Theorem iundom 4812

Description: An upper bound for the cardinality of an indexed union. C depends on x and should be thought of as C(x).

Hypotheses

Ref	Expression
iundom.1	|- A e. V
iundom.2	|- B e. V
iundom.3	|- C e. V

Assertion

Ref	Expression
iundom	|- (A.x e. A C ~<_ B -> U_x e. A C ~<_ (A X. B))

Distinct variable groups:   x,A   x,B

Proof of Theorem iundom

Step	Hyp	Ref	Expression
1		iundom.3	. . . . 5 |- C e. V
2		fvopab2 3791	. . . . 5 |- ((x e. A /\ C e. V) -> ({<.x, y>. | (x e. A /\ y = C)}` x) = C)
3	1, 2	mpan2 696	. . . 4 |- (x e. A -> ({<.x, y>. | (x e. A /\ y = C)}` x) = C)
4	3	breq1d 2629	. . 3 |- (x e. A -> (({<.x, y>. | (x e. A /\ y = C)}` x) ~<_ B <-> C ~<_ B))
5	4	ralbiia 1673	. 2 |- (A.x e. A ({<.x, y>. | (x e. A /\ y = C)}` x) ~<_ B <-> A.x e. A C ~<_ B)
6		eqid 1475	. . . . . 6 |- {<.x, y>. | (x e. A /\ y = C)} = {<.x, y>. | (x e. A /\ y = C)}
7	1, 6	fnopab2 3618	. . . . 5 |- {<.x, y>. | (x e. A /\ y = C)} Fn A
8		fnfun 3585	. . . . 5 |- ({<.x, y>. | (x e. A /\ y = C)} Fn A -> Fun {<.x, y>. | (x e. A /\ y = C)})
9	7, 8	ax-mp 7	. . . 4 |- Fun {<.x, y>. | (x e. A /\ y = C)}
10		hbopab1 2813	. . . . 5 |- (z e. {<.x, y>. | (x e. A /\ y = C)} -> A.x z e. {<.x, y>. | (x e. A /\ y = C)})
11		iundom.1	. . . . 5 |- A e. V
12		iundom.2	. . . . 5 |- B e. V
13	10, 11, 12	uniimadomf 4811	. . . 4 |- ((Fun {<.x, y>. | (x e. A /\ y = C)} /\ A.x e. A ({<.x, y>. | (x e. A /\ y = C)}` x) ~<_ B) -> U.({<.x, y>. | (x e. A /\ y = C)}"A) ~<_ (A X. B))
14	9, 13	mpan 695	. . 3 |- (A.x e. A ({<.x, y>. | (x e. A /\ y = C)}` x) ~<_ B -> U.({<.x, y>. | (x e. A /\ y = C)}"A) ~<_ (A X. B))
15	3	iuneq2i 2580	. . . 4 |- U_x e. A ({<.x, y>. | (x e. A /\ y = C)}` x) = U_x e. A C
16	10	funiunfvf 3870	. . . . 5 |- (Fun {<.x, y>. | (x e. A /\ y = C)} -> U_x e. A ({<.x, y>. | (x e. A /\ y = C)}` x) = U.({<.x, y>. | (x e. A /\ y = C)}"A))
17	9, 16	ax-mp 7	. . . 4 |- U_x e. A ({<.x, y>. | (x e. A /\ y = C)}` x) = U.({<.x, y>. | (x e. A /\ y = C)}"A)
18	15, 17	eqtr3 1497	. . 3 |- U_x e. A C = U.({<.x, y>. | (x e. A /\ y = C)}"A)
19	14, 18	syl5eqbr 2648	. 2 |- (A.x e. A ({<.x, y>. | (x e. A /\ y = C)}` x) ~<_ B -> U_x e. A C ~<_ (A X. B))
20	5, 19	sylbir 201	1 |- (A.x e. A C ~<_ B -> U_x e. A C ~<_ (A X. B))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 956   e. wcel 958  A.wral 1645  Vcvv 1811  U.cuni 2503  U_ciun 2566   class class class wbr 2619  {copab 2666   X. cxp 3168  "cima 3173  Fun wfun 3176   Fn wfn 3177  ` cfv 3182   ~<_ cdom 4365

This theorem is referenced by:  iunctb 7575

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-nul 2710  ax-pow 2742  ax-pr 2779  ax-un 2866  ax-reg 4593  ax-inf2 4625  ax-ac 4744

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  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-reu 1651  df-rab 1652  df-v 1812  df-sbc 1942  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-if 2362  df-pw 2402  df-sn 2412  df-pr 2413  df-tp 2415  df-op 2416  df-uni 2504  df-int 2534  df-iun 2568  df-iin 2569  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-id 2835  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951  df-on 2952  df-lim 2953  df-suc 2954  df-om 3132  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  df-rdg 3932  df-en 4368  df-dom 4369  df-r1 4643  df-rank 4644

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