Theorem bclelnu 14495

Description: The base class of an element of a nuple.

Hypothesis

Ref	Expression
bcelnu.1	|- (x = I -> B = C)

Assertion

Ref	Expression
bclelnu	|- ((F e. X_x e. A B /\ I e. A) -> (F` I) e. C)

Distinct variable groups:   x,A   x,C   x,F   x,I

Proof of Theorem bclelnu

Step	Hyp	Ref	Expression
1		elixp2b 14494	. 2 |- (F e. X_x e. A B -> A.x e. A (F` x) e. B)
2		fveq2 4681	. . . . 5 |- (x = I -> (F` x) = (F` I))
3		bcelnu.1	. . . . 5 |- (x = I -> B = C)
4	2, 3	eleq12d 1965	. . . 4 |- (x = I -> ((F` x) e. B <-> (F` I) e. C))
5	4	rcla4va 2378	. . 3 |- ((I e. A /\ A.x e. A (F` x) e. B) -> (F` I) e. C)
6	5	ex 402	. 2 |- (I e. A -> (A.x e. A (F` x) e. B -> (F` I) e. C))
7	1, 6	mpan9 521	1 |- ((F e. X_x e. A B /\ I e. A) -> (F` I) e. C)

Syntax hints:   -> wi 3   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  ` cfv 3998  X_cixp 5406

This theorem is referenced by:  prmapcp2 14497

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-14 1312  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865  ax-sep 3438  ax-nul 3445  ax-pow 3481  ax-pr 3524

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-3an 860  df-ex 1327  df-sb 1536  df-eu 1775  df-mo 1776  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-ral 2109  df-rex 2110  df-v 2294  df-dif 2597  df-un 2600  df-in 2603  df-ss 2605  df-nul 2876  df-pw 3035  df-sn 3049  df-pr 3050  df-op 3053  df-uni 3178  df-br 3339  df-opab 3396  df-id 3586  df-xp 4000  df-rel 4001  df-cnv 4002  df-co 4003  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fun 4008  df-fn 4009  df-fv 4014  df-ixp 5407

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