Theorem raleqfn 15672

Description: Change the domain of quantification by a function.

Hypothesis

Ref	Expression
raleqfn.1	|- (y = (F` x) -> (ph <-> ps))

Assertion

Ref	Expression
raleqfn	|- ((F Fn A /\ B C_ A) -> (A.x e. B ps <-> A.y e. (F"B)ph))

Distinct variable groups:   x,F,y   x,A,y   x,B,y   ph,x   ps,y

Proof of Theorem raleqfn

Step	Hyp	Ref	Expression
1		fvelimab 4725	. . . . 5 |- ((F Fn A /\ B C_ A) -> (y e. (F"B) <-> E.x e. B (F` x) = y))
2		r19.29r 2229	. . . . . . 7 |- ((E.x e. B (F` x) = y /\ A.x e. B ps) -> E.x e. B ((F` x) = y /\ ps))
3		raleqfn.1	. . . . . . . . . . 11 |- (y = (F` x) -> (ph <-> ps))
4	3	eqcoms 1887	. . . . . . . . . 10 |- ((F` x) = y -> (ph <-> ps))
5	4	biimpar 461	. . . . . . . . 9 |- (((F` x) = y /\ ps) -> ph)
6	5	adantl 424	. . . . . . . 8 |- ((x e. B /\ ((F` x) = y /\ ps)) -> ph)
7	6	r19.23aiva 2212	. . . . . . 7 |- (E.x e. B ((F` x) = y /\ ps) -> ph)
8	2, 7	syl 12	. . . . . 6 |- ((E.x e. B (F` x) = y /\ A.x e. B ps) -> ph)
9	8	ex 402	. . . . 5 |- (E.x e. B (F` x) = y -> (A.x e. B ps -> ph))
10	1, 9	syl6bi 231	. . . 4 |- ((F Fn A /\ B C_ A) -> (y e. (F"B) -> (A.x e. B ps -> ph)))
11	10	com23 36	. . 3 |- ((F Fn A /\ B C_ A) -> (A.x e. B ps -> (y e. (F"B) -> ph)))
12	11	r19.21adv 2181	. 2 |- ((F Fn A /\ B C_ A) -> (A.x e. B ps -> A.y e. (F"B)ph))
13		fnfun 4510	. . . . . . 7 |- (F Fn A -> Fun F)
14	13	adantr 425	. . . . . 6 |- ((F Fn A /\ B C_ A) -> Fun F)
15		fndm 4512	. . . . . . . 8 |- (F Fn A -> dom F = A)
16	15	sseq2d 2645	. . . . . . 7 |- (F Fn A -> (B C_ dom F <-> B C_ A))
17	16	biimpar 461	. . . . . 6 |- ((F Fn A /\ B C_ A) -> B C_ dom F)
18		funfvima2 4829	. . . . . 6 |- ((Fun F /\ B C_ dom F) -> (x e. B -> (F` x) e. (F"B)))
19	14, 17, 18	syl11anc 524	. . . . 5 |- ((F Fn A /\ B C_ A) -> (x e. B -> (F` x) e. (F"B)))
20	3	rcla4v 2376	. . . . 5 |- ((F` x) e. (F"B) -> (A.y e. (F"B)ph -> ps))
21	19, 20	syl6 25	. . . 4 |- ((F Fn A /\ B C_ A) -> (x e. B -> (A.y e. (F"B)ph -> ps)))
22	21	com23 36	. . 3 |- ((F Fn A /\ B C_ A) -> (A.y e. (F"B)ph -> (x e. B -> ps)))
23	22	r19.21adv 2181	. 2 |- ((F Fn A /\ B C_ A) -> (A.y e. (F"B)ph -> A.x e. B ps))
24	12, 23	impbid 574	1 |- ((F Fn A /\ B C_ A) -> (A.x e. B ps <-> A.y e. (F"B)ph))

Syntax hints:   -> wi 3   <-> wb 163   /\ wa 240   = wceq 1298   e. wcel 1300  A.wral 2105  E.wrex 2106   C_ wss 2593  dom cdm 3986  "cima 3989  Fun wfun 3992   Fn wfn 3993  ` cfv 3998

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-13 1311  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  ax-un 3790

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-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

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