Theorem oprprc2 3985

Description: The value of an operation when the second argument is a proper class. Note: this theorem is dependent on our particular definitions of operation value, function value, and ordered pair.

Assertion

Ref	Expression
oprprc2	|- (-. B e. V -> (AFB) = (AFA))

Proof of Theorem oprprc2

Step	Hyp	Ref	Expression
1		opprc2 2499	. . 3 |- (-. B e. V -> <.A, B>. = <.A, A>.)
2	1	fveq2d 3728	. 2 |- (-. B e. V -> (F` <.A, B>.) = (F` <.A, A>.))
3		df-opr 3965	. 2 |- (AFB) = (F` <.A, B>.)
4		df-opr 3965	. 2 |- (AFA) = (F` <.A, A>.)
5	2, 3, 4	3eqtr4g 1531	1 |- (-. B e. V -> (AFB) = (AFA))

Syntax hints:  -. wn 2   -> wi 3   = wceq 956   e. wcel 958  Vcvv 1811  <.cop 2411  ` cfv 3182  (class class class)co 3963

This theorem is referenced by:  ndmoprcl 4044  ndmioo 6370  eliooret 6386  elfzlem 6473

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-xp 3184  df-cnv 3186  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fv 3198  df-opr 3965

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