Theorem 1stcof 4101

Description: Composition of the first member function with another function.

Assertion

Ref	Expression
1stcof	|- (F:A-->(B X. C) -> (1st o. F):A-->B)

Proof of Theorem 1stcof

Step	Hyp	Ref	Expression
1		ffn 3627	. . . . 5 |- (F:A-->(B X. C) -> F Fn A)
2		fnf 3628	. . . . 5 |- (F Fn A <-> F:A-->V)
3	1, 2	sylib 198	. . . 4 |- (F:A-->(B X. C) -> F:A-->V)
4		fo1st 4091	. . . . . . 7 |- 1st:V-onto->V
5		fof 3672	. . . . . . 7 |- (1st:V-onto->V -> 1st:V-->V)
6	4, 5	ax-mp 7	. . . . . 6 |- 1st:V-->V
7		ffn 3627	. . . . . 6 |- (1st:V-->V -> 1st Fn V)
8	6, 7	ax-mp 7	. . . . 5 |- 1st Fn V
9		fnfco 3642	. . . . 5 |- ((1st Fn V /\ F:A-->V) -> (1st o. F) Fn A)
10	8, 9	mpan 695	. . . 4 |- (F:A-->V -> (1st o. F) Fn A)
11	3, 10	syl 10	. . 3 |- (F:A-->(B X. C) -> (1st o. F) Fn A)
12		frn 3633	. . . . . 6 |- (F:A-->(B X. C) -> ran F (_ (B X. C))
13		ssres2 3386	. . . . . 6 |- (ran F (_ (B X. C) -> (1st |` ran F) (_ (1st |` (B X. C)))
14		rnss 3342	. . . . . 6 |- ((1st |` ran F) (_ (1st |` (B X. C)) -> ran (1st |` ran F) (_ ran (1st |` (B X. C)))
15	12, 13, 14	3syl 20	. . . . 5 |- (F:A-->(B X. C) -> ran (1st |` ran F) (_ ran (1st |` (B X. C)))
16		f1stres 4093	. . . . . . 7 |- (1st |` (B X. C)):(B X. C)-->B
17		frn 3633	. . . . . . 7 |- ((1st |` (B X. C)):(B X. C)-->B -> ran (1st |` (B X. C)) (_ B)
18	16, 17	ax-mp 7	. . . . . 6 |- ran (1st |` (B X. C)) (_ B
19	18	a1i 8	. . . . 5 |- (F:A-->(B X. C) -> ran (1st |` (B X. C)) (_ B)
20	15, 19	sstrd 2074	. . . 4 |- (F:A-->(B X. C) -> ran (1st |` ran F) (_ B)
21		rnco 3502	. . . 4 |- ran (1st o. F) = ran (1st |` ran F)
22	20, 21	syl5ss 2105	. . 3 |- (F:A-->(B X. C) -> ran (1st o. F) (_ B)
23	11, 22	jca 288	. 2 |- (F:A-->(B X. C) -> ((1st o. F) Fn A /\ ran (1st o. F) (_ B))
24		df-f 3194	. 2 |- ((1st o. F):A-->B <-> ((1st o. F) Fn A /\ ran (1st o. F) (_ B))
25	23, 24	sylibr 200	1 |- (F:A-->(B X. C) -> (1st o. F):A-->B)

Syntax hints:   -> wi 3   /\ wa 223  Vcvv 1811   (_ wss 2047   X. cxp 3168  ran crn 3171   |` cres 3172   o. ccom 3174   Fn wfn 3177  -->wf 3178  -onto->wfo 3180  1stc1st 4077

This theorem is referenced by:  bcthlem22 8020

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-3an 777  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-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-fn 3193  df-f 3194  df-fo 3196  df-fv 3198  df-1st 4079

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