Theorem xpnum 9943

Description: The cartesian product of numerable sets is numerable. (Contributed by Mario Carneiro, 3-Mar-2013.) (Revised by Mario Carneiro, 29-Apr-2015.)

Assertion

Ref	Expression
xpnum	⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴 × 𝐵) ∈ dom card)

Proof of Theorem xpnum

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

Step	Hyp	Ref	Expression
1		isnum2 9937	. 2 ⊢ (𝐴 ∈ dom card ↔ ∃𝑥 ∈ On 𝑥 ≈ 𝐴)
2		isnum2 9937	. 2 ⊢ (𝐵 ∈ dom card ↔ ∃𝑦 ∈ On 𝑦 ≈ 𝐵)
3		reeanv 3227	. . 3 ⊢ (∃𝑥 ∈ On ∃𝑦 ∈ On (𝑥 ≈ 𝐴 ∧ 𝑦 ≈ 𝐵) ↔ (∃𝑥 ∈ On 𝑥 ≈ 𝐴 ∧ ∃𝑦 ∈ On 𝑦 ≈ 𝐵))
4		omcl 8533	. . . . . 6 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥 ·o 𝑦) ∈ On)
5		omxpen 9071	. . . . . . 7 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥 ·o 𝑦) ≈ (𝑥 × 𝑦))
6		xpen 9137	. . . . . . 7 ⊢ ((𝑥 ≈ 𝐴 ∧ 𝑦 ≈ 𝐵) → (𝑥 × 𝑦) ≈ (𝐴 × 𝐵))
7		entr 8999	. . . . . . 7 ⊢ (((𝑥 ·o 𝑦) ≈ (𝑥 × 𝑦) ∧ (𝑥 × 𝑦) ≈ (𝐴 × 𝐵)) → (𝑥 ·o 𝑦) ≈ (𝐴 × 𝐵))
8	5, 6, 7	syl2an 597	. . . . . 6 ⊢ (((𝑥 ∈ On ∧ 𝑦 ∈ On) ∧ (𝑥 ≈ 𝐴 ∧ 𝑦 ≈ 𝐵)) → (𝑥 ·o 𝑦) ≈ (𝐴 × 𝐵))
9		isnumi 9938	. . . . . 6 ⊢ (((𝑥 ·o 𝑦) ∈ On ∧ (𝑥 ·o 𝑦) ≈ (𝐴 × 𝐵)) → (𝐴 × 𝐵) ∈ dom card)
10	4, 8, 9	syl2an2r 684	. . . . 5 ⊢ (((𝑥 ∈ On ∧ 𝑦 ∈ On) ∧ (𝑥 ≈ 𝐴 ∧ 𝑦 ≈ 𝐵)) → (𝐴 × 𝐵) ∈ dom card)
11	10	ex 414	. . . 4 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → ((𝑥 ≈ 𝐴 ∧ 𝑦 ≈ 𝐵) → (𝐴 × 𝐵) ∈ dom card))
12	11	rexlimivv 3200	. . 3 ⊢ (∃𝑥 ∈ On ∃𝑦 ∈ On (𝑥 ≈ 𝐴 ∧ 𝑦 ≈ 𝐵) → (𝐴 × 𝐵) ∈ dom card)
13	3, 12	sylbir 234	. 2 ⊢ ((∃𝑥 ∈ On 𝑥 ≈ 𝐴 ∧ ∃𝑦 ∈ On 𝑦 ≈ 𝐵) → (𝐴 × 𝐵) ∈ dom card)
14	1, 2, 13	syl2anb 599	1 ⊢ ((𝐴 ∈ dom card ∧ 𝐵 ∈ dom card) → (𝐴 × 𝐵) ∈ dom card)

Syntax hints:   → wi 4   ∧ wa 397   ∈ wcel 2107  ∃wrex 3071   class class class wbr 5148   × cxp 5674  dom cdm 5676  Oncon0 6362  (class class class)co 7406   ·_o comu 8461   ≈ cen 8933  cardccrd 9927

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-rep 5285  ax-sep 5299  ax-nul 5306  ax-pow 5363  ax-pr 5427  ax-un 7722

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3377  df-reu 3378  df-rab 3434  df-v 3477  df-sbc 3778  df-csb 3894  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-pss 3967  df-nul 4323  df-if 4529  df-pw 4604  df-sn 4629  df-pr 4631  df-op 4635  df-uni 4909  df-int 4951  df-iun 4999  df-br 5149  df-opab 5211  df-mpt 5232  df-tr 5266  df-id 5574  df-eprel 5580  df-po 5588  df-so 5589  df-fr 5631  df-we 5633  df-xp 5682  df-rel 5683  df-cnv 5684  df-co 5685  df-dm 5686  df-rn 5687  df-res 5688  df-ima 5689  df-pred 6298  df-ord 6365  df-on 6366  df-lim 6367  df-suc 6368  df-iota 6493  df-fun 6543  df-fn 6544  df-f 6545  df-f1 6546  df-fo 6547  df-f1o 6548  df-fv 6549  df-ov 7409  df-oprab 7410  df-mpo 7411  df-om 7853  df-1st 7972  df-2nd 7973  df-frecs 8263  df-wrecs 8294  df-recs 8368  df-rdg 8407  df-1o 8463  df-oadd 8467  df-omul 8468  df-er 8700  df-en 8937  df-dom 8938  df-card 9931

This theorem is referenced by:  iunfictbso  10106  znnen  16152  qnnen  16153  ptcmplem2  23549  finixpnum  36462  poimirlem32  36509  isnumbasgrplem2  41832