Theorem fopabap 3841

Description: Append an additional value to a function.

Hypotheses

Ref	Expression
fopabap.0a	|- A e. V
fopabap.0b	|- B e. V
fopabap.1	|- (R u. {A}) = S
fopabap.2	|- (x = A -> C = B)

Assertion

Ref	Expression
fopabap	|- ({<.x, y>. | (x e. R /\ y = C)} u. {<.A, B>.}) = {<.x, y>. | (x e. S /\ y = C)}

Distinct variable groups:   x,y,A   x,B,y

Proof of Theorem fopabap

Step	Hyp	Ref	Expression
1		fopabap.0a	. . . 4 |- A e. V
2		fopabap.0b	. . . 4 |- B e. V
3	1, 2	fopabsn 3840	. . 3 |- {<.A, B>.} = {<.x, y>. | (x e. {A} /\ y = B)}
4	3	uneq2i 2181	. 2 |- ({<.x, y>. | (x e. R /\ y = C)} u. {<.A, B>.}) = ({<.x, y>. | (x e. R /\ y = C)} u. {<.x, y>. | (x e. {A} /\ y = B)})
5		unopab 2679	. 2 |- ({<.x, y>. | (x e. R /\ y = C)} u. {<.x, y>. | (x e. {A} /\ y = B)}) = {<.x, y>. | ((x e. R /\ y = C) \/ (x e. {A} /\ y = B))}
6		fopabap.1	. . . . . . 7 |- (R u. {A}) = S
7	6	eleq2i 1538	. . . . . 6 |- (x e. (R u. {A}) <-> x e. S)
8		elun 2173	. . . . . 6 |- (x e. (R u. {A}) <-> (x e. R \/ x e. {A}))
9	7, 8	bitr3 175	. . . . 5 |- (x e. S <-> (x e. R \/ x e. {A}))
10	9	anbi1i 481	. . . 4 |- ((x e. S /\ y = C) <-> ((x e. R \/ x e. {A}) /\ y = C))
11		andir 605	. . . 4 |- (((x e. R \/ x e. {A}) /\ y = C) <-> ((x e. R /\ y = C) \/ (x e. {A} /\ y = C)))
12		elsn 2421	. . . . . . . 8 |- (x e. {A} <-> x = A)
13		fopabap.2	. . . . . . . 8 |- (x = A -> C = B)
14	12, 13	sylbi 199	. . . . . . 7 |- (x e. {A} -> C = B)
15	14	eqeq2d 1486	. . . . . 6 |- (x e. {A} -> (y = C <-> y = B))
16	15	pm5.32i 645	. . . . 5 |- ((x e. {A} /\ y = C) <-> (x e. {A} /\ y = B))
17	16	orbi2i 255	. . . 4 |- (((x e. R /\ y = C) \/ (x e. {A} /\ y = C)) <-> ((x e. R /\ y = C) \/ (x e. {A} /\ y = B)))
18	10, 11, 17	3bitrr 178	. . 3 |- (((x e. R /\ y = C) \/ (x e. {A} /\ y = B)) <-> (x e. S /\ y = C))
19	18	opabbii 2671	. 2 |- {<.x, y>. | ((x e. R /\ y = C) \/ (x e. {A} /\ y = B))} = {<.x, y>. | (x e. S /\ y = C)}
20	4, 5, 19	3eqtr 1499	1 |- ({<.x, y>. | (x e. R /\ y = C)} u. {<.A, B>.}) = {<.x, y>. | (x e. S /\ y = C)}

Syntax hints:   -> wi 3   \/ wo 222   /\ wa 223   = wceq 956   e. wcel 958  Vcvv 1811   u. cun 2045  {csn 2409  <.cop 2411  {copab 2666

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

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-f1 3195  df-fo 3196  df-f1o 3197  df-fv 3198

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