Theorem tz6.12 2843

Description: Theorem 6.12(1) of [TakeutiZaring] p. 27.

Hypothesis

Ref	Expression
tz6.12.1	⊢ A ∈ V

Assertion

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

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

Proof of Theorem tz6.12

Step	Hyp	Ref	Expression
1		df-br 2063	. . 3 ⊢ (AFy ↔ ⟨A, y⟩ ∈ F)
2	1	bieu 1014	. . 3 ⊢ (∃!y AFy ↔ ∃!y⟨A, y⟩ ∈ F)
3	1, 2	anbi12i 369	. 2 ⊢ ((AFy ∧ ∃!y AFy) ↔ (⟨A, y⟩ ∈ F ∧ ∃!y⟨A, y⟩ ∈ F))
4		tz6.12.1	. . 3 ⊢ A ∈ V
5	4	tz6.12-1 2842	. 2 ⊢ ((AFy ∧ ∃!y AFy) → (F ‘A) = y)
6	3, 5	sylbir 176	1 ⊢ ((⟨A, y⟩ ∈ F ∧ ∃!y⟨A, y⟩ ∈ F) → (F ‘A) = y)

Syntax hints:   → wi 2   ∧ wa 196  ∃!weu 1007   = wceq 1091   ∈ wcel 1092  Vcvv 1348  ⟨cop 1810   class class class wbr 2054   ‘cfv 2422

This theorem is referenced by:  tz6.12f 2844  aceq5lem5 3562

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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