 ⊢ x ∈ V
14	13	tz6.12 2843	. 2 ⊢ ((⟨x, z⟩ ∈ F ∧ ∃!z⟨x, z⟩ ∈ F) → (F ‘x) = z)
15	1, 12, 14	chv2 850	1 ⊢ ((⟨x, y⟩ ∈ F ∧ ∃!y⟨x, y⟩ ∈ F) → (F ‘x) = y)

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672   = weq 797  ∃!weu 1007   = wceq 1091   ∈ wcel 1092  ⟨cop 1810   ‘cfv 2422

This theorem is referenced by:  fvopab2 2878

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-mo 1010  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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