Theorem cardiun 9942

Description: The indexed union of a set of cardinals is a cardinal. (Contributed by NM, 3-Nov-2003.)

Assertion

Ref	Expression
cardiun	⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ∈ 𝐴 (card‘𝐵) = 𝐵 → (card‘∪ 𝑥 ∈ 𝐴 𝐵) = ∪ 𝑥 ∈ 𝐴 𝐵))

Distinct variable group:   𝑥,𝐴

Allowed substitution hints:   𝐵(𝑥)   𝑉(𝑥)

Proof of Theorem cardiun

Dummy variables 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		abrexexg 7944	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (card‘𝐵)} ∈ V)
2		vex 3460	. . . . . . . . 9 ⊢ 𝑦 ∈ V
3		eqeq1 2768	. . . . . . . . . 10 ⊢ (𝑧 = 𝑦 → (𝑧 = (card‘𝐵) ↔ 𝑦 = (card‘𝐵)))
4	3	rexbidv 3188	. . . . . . . . 9 ⊢ (𝑧 = 𝑦 → (∃𝑥 ∈ 𝐴 𝑧 = (card‘𝐵) ↔ ∃𝑥 ∈ 𝐴 𝑦 = (card‘𝐵)))
5	2, 4	elab 3640	. . . . . . . 8 ⊢ (𝑦 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (card‘𝐵)} ↔ ∃𝑥 ∈ 𝐴 𝑦 = (card‘𝐵))
6		cardidm 9919	. . . . . . . . . 10 ⊢ (card‘(card‘𝐵)) = (card‘𝐵)
7		fveq2 6869	. . . . . . . . . 10 ⊢ (𝑦 = (card‘𝐵) → (card‘𝑦) = (card‘(card‘𝐵)))
8		id 22	. . . . . . . . . 10 ⊢ (𝑦 = (card‘𝐵) → 𝑦 = (card‘𝐵))
9	6, 7, 8	3eqtr4a 2825	. . . . . . . . 9 ⊢ (𝑦 = (card‘𝐵) → (card‘𝑦) = 𝑦)
10	9	rexlimivw 3161	. . . . . . . 8 ⊢ (∃𝑥 ∈ 𝐴 𝑦 = (card‘𝐵) → (card‘𝑦) = 𝑦)
11	5, 10	sylbi 219	. . . . . . 7 ⊢ (𝑦 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (card‘𝐵)} → (card‘𝑦) = 𝑦)
12	11	rgen 3080	. . . . . 6 ⊢ ∀𝑦 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (card‘𝐵)} (card‘𝑦) = 𝑦
13		carduni 9941	. . . . . 6 ⊢ ({𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (card‘𝐵)} ∈ V → (∀𝑦 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (card‘𝐵)} (card‘𝑦) = 𝑦 → (card‘∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (card‘𝐵)}) = ∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (card‘𝐵)}))
14	1, 12, 13	mpisyl 21	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (card‘∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (card‘𝐵)}) = ∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (card‘𝐵)})
15		fvex 6882	. . . . . . 7 ⊢ (card‘𝐵) ∈ V
16	15	dfiun2 4991	. . . . . 6 ⊢ ∪ 𝑥 ∈ 𝐴 (card‘𝐵) = ∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (card‘𝐵)}
17	16	fveq2i 6872	. . . . 5 ⊢ (card‘∪ 𝑥 ∈ 𝐴 (card‘𝐵)) = (card‘∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (card‘𝐵)})
18	14, 17, 16	3eqtr4g 2824	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (card‘∪ 𝑥 ∈ 𝐴 (card‘𝐵)) = ∪ 𝑥 ∈ 𝐴 (card‘𝐵))
19	18	adantr 484	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 (card‘𝐵) = 𝐵) → (card‘∪ 𝑥 ∈ 𝐴 (card‘𝐵)) = ∪ 𝑥 ∈ 𝐴 (card‘𝐵))
20		iuneq2 4971	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (card‘𝐵) = 𝐵 → ∪ 𝑥 ∈ 𝐴 (card‘𝐵) = ∪ 𝑥 ∈ 𝐴 𝐵)
21	20	adantl 485	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 (card‘𝐵) = 𝐵) → ∪ 𝑥 ∈ 𝐴 (card‘𝐵) = ∪ 𝑥 ∈ 𝐴 𝐵)
22	21	fveq2d 6873	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 (card‘𝐵) = 𝐵) → (card‘∪ 𝑥 ∈ 𝐴 (card‘𝐵)) = (card‘∪ 𝑥 ∈ 𝐴 𝐵))
23	19, 22, 21	3eqtr3d 2807	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 (card‘𝐵) = 𝐵) → (card‘∪ 𝑥 ∈ 𝐴 𝐵) = ∪ 𝑥 ∈ 𝐴 𝐵)
24	23	ex 416	1 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ∈ 𝐴 (card‘𝐵) = 𝐵 → (card‘∪ 𝑥 ∈ 𝐴 𝐵) = ∪ 𝑥 ∈ 𝐴 𝐵))

Syntax hints:   → wi 4   ∧ wa 399   = wceq 1562   ∈ wcel 2144  {cab 2742  ∀wral 3078  ∃wrex 3088  Vcvv 3456  ∪ cuni 4867  ∪ ciun 4951  ‘cfv 6523  cardccrd 9895

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1817  ax-4 1831  ax-5 1932  ax-6 1989  ax-7 2030  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2214  ax-ext 2736  ax-rep 5229  ax-sep 5248  ax-nul 5258  ax-pow 5324  ax-pr 5392  ax-un 7720

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-3or 1100  df-3an 1101  df-tru 1565  df-fal 1575  df-ex 1802  df-nf 1806  df-sb 2093  df-mo 2568  df-eu 2598  df-clab 2743  df-cleq 2756  df-clel 2839  df-nfc 2913  df-ne 2960  df-ral 3079  df-rex 3089  df-rab 3417  df-v 3458  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-pss 3926  df-nul 4288  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-int 4908  df-iun 4953  df-br 5103  df-opab 5165  df-mpt 5184  df-tr 5210  df-id 5544  df-eprel 5549  df-po 5557  df-so 5558  df-fr 5602  df-we 5604  df-xp 5655  df-rel 5656  df-cnv 5657  df-co 5658  df-dm 5659  df-rn 5660  df-res 5661  df-ima 5662  df-ord 6351  df-on 6352  df-iota 6479  df-fun 6525  df-fn 6526  df-f 6527  df-f1 6528  df-fo 6529  df-f1o 6530  df-fv 6531  df-er 8680  df-en 8930  df-dom 8931  df-sdom 8932  df-card 9899

This theorem is referenced by:  alephcard  10028