Theorem iundom 4822

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 ∈ V
iundom.2	⊢ B ∈ V
iundom.3	⊢ C ∈ V

Assertion

Ref	Expression
iundom	⊢ (∀x ∈ A C ≼ B → ∪x ∈ A C ≼ (A × B))

Distinct variable groups:   x,A   x,B

Proof of Theorem iundom

Step	Hyp	Ref	Expression
1		iundom.3	. . . . 5 ⊢ C ∈ V
2		fvopab2 3797	. . . . 5 ⊢ ((x ∈ A ⋀ C ∈ V) → ({⟨x, y⟩∣(x ∈ A ⋀ y = C)} ‘x) = C)
3	1, 2	mpan2 698	. . . 4 ⊢ (x ∈ A → ({⟨x, y⟩∣(x ∈ A ⋀ y = C)} ‘x) = C)
4	3	breq1d 2634	. . 3 ⊢ (x ∈ A → (({⟨x, y⟩∣(x ∈ A ⋀ y = C)} ‘x) ≼ B ↔ C ≼ B))
5	4	ralbiia 1676	. 2 ⊢ (∀x ∈ A ({⟨x, y⟩∣(x ∈ A ⋀ y = C)} ‘x) ≼ B ↔ ∀x ∈ A C ≼ B)
6		eqid 1478	. . . . . 6 ⊢ {⟨x, y⟩∣(x ∈ A ⋀ y = C)} = {⟨x, y⟩∣(x ∈ A ⋀ y = C)}
7	1, 6	fnopab2 3624	. . . . 5 ⊢ {⟨x, y⟩∣(x ∈ A ⋀ y = C)} Fn A
8		fnfun 3591	. . . . 5 ⊢ ({⟨x, y⟩∣(x ∈ A ⋀ y = C)} Fn A → Fun {⟨x, y⟩∣(x ∈ A ⋀ y = C)})
9	7, 8	ax-mp 7	. . . 4 ⊢ Fun {⟨x, y⟩∣(x ∈ A ⋀ y = C)}
10		hbopab1 2819	. . . . 5 ⊢ (z ∈ {⟨x, y⟩∣(x ∈ A ⋀ y = C)} → ∀x z ∈ {⟨x, y⟩∣(x ∈ A ⋀ y = C)})
11		iundom.1	. . . . 5 ⊢ A ∈ V
12		iundom.2	. . . . 5 ⊢ B ∈ V
13	10, 11, 12	uniimadomf 4821	. . . 4 ⊢ ((Fun {⟨x, y⟩∣(x ∈ A ⋀ y = C)} ⋀ ∀x ∈ A ({⟨x, y⟩∣(x ∈ A ⋀ y = C)} ‘x) ≼ B) → ∪({⟨x, y⟩∣(x ∈ A ⋀ y = C)} “ A) ≼ (A × B))
14	9, 13	mpan 697	. . 3 ⊢ (∀x ∈ A ({⟨x, y⟩∣(x ∈ A ⋀ y = C)} ‘x) ≼ B → ∪({⟨x, y⟩∣(x ∈ A ⋀ y = C)} “ A) ≼ (A × B))
15	3	iuneq2i 2584	. . . 4 ⊢ ∪x ∈ A ({⟨x, y⟩∣(x ∈ A ⋀ y = C)} ‘x) = ∪x ∈ A C
16	10	funiunfvf 3876	. . . . 5 ⊢ (Fun {⟨x, y⟩∣(x ∈ A ⋀ y = C)} → ∪x ∈ A ({⟨x, y⟩∣(x ∈ A ⋀ y = C)} ‘x) = ∪({⟨x, y⟩∣(x ∈ A ⋀ y = C)} “ A))
17	9, 16	ax-mp 7	. . . 4 ⊢ ∪x ∈ A ({⟨x, y⟩∣(x ∈ A ⋀ y = C)} ‘x) = ∪({⟨x, y⟩∣(x ∈ A ⋀ y = C)} “ A)
18	15, 17	eqtr3 1500	. . 3 ⊢ ∪x ∈ A C = ∪({⟨x, y⟩∣(x ∈ A ⋀ y = C)} “ A)
19	14, 18	syl5eqbr 2653	. 2 ⊢ (∀x ∈ A ({⟨x, y⟩∣(x ∈ A ⋀ y = C)} ‘x) ≼ B → ∪x ∈ A C ≼ (A × B))
20	5, 19	sylbir 201	1 ⊢ (∀x ∈ A C ≼ B → ∪x ∈ A C ≼ (A × B))

Syntax hints:   → wi 3   ⋀ wa 223   = wceq 958   ∈ wcel 960  ∀wral 1648  Vcvv 1814  ∪cuni 2507  ∪ciun 2570   class class class wbr 2624  {copab 2671   × cxp 3174   “ cima 3179  Fun wfun 3182   Fn wfn 3183   ‘cfv 3188   ≼ cdom 4371

This theorem is referenced by:  iunctb 7576

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-9 967  ax-10 968  ax-11 969  ax-12 970  ax-13 971  ax-14 972  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462  ax-rep 2698  ax-sep 2708  ax-nul 2715  ax-pow 2748  ax-pr 2785  ax-un 2872  ax-reg 4602  ax-inf2 4634  ax-ac 4754

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 778  df-3an 779  df-ex 983  df-sb 1174  df-eu 1384  df-mo 1385  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-ral 1652  df-rex 1653  df-reu 1654  df-rab 1655  df-v 1815  df-sbc 1945  df-dif 2052  df-un 2053  df-in 2054  df-ss 2056  df-nul 2284  df-if 2366  df-pw 2406  df-sn 2416  df-pr 2417  df-tp 2419  df-op 2420  df-uni 2508  df-int 2538  df-iun 2572  df-iin 2573  df-br 2625  df-opab 2672  df-tr 2686  df-eprel 2838  df-id 2841  df-po 2846  df-so 2856  df-fr 2923  df-we 2940  df-ord 2957  df-on 2958  df-lim 2959  df-suc 2960  df-om 3138  df-xp 3190  df-rel 3191  df-cnv 3192  df-co 3193  df-dm 3194  df-rn 3195  df-res 3196  df-ima 3197  df-fun 3198  df-fn 3199  df-f 3200  df-f1 3201  df-fo 3202  df-f1o 3203  df-fv 3204  df-rdg 3938  df-en 4374  df-dom 4375  df-r1 4653  df-rank 4654

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