Theorem funeu2 3540

Description: There is exactly one value of a function.

Assertion

Ref	Expression
funeu2	⊢ ((Fun F ⋀ ⟨x, y⟩ ∈ F) → ∃!y⟨x, y⟩ ∈ F)

Distinct variable group:   x,y,F

Proof of Theorem funeu2

Step	Hyp	Ref	Expression
1		funeu 3539	. 2 ⊢ ((Fun F ⋀ xFy) → ∃!y xFy)
2		df-br 2616	. . 3 ⊢ (xFy ↔ ⟨x, y⟩ ∈ F)
3	2	anbi2i 480	. 2 ⊢ ((Fun F ⋀ xFy) ↔ (Fun F ⋀ ⟨x, y⟩ ∈ F))
4	2	eubii 1385	. 2 ⊢ (∃!y xFy ↔ ∃!y⟨x, y⟩ ∈ F)
5	1, 3, 4	3imtr3 218	1 ⊢ ((Fun F ⋀ ⟨x, y⟩ ∈ F) → ∃!y⟨x, y⟩ ∈ F)

Syntax hints:   → wi 3   ⋀ wa 223   ∈ wcel 956  ∃!weu 1378  ⟨cop 2407   class class class wbr 2615  Fun wfun 3176

This theorem is referenced by:  dffun7 3542  funssres 3555

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 2699  ax-pow 2738  ax-pr 2775

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-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-br 2616  df-opab 2663  df-id 2832  df-cnv 3186  df-co 3187  df-fun 3192

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