Theorem xpen 6692

Description: Equinumerosity law for Cartesian product. Proposition 4.22(b) of [Mendelson] p. 254. (Contributed by NM, 24-Jul-2004.)

Assertion

Ref	Expression
xpen	⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (𝐴 × 𝐶) ≈ (𝐵 × 𝐷))

Proof of Theorem xpen

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

Step	Hyp	Ref	Expression
1		bren 6595	. . . 4 ⊢ (𝐴 ≈ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝐵)
2	1	biimpi 119	. . 3 ⊢ (𝐴 ≈ 𝐵 → ∃𝑓 𝑓:𝐴–1-1-onto→𝐵)
3	2	adantr 272	. 2 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → ∃𝑓 𝑓:𝐴–1-1-onto→𝐵)
4		bren 6595	. . . . 5 ⊢ (𝐶 ≈ 𝐷 ↔ ∃𝑔 𝑔:𝐶–1-1-onto→𝐷)
5	4	biimpi 119	. . . 4 ⊢ (𝐶 ≈ 𝐷 → ∃𝑔 𝑔:𝐶–1-1-onto→𝐷)
6	5	ad2antlr 478	. . 3 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) ∧ 𝑓:𝐴–1-1-onto→𝐵) → ∃𝑔 𝑔:𝐶–1-1-onto→𝐷)
7		relen 6592	. . . . . . 7 ⊢ Rel ≈
8	7	brrelex1i 4542	. . . . . 6 ⊢ (𝐴 ≈ 𝐵 → 𝐴 ∈ V)
9	7	brrelex1i 4542	. . . . . 6 ⊢ (𝐶 ≈ 𝐷 → 𝐶 ∈ V)
10		xpexg 4613	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝐶 ∈ V) → (𝐴 × 𝐶) ∈ V)
11	8, 9, 10	syl2an 285	. . . . 5 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (𝐴 × 𝐶) ∈ V)
12	11	ad2antrr 477	. . . 4 ⊢ ((((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) ∧ 𝑓:𝐴–1-1-onto→𝐵) ∧ 𝑔:𝐶–1-1-onto→𝐷) → (𝐴 × 𝐶) ∈ V)
13		simplr 502	. . . . . 6 ⊢ ((((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) ∧ 𝑓:𝐴–1-1-onto→𝐵) ∧ 𝑔:𝐶–1-1-onto→𝐷) → 𝑓:𝐴–1-1-onto→𝐵)
14		f1ofn 5324	. . . . . . . 8 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝑓 Fn 𝐴)
15		dffn5im 5421	. . . . . . . 8 ⊢ (𝑓 Fn 𝐴 → 𝑓 = (𝑥 ∈ 𝐴 ↦ (𝑓‘𝑥)))
16	14, 15	syl 14	. . . . . . 7 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝑓 = (𝑥 ∈ 𝐴 ↦ (𝑓‘𝑥)))
17		f1oeq1 5314	. . . . . . 7 ⊢ (𝑓 = (𝑥 ∈ 𝐴 ↦ (𝑓‘𝑥)) → (𝑓:𝐴–1-1-onto→𝐵 ↔ (𝑥 ∈ 𝐴 ↦ (𝑓‘𝑥)):𝐴–1-1-onto→𝐵))
18	13, 16, 17	3syl 17	. . . . . 6 ⊢ ((((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) ∧ 𝑓:𝐴–1-1-onto→𝐵) ∧ 𝑔:𝐶–1-1-onto→𝐷) → (𝑓:𝐴–1-1-onto→𝐵 ↔ (𝑥 ∈ 𝐴 ↦ (𝑓‘𝑥)):𝐴–1-1-onto→𝐵))
19	13, 18	mpbid 146	. . . . 5 ⊢ ((((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) ∧ 𝑓:𝐴–1-1-onto→𝐵) ∧ 𝑔:𝐶–1-1-onto→𝐷) → (𝑥 ∈ 𝐴 ↦ (𝑓‘𝑥)):𝐴–1-1-onto→𝐵)
20		simpr 109	. . . . . 6 ⊢ ((((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) ∧ 𝑓:𝐴–1-1-onto→𝐵) ∧ 𝑔:𝐶–1-1-onto→𝐷) → 𝑔:𝐶–1-1-onto→𝐷)
21		f1ofn 5324	. . . . . . . 8 ⊢ (𝑔:𝐶–1-1-onto→𝐷 → 𝑔 Fn 𝐶)
22		dffn5im 5421	. . . . . . . 8 ⊢ (𝑔 Fn 𝐶 → 𝑔 = (𝑦 ∈ 𝐶 ↦ (𝑔‘𝑦)))
23	21, 22	syl 14	. . . . . . 7 ⊢ (𝑔:𝐶–1-1-onto→𝐷 → 𝑔 = (𝑦 ∈ 𝐶 ↦ (𝑔‘𝑦)))
24		f1oeq1 5314	. . . . . . 7 ⊢ (𝑔 = (𝑦 ∈ 𝐶 ↦ (𝑔‘𝑦)) → (𝑔:𝐶–1-1-onto→𝐷 ↔ (𝑦 ∈ 𝐶 ↦ (𝑔‘𝑦)):𝐶–1-1-onto→𝐷))
25	20, 23, 24	3syl 17	. . . . . 6 ⊢ ((((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) ∧ 𝑓:𝐴–1-1-onto→𝐵) ∧ 𝑔:𝐶–1-1-onto→𝐷) → (𝑔:𝐶–1-1-onto→𝐷 ↔ (𝑦 ∈ 𝐶 ↦ (𝑔‘𝑦)):𝐶–1-1-onto→𝐷))
26	20, 25	mpbid 146	. . . . 5 ⊢ ((((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) ∧ 𝑓:𝐴–1-1-onto→𝐵) ∧ 𝑔:𝐶–1-1-onto→𝐷) → (𝑦 ∈ 𝐶 ↦ (𝑔‘𝑦)):𝐶–1-1-onto→𝐷)
27	19, 26	xpf1o 6691	. . . 4 ⊢ ((((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) ∧ 𝑓:𝐴–1-1-onto→𝐵) ∧ 𝑔:𝐶–1-1-onto→𝐷) → (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐶 ↦ ⟨(𝑓‘𝑥), (𝑔‘𝑦)⟩):(𝐴 × 𝐶)–1-1-onto→(𝐵 × 𝐷))
28		f1oeng 6605	. . . 4 ⊢ (((𝐴 × 𝐶) ∈ V ∧ (𝑥 ∈ 𝐴, 𝑦 ∈ 𝐶 ↦ ⟨(𝑓‘𝑥), (𝑔‘𝑦)⟩):(𝐴 × 𝐶)–1-1-onto→(𝐵 × 𝐷)) → (𝐴 × 𝐶) ≈ (𝐵 × 𝐷))
29	12, 27, 28	syl2anc 406	. . 3 ⊢ ((((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) ∧ 𝑓:𝐴–1-1-onto→𝐵) ∧ 𝑔:𝐶–1-1-onto→𝐷) → (𝐴 × 𝐶) ≈ (𝐵 × 𝐷))
30	6, 29	exlimddv 1852	. 2 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) ∧ 𝑓:𝐴–1-1-onto→𝐵) → (𝐴 × 𝐶) ≈ (𝐵 × 𝐷))
31	3, 30	exlimddv 1852	1 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (𝐴 × 𝐶) ≈ (𝐵 × 𝐷))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1314  ∃wex 1451   ∈ wcel 1463  Vcvv 2657  ⟨cop 3496   class class class wbr 3895   ↦ cmpt 3949   × cxp 4497   Fn wfn 5076  –1-1-onto→wf1o 5080  ‘cfv 5081   ∈ cmpo 5730   ≈ cen 6586

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4003  ax-sep 4006  ax-pow 4058  ax-pr 4091  ax-un 4315

This theorem depends on definitions:  df-bi 116  df-3an 947  df-tru 1317  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-ral 2395  df-rex 2396  df-reu 2397  df-rab 2399  df-v 2659  df-sbc 2879  df-csb 2972  df-un 3041  df-in 3043  df-ss 3050  df-pw 3478  df-sn 3499  df-pr 3500  df-op 3502  df-uni 3703  df-iun 3781  df-br 3896  df-opab 3950  df-mpt 3951  df-id 4175  df-xp 4505  df-rel 4506  df-cnv 4507  df-co 4508  df-dm 4509  df-rn 4510  df-res 4511  df-ima 4512  df-iota 5046  df-fun 5083  df-fn 5084  df-f 5085  df-f1 5086  df-fo 5087  df-f1o 5088  df-fv 5089  df-oprab 5732  df-mpo 5733  df-1st 5992  df-2nd 5993  df-en 6589

This theorem is referenced by:  xpdjuen  7022  xpnnen  11752  xpomen  11753  qnnen  11789