Theorem 1stval2 4089

Description: Alternate value of the function that extracts the first member of an ordered pair. Definition 5.13 (i) of [Monk1] p. 52.

Assertion

Ref	Expression
1stval2	|- (A e. (V X. V) -> (1st` A) = |^||^|A)

Proof of Theorem 1stval2

Step	Hyp	Ref	Expression
1		elvv 3228	. 2 |- (A e. (V X. V) <-> E.xE.y A = <.x, y>.)
2		visset 1813	. . . . . 6 |- x e. V
3	2	op1st 4085	. . . . 5 |- (1st` <.x, y>.) = x
4	2	op1stb 2913	. . . . 5 |- |^||^|<.x, y>. = x
5	3, 4	eqtr4 1498	. . . 4 |- (1st` <.x, y>.) = |^||^|<.x, y>.
6		fveq2 3724	. . . 4 |- (A = <.x, y>. -> (1st` A) = (1st` <.x, y>.))
7		inteq 2536	. . . . 5 |- (A = <.x, y>. -> |^|A = |^|<.x, y>.)
8	7	inteqd 2538	. . . 4 |- (A = <.x, y>. -> |^||^|A = |^||^|<.x, y>.)
9	5, 6, 8	3eqtr4a 1532	. . 3 |- (A = <.x, y>. -> (1st` A) = |^||^|A)
10	9	19.23aivv 1296	. 2 |- (E.xE.y A = <.x, y>. -> (1st` A) = |^||^|A)
11	1, 10	sylbi 199	1 |- (A e. (V X. V) -> (1st` A) = |^||^|A)

Syntax hints:   -> wi 3   = wceq 956   e. wcel 958  E.wex 980  Vcvv 1811  <.cop 2411  |^|cint 2533   X. cxp 3168  ` cfv 3182  1stc1st 4077

This theorem is referenced by:  1stdm 4109

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-9 965  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-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866

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-ral 1649  df-rex 1650  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-int 2534  df-br 2620  df-opab 2667  df-id 2835  df-xp 3184  df-rel 3185  df-cnv 3186  df-co 3187  df-dm 3188  df-rn 3189  df-res 3190  df-ima 3191  df-fun 3192  df-fv 3198  df-1st 4079

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