Theorem cardiun 9908

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 7917	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (card‘𝐵)} ∈ V)
2		vex 3446	. . . . . . . . 9 ⊢ 𝑦 ∈ V
3		eqeq1 2741	. . . . . . . . . 10 ⊢ (𝑧 = 𝑦 → (𝑧 = (card‘𝐵) ↔ 𝑦 = (card‘𝐵)))
4	3	rexbidv 3162	. . . . . . . . 9 ⊢ (𝑧 = 𝑦 → (∃𝑥 ∈ 𝐴 𝑧 = (card‘𝐵) ↔ ∃𝑥 ∈ 𝐴 𝑦 = (card‘𝐵)))
5	2, 4	elab 3636	. . . . . . . 8 ⊢ (𝑦 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (card‘𝐵)} ↔ ∃𝑥 ∈ 𝐴 𝑦 = (card‘𝐵))
6		cardidm 9885	. . . . . . . . . 10 ⊢ (card‘(card‘𝐵)) = (card‘𝐵)
7		fveq2 6844	. . . . . . . . . 10 ⊢ (𝑦 = (card‘𝐵) → (card‘𝑦) = (card‘(card‘𝐵)))
8		id 22	. . . . . . . . . 10 ⊢ (𝑦 = (card‘𝐵) → 𝑦 = (card‘𝐵))
9	6, 7, 8	3eqtr4a 2798	. . . . . . . . 9 ⊢ (𝑦 = (card‘𝐵) → (card‘𝑦) = 𝑦)
10	9	rexlimivw 3135	. . . . . . . 8 ⊢ (∃𝑥 ∈ 𝐴 𝑦 = (card‘𝐵) → (card‘𝑦) = 𝑦)
11	5, 10	sylbi 217	. . . . . . 7 ⊢ (𝑦 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (card‘𝐵)} → (card‘𝑦) = 𝑦)
12	11	rgen 3054	. . . . . 6 ⊢ ∀𝑦 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (card‘𝐵)} (card‘𝑦) = 𝑦
13		carduni 9907	. . . . . 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 6857	. . . . . . 7 ⊢ (card‘𝐵) ∈ V
16	15	dfiun2 4989	. . . . . 6 ⊢ ∪ 𝑥 ∈ 𝐴 (card‘𝐵) = ∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (card‘𝐵)}
17	16	fveq2i 6847	. . . . 5 ⊢ (card‘∪ 𝑥 ∈ 𝐴 (card‘𝐵)) = (card‘∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (card‘𝐵)})
18	14, 17, 16	3eqtr4g 2797	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (card‘∪ 𝑥 ∈ 𝐴 (card‘𝐵)) = ∪ 𝑥 ∈ 𝐴 (card‘𝐵))
19	18	adantr 480	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 (card‘𝐵) = 𝐵) → (card‘∪ 𝑥 ∈ 𝐴 (card‘𝐵)) = ∪ 𝑥 ∈ 𝐴 (card‘𝐵))
20		iuneq2 4968	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (card‘𝐵) = 𝐵 → ∪ 𝑥 ∈ 𝐴 (card‘𝐵) = ∪ 𝑥 ∈ 𝐴 𝐵)
21	20	adantl 481	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 (card‘𝐵) = 𝐵) → ∪ 𝑥 ∈ 𝐴 (card‘𝐵) = ∪ 𝑥 ∈ 𝐴 𝐵)
22	21	fveq2d 6848	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 (card‘𝐵) = 𝐵) → (card‘∪ 𝑥 ∈ 𝐴 (card‘𝐵)) = (card‘∪ 𝑥 ∈ 𝐴 𝐵))
23	19, 22, 21	3eqtr3d 2780	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 (card‘𝐵) = 𝐵) → (card‘∪ 𝑥 ∈ 𝐴 𝐵) = ∪ 𝑥 ∈ 𝐴 𝐵)
24	23	ex 412	1 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ∈ 𝐴 (card‘𝐵) = 𝐵 → (card‘∪ 𝑥 ∈ 𝐴 𝐵) = ∪ 𝑥 ∈ 𝐴 𝐵))

Syntax hints:   → wi 4   ∧ wa 395   = wceq 1542   ∈ wcel 2114  {cab 2715  ∀wral 3052  ∃wrex 3062  Vcvv 3442  ∪ cuni 4865  ∪ ciun 4948  ‘cfv 6502  cardccrd 9861

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-10 2147  ax-11 2163  ax-12 2185  ax-ext 2709  ax-rep 5226  ax-sep 5245  ax-nul 5255  ax-pow 5314  ax-pr 5381  ax-un 7692

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3or 1088  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-nf 1786  df-sb 2069  df-mo 2540  df-eu 2570  df-clab 2716  df-cleq 2729  df-clel 2812  df-nfc 2886  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3402  df-v 3444  df-dif 3906  df-un 3908  df-in 3910  df-ss 3920  df-pss 3923  df-nul 4288  df-if 4482  df-pw 4558  df-sn 4583  df-pr 4585  df-op 4589  df-uni 4866  df-int 4905  df-iun 4950  df-br 5101  df-opab 5163  df-mpt 5182  df-tr 5208  df-id 5529  df-eprel 5534  df-po 5542  df-so 5543  df-fr 5587  df-we 5589  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-res 5646  df-ima 5647  df-ord 6330  df-on 6331  df-iota 6458  df-fun 6504  df-fn 6505  df-f 6506  df-f1 6507  df-fo 6508  df-f1o 6509  df-fv 6510  df-er 8647  df-en 8898  df-dom 8899  df-sdom 8900  df-card 9865

This theorem is referenced by:  alephcard  9994