Theorem en2d 4400

Description: Equinumerosity inference from an implicit one-to-one onto function.

Hypotheses

Ref	Expression
en2d.1	|- (ph -> A e. V)
en2d.2	|- (ph -> (x e. A -> C e. V))
en2d.3	|- (ph -> (y e. B -> D e. V))
en2d.4	|- (ph -> ((x e. A /\ y = C) <-> (y e. B /\ x = D)))

Assertion

Ref	Expression
en2d	|- (ph -> A ~~ B)

Distinct variable groups:   x,y,A   x,B,y   y,C   x,D   ph,x,y

Proof of Theorem en2d

Step	Hyp	Ref	Expression
1		f1oeng 4395	. 2 |- ((A e. V /\ {<.x, y>. | (y e. B /\ x = D)}:A-1-1-onto->B) -> A ~~ B)
2		en2d.1	. 2 |- (ph -> A e. V)
3		en2d.2	. . . . . . . 8 |- (ph -> (x e. A -> C e. V))
4		eueq 1916	. . . . . . . 8 |- (C e. V <-> E!y y = C)
5	3, 4	syl6ib 212	. . . . . . 7 |- (ph -> (x e. A -> E!y y = C))
6	5	r19.21aiv 1713	. . . . . 6 |- (ph -> A.x e. A E!y y = C)
7		eqid 1475	. . . . . . 7 |- {<.x, y>. | (x e. A /\ y = C)} = {<.x, y>. | (x e. A /\ y = C)}
8	7	fnopabg 3615	. . . . . 6 |- (A.x e. A E!y y = C <-> {<.x, y>. | (x e. A /\ y = C)} Fn A)
9	6, 8	sylib 198	. . . . 5 |- (ph -> {<.x, y>. | (x e. A /\ y = C)} Fn A)
10		en2d.4	. . . . . . 7 |- (ph -> ((x e. A /\ y = C) <-> (y e. B /\ x = D)))
11	10	opabbidv 2670	. . . . . 6 |- (ph -> {<.x, y>. | (x e. A /\ y = C)} = {<.x, y>. | (y e. B /\ x = D)})
12		fneq1 3582	. . . . . 6 |- ({<.x, y>. | (x e. A /\ y = C)} = {<.x, y>. | (y e. B /\ x = D)} -> ({<.x, y>. | (x e. A /\ y = C)} Fn A <-> {<.x, y>. | (y e. B /\ x = D)} Fn A))
13	11, 12	syl 10	. . . . 5 |- (ph -> ({<.x, y>. | (x e. A /\ y = C)} Fn A <-> {<.x, y>. | (y e. B /\ x = D)} Fn A))
14	9, 13	mpbid 195	. . . 4 |- (ph -> {<.x, y>. | (y e. B /\ x = D)} Fn A)
15		en2d.3	. . . . . . 7 |- (ph -> (y e. B -> D e. V))
16		eueq 1916	. . . . . . 7 |- (D e. V <-> E!x x = D)
17	15, 16	syl6ib 212	. . . . . 6 |- (ph -> (y e. B -> E!x x = D))
18	17	r19.21aiv 1713	. . . . 5 |- (ph -> A.y e. B E!x x = D)
19		cnvopab 3445	. . . . . 6 |- `'{<.x, y>. | (y e. B /\ x = D)} = {<.y, x>. | (y e. B /\ x = D)}
20	19	fnopabg 3615	. . . . 5 |- (A.y e. B E!x x = D <-> `'{<.x, y>. | (y e. B /\ x = D)} Fn B)
21	18, 20	sylib 198	. . . 4 |- (ph -> `'{<.x, y>. | (y e. B /\ x = D)} Fn B)
22	14, 21	jca 288	. . 3 |- (ph -> ({<.x, y>. | (y e. B /\ x = D)} Fn A /\ `'{<.x, y>. | (y e. B /\ x = D)} Fn B))
23		f1o4 3696	. . 3 |- ({<.x, y>. | (y e. B /\ x = D)}:A-1-1-onto->B <-> ({<.x, y>. | (y e. B /\ x = D)} Fn A /\ `'{<.x, y>. | (y e. B /\ x = D)} Fn B))
24	22, 23	sylibr 200	. 2 |- (ph -> {<.x, y>. | (y e. B /\ x = D)}:A-1-1-onto->B)
25	1, 2, 24	sylanc 471	1 |- (ph -> A ~~ B)

Syntax hints:   -> wi 3   <-> wb 146   /\ wa 223   = wceq 956   e. wcel 958  E!weu 1380  A.wral 1645  Vcvv 1811   class class class wbr 2619  {copab 2666  `'ccnv 3169   Fn wfn 3177  -1-1-onto->wf1o 3181   ~~ cen 4364

This theorem is referenced by:  en3d 4401  en2 4402

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-rep 2693  ax-sep 2703  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-f1 3195  df-fo 3196  df-f1o 3197  df-en 4368

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