Theorem tz6.12 3728

Description: Function value. 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 groups:   y,F   y,A

Proof of Theorem tz6.12

Step	Hyp	Ref	Expression
1		tz6.12.1	. . 3 ⊢ A ∈ V
2	1	tz6.12-1 3727	. 2 ⊢ ((AFy ⋀ ∃!y AFy) → (F ‘A) = y)
3		df-br 2615	. 2 ⊢ (AFy ↔ ⟨A, y⟩ ∈ F)
4	3	eubii 1385	. 2 ⊢ (∃!y AFy ↔ ∃!y⟨A, y⟩ ∈ F)
5	2, 3, 4	syl2anbr 456	1 ⊢ ((⟨A, y⟩ ∈ F ⋀ ∃!y⟨A, y⟩ ∈ F) → (F ‘A) = y)

Syntax hints:   → wi 3   ⋀ wa 223   = wceq 954   ∈ wcel 956  ∃!weu 1378  Vcvv 1807  ⟨cop 2407   class class class wbr 2614   ‘cfv 3177

This theorem is referenced by:  tz6.12f 3729  aceq5lem5 4719

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-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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