Theorem xpen 7135

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 7020	. . . 4 ⊢ (𝐴 ≈ 𝐵 ↔ ∃𝑓 𝑓:𝐴–1-1-onto→𝐵)
2	1	biimpi 120	. . 3 ⊢ (𝐴 ≈ 𝐵 → ∃𝑓 𝑓:𝐴–1-1-onto→𝐵)
3	2	adantr 276	. 2 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → ∃𝑓 𝑓:𝐴–1-1-onto→𝐵)
4		bren 7020	. . . . 5 ⊢ (𝐶 ≈ 𝐷 ↔ ∃𝑔 𝑔:𝐶–1-1-onto→𝐷)
5	4	biimpi 120	. . . 4 ⊢ (𝐶 ≈ 𝐷 → ∃𝑔 𝑔:𝐶–1-1-onto→𝐷)
6	5	ad2antlr 493	. . 3 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) ∧ 𝑓:𝐴–1-1-onto→𝐵) → ∃𝑔 𝑔:𝐶–1-1-onto→𝐷)
7		relen 7016	. . . . . . 7 ⊢ Rel ≈
8	7	brrelex1i 4813	. . . . . 6 ⊢ (𝐴 ≈ 𝐵 → 𝐴 ∈ V)
9	7	brrelex1i 4813	. . . . . 6 ⊢ (𝐶 ≈ 𝐷 → 𝐶 ∈ V)
10		xpexg 4884	. . . . . 6 ⊢ ((𝐴 ∈ V ∧ 𝐶 ∈ V) → (𝐴 × 𝐶) ∈ V)
11	8, 9, 10	syl2an 289	. . . . 5 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (𝐴 × 𝐶) ∈ V)
12	11	ad2antrr 492	. . . 4 ⊢ ((((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) ∧ 𝑓:𝐴–1-1-onto→𝐵) ∧ 𝑔:𝐶–1-1-onto→𝐷) → (𝐴 × 𝐶) ∈ V)
13		simplr 533	. . . . . 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 5635	. . . . . . . 8 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝑓 Fn 𝐴)
15		dffn5im 5742	. . . . . . . 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 5622	. . . . . . 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 147	. . . . 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 110	. . . . . 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 5635	. . . . . . . 8 ⊢ (𝑔:𝐶–1-1-onto→𝐷 → 𝑔 Fn 𝐶)
22		dffn5im 5742	. . . . . . . 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 5622	. . . . . . 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 147	. . . . 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 7134	. . . 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 7033	. . . 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 415	. . 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 1954	. 2 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) ∧ 𝑓:𝐴–1-1-onto→𝐵) → (𝐴 × 𝐶) ≈ (𝐵 × 𝐷))
31	3, 30	exlimddv 1954	1 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (𝐴 × 𝐶) ≈ (𝐵 × 𝐷))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1402  ∃wex 1545   ∈ wcel 2209  Vcvv 2821  ⟨cop 3708   class class class wbr 4125   ↦ cmpt 4187   × cxp 4767   Fn wfn 5367  –1-1-onto→wf1o 5371  ‘cfv 5372   ∈ cmpo 6077   ≈ cen 7010

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-coll 4241  ax-sep 4244  ax-pow 4306  ax-pr 4341  ax-un 4573

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-eu 2089  df-mo 2090  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-rex 2534  df-reu 2535  df-rab 2537  df-v 2823  df-sbc 3052  df-csb 3148  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-uni 3931  df-iun 4009  df-br 4126  df-opab 4188  df-mpt 4189  df-id 4433  df-xp 4775  df-rel 4776  df-cnv 4777  df-co 4778  df-dm 4779  df-rn 4780  df-res 4781  df-ima 4782  df-iota 5332  df-fun 5374  df-fn 5375  df-f 5376  df-f1 5377  df-fo 5378  df-f1o 5379  df-fv 5380  df-oprab 6079  df-mpo 6080  df-1st 6364  df-2nd 6365  df-en 7013

This theorem is referenced by:  xpdjuen  7564  xpnnen  13263  xpomen  13264  qnnen  13300