Theorem xpen 7002

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 6893	. . . 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 6893	. . . . 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 489	. . 3 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) ∧ 𝑓:𝐴–1-1-onto→𝐵) → ∃𝑔 𝑔:𝐶–1-1-onto→𝐷)
7		relen 6889	. . . . . . 7 ⊢ Rel ≈
8	7	brrelex1i 4761	. . . . . 6 ⊢ (𝐴 ≈ 𝐵 → 𝐴 ∈ V)
9	7	brrelex1i 4761	. . . . . 6 ⊢ (𝐶 ≈ 𝐷 → 𝐶 ∈ V)
10		xpexg 4832	. . . . . 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 488	. . . 4 ⊢ ((((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) ∧ 𝑓:𝐴–1-1-onto→𝐵) ∧ 𝑔:𝐶–1-1-onto→𝐷) → (𝐴 × 𝐶) ∈ V)
13		simplr 528	. . . . . 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 5572	. . . . . . . 8 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝑓 Fn 𝐴)
15		dffn5im 5678	. . . . . . . 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 5559	. . . . . . 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 5572	. . . . . . . 8 ⊢ (𝑔:𝐶–1-1-onto→𝐷 → 𝑔 Fn 𝐶)
22		dffn5im 5678	. . . . . . . 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 5559	. . . . . . 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 7001	. . . 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 6906	. . . 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 411	. . 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 1945	. 2 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) ∧ 𝑓:𝐴–1-1-onto→𝐵) → (𝐴 × 𝐶) ≈ (𝐵 × 𝐷))
31	3, 30	exlimddv 1945	1 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (𝐴 × 𝐶) ≈ (𝐵 × 𝐷))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1395  ∃wex 1538   ∈ wcel 2200  Vcvv 2799  ⟨cop 3669   class class class wbr 4082   ↦ cmpt 4144   × cxp 4716   Fn wfn 5312  –1-1-onto→wf1o 5316  ‘cfv 5317   ∈ cmpo 6002   ≈ cen 6883

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 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-coll 4198  ax-sep 4201  ax-pow 4257  ax-pr 4292  ax-un 4523

This theorem depends on definitions:  df-bi 117  df-3an 1004  df-tru 1398  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-csb 3125  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3888  df-iun 3966  df-br 4083  df-opab 4145  df-mpt 4146  df-id 4383  df-xp 4724  df-rel 4725  df-cnv 4726  df-co 4727  df-dm 4728  df-rn 4729  df-res 4730  df-ima 4731  df-iota 5277  df-fun 5319  df-fn 5320  df-f 5321  df-f1 5322  df-fo 5323  df-f1o 5324  df-fv 5325  df-oprab 6004  df-mpo 6005  df-1st 6284  df-2nd 6285  df-en 6886

This theorem is referenced by:  xpdjuen  7396  xpnnen  12960  xpomen  12961  qnnen  12997