Theorem bnj549 13275

Description: Technical lemma of bnj75 13310. This lemma may no longer be used or have become an indirect lemma of the theorem in question (i.e. a lemma of a lemma... of the theorem).

Hypotheses

Ref	Expression
bnj549.1	|- B = U_y e. (f` i) pred(y, A, R)
bnj549.2	|- C = U_y e. (f` p) pred(y, A, R)

Assertion

Ref	Expression
bnj549	|- (p = i -> B = C)

Distinct variable groups:   y,f   y,i   y,p

Proof of Theorem bnj549

Step	Hyp	Ref	Expression
1		fveq2 4681	. . 3 |- (p = i -> (f` p) = (f` i))
2		iuneq1 3269	. . 3 |- ((f` p) = (f` i) -> U_y e. (f` p) pred(y, A, R) = U_y e. (f` i) pred(y, A, R))
3	1, 2	syl 12	. 2 |- (p = i -> U_y e. (f` p) pred(y, A, R) = U_y e. (f` i) pred(y, A, R))
4		bnj549.1	. . . 4 |- B = U_y e. (f` i) pred(y, A, R)
5		bnj549.2	. . . 4 |- C = U_y e. (f` p) pred(y, A, R)
6	4, 5	eqeq12i 1897	. . 3 |- (B = C <-> U_y e. (f` i) pred(y, A, R) = U_y e. (f` p) pred(y, A, R))
7		eqcom 1886	. . 3 |- (U_y e. (f` i) pred(y, A, R) = U_y e. (f` p) pred(y, A, R) <-> U_y e. (f` p) pred(y, A, R) = U_y e. (f` i) pred(y, A, R))
8	6, 7	bitri 190	. 2 |- (B = C <-> U_y e. (f` p) pred(y, A, R) = U_y e. (f` i) pred(y, A, R))
9	3, 8	sylibr 217	1 |- (p = i -> B = C)

Syntax hints:   -> wi 3   = wceq 1298  U_ciun 3255  ` cfv 3998   predsyn-bnj14 12023

This theorem is referenced by:  bnj553 13278

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-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-iun 3257  df-br 3339  df-opab 3396  df-xp 4000  df-cnv 4002  df-dm 4004  df-rn 4005  df-res 4006  df-ima 4007  df-fv 4014

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