Theorem cardiun 10063

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 7973	. . . . . 6 ⊢ (𝐴 ∈ 𝑉 → {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (card‘𝐵)} ∈ V)
2		vex 3454	. . . . . . . . 9 ⊢ 𝑦 ∈ V
3		eqeq1 2764	. . . . . . . . . 10 ⊢ (𝑧 = 𝑦 → (𝑧 = (card‘𝐵) ↔ 𝑦 = (card‘𝐵)))
4	3	rexbidv 3186	. . . . . . . . 9 ⊢ (𝑧 = 𝑦 → (∃𝑥 ∈ 𝐴 𝑧 = (card‘𝐵) ↔ ∃𝑥 ∈ 𝐴 𝑦 = (card‘𝐵)))
5	2, 4	elab 3632	. . . . . . . 8 ⊢ (𝑦 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (card‘𝐵)} ↔ ∃𝑥 ∈ 𝐴 𝑦 = (card‘𝐵))
6		cardidm 10040	. . . . . . . . . 10 ⊢ (card‘(card‘𝐵)) = (card‘𝐵)
7		fveq2 6885	. . . . . . . . . 10 ⊢ (𝑦 = (card‘𝐵) → (card‘𝑦) = (card‘(card‘𝐵)))
8		id 23	. . . . . . . . . 10 ⊢ (𝑦 = (card‘𝐵) → 𝑦 = (card‘𝐵))
9	6, 7, 8	3eqtr4a 2821	. . . . . . . . 9 ⊢ (𝑦 = (card‘𝐵) → (card‘𝑦) = 𝑦)
10	9	rexlimivw 3159	. . . . . . . 8 ⊢ (∃𝑥 ∈ 𝐴 𝑦 = (card‘𝐵) → (card‘𝑦) = 𝑦)
11	5, 10	sylbi 220	. . . . . . 7 ⊢ (𝑦 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (card‘𝐵)} → (card‘𝑦) = 𝑦)
12	11	rgen 3078	. . . . . 6 ⊢ ∀𝑦 ∈ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (card‘𝐵)} (card‘𝑦) = 𝑦
13		carduni 10062	. . . . . 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 22	. . . . 5 ⊢ (𝐴 ∈ 𝑉 → (card‘∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (card‘𝐵)}) = ∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (card‘𝐵)})
15		fvex 6898	. . . . . . 7 ⊢ (card‘𝐵) ∈ V
16	15	dfiun2 4989	. . . . . 6 ⊢ ∪ 𝑥 ∈ 𝐴 (card‘𝐵) = ∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (card‘𝐵)}
17	16	fveq2i 6888	. . . . 5 ⊢ (card‘∪ 𝑥 ∈ 𝐴 (card‘𝐵)) = (card‘∪ {𝑧 ∣ ∃𝑥 ∈ 𝐴 𝑧 = (card‘𝐵)})
18	14, 17, 16	3eqtr4g 2820	. . . 4 ⊢ (𝐴 ∈ 𝑉 → (card‘∪ 𝑥 ∈ 𝐴 (card‘𝐵)) = ∪ 𝑥 ∈ 𝐴 (card‘𝐵))
19	18	adantr 486	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 (card‘𝐵) = 𝐵) → (card‘∪ 𝑥 ∈ 𝐴 (card‘𝐵)) = ∪ 𝑥 ∈ 𝐴 (card‘𝐵))
20		iuneq2 4970	. . . . 5 ⊢ (∀𝑥 ∈ 𝐴 (card‘𝐵) = 𝐵 → ∪ 𝑥 ∈ 𝐴 (card‘𝐵) = ∪ 𝑥 ∈ 𝐴 𝐵)
21	20	adantl 487	. . . 4 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 (card‘𝐵) = 𝐵) → ∪ 𝑥 ∈ 𝐴 (card‘𝐵) = ∪ 𝑥 ∈ 𝐴 𝐵)
22	21	fveq2d 6889	. . 3 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 (card‘𝐵) = 𝐵) → (card‘∪ 𝑥 ∈ 𝐴 (card‘𝐵)) = (card‘∪ 𝑥 ∈ 𝐴 𝐵))
23	19, 22, 21	3eqtr3d 2803	. 2 ⊢ ((𝐴 ∈ 𝑉 ∧ ∀𝑥 ∈ 𝐴 (card‘𝐵) = 𝐵) → (card‘∪ 𝑥 ∈ 𝐴 𝐵) = ∪ 𝑥 ∈ 𝐴 𝐵)
24	23	ex 418	1 ⊢ (𝐴 ∈ 𝑉 → (∀𝑥 ∈ 𝐴 (card‘𝐵) = 𝐵 → (card‘∪ 𝑥 ∈ 𝐴 𝐵) = ∪ 𝑥 ∈ 𝐴 𝐵))

This proof depends on syntax axioms:   → wi 4   ∧ wa 401   = wceq 1569   ∈ wcel 2144  {cab 2738  ∀wral 3076  ∃wrex 3086  Vcvv 3450  ∪ cuni 4866  ∪ ciun 4950  ‘cfv 6538  cardccrd 10016

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-rep 5231  ax-sep 5248  ax-nul 5259  ax-pow 5326  ax-pr 5390  ax-un 7751

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-eu 2594  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-pss 4014  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-pw 4558  df-sn 4584  df-pr 4586  df-op 4590  df-uni 4867  df-int 4907  df-iun 4952  df-br 5103  df-opab 5167  df-mpt 5186  df-tr 5212  df-id 5545  df-eprel 5550  df-po 5558  df-so 5559  df-fr 5603  df-we 5605  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-ord 6365  df-on 6366  df-iota 6494  df-fun 6540  df-fn 6541  df-f 6542  df-f1 6543  df-fo 6544  df-f1o 6545  df-fv 6546  df-er 8717  df-en 8974  df-dom 8975  df-sdom 8976  df-card 10020

This theorem is used by:  alephcard  10149