Theorem bcthlem3 7998

Description: Lemma for bcth 8029. Two ways to express the first component of a ball (expressed as an ordered pair) in the sequence of balls g.

Assertion

Ref	Expression
bcthlem3	|- ((g:NN-->A /\ N e. NN) -> ((1st o. g)` N) = (1st` (g` N)))

Proof of Theorem bcthlem3

Step	Hyp	Ref	Expression
1		fo1st 4097	. . 3 |- 1st:V-onto->V
2		fofun 3679	. . 3 |- (1st:V-onto->V -> Fun 1st)
3	1, 2	ax-mp 7	. 2 |- Fun 1st
4		fvco3 3782	. 2 |- ((Fun 1st /\ g:NN-->A /\ N e. NN) -> ((1st o. g)` N) = (1st` (g` N)))
5	3, 4	mp3an1 905	1 |- ((g:NN-->A /\ N e. NN) -> ((1st o. g)` N) = (1st` (g` N)))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 958   e. wcel 960  Vcvv 1814   o. ccom 3180  Fun wfun 3182  -->wf 3184  -onto->wfo 3186  ` cfv 3188  1stc1st 4083  NNcn 5308

This theorem is referenced by:  bcthlem11 8006  bcthlem21 8016  bcthlem24 8019  bcthlem25 8020  bcthlem26 8021

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-v 1815  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-pw 2406  df-sn 2416  df-pr 2417  df-op 2420  df-uni 2508  df-br 2625  df-opab 2672  df-id 2841  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-fo 3202  df-fv 3204  df-1st 4085

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