Theorem xpen 7074

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 6960	. . . 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 6960	. . . . 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 6956	. . . . . . 7 ⊢ Rel ≈
8	7	brrelex1i 4775	. . . . . 6 ⊢ (𝐴 ≈ 𝐵 → 𝐴 ∈ V)
9	7	brrelex1i 4775	. . . . . 6 ⊢ (𝐶 ≈ 𝐷 → 𝐶 ∈ V)
10		xpexg 4846	. . . . . 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 529	. . . . . 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 5593	. . . . . . . 8 ⊢ (𝑓:𝐴–1-1-onto→𝐵 → 𝑓 Fn 𝐴)
15		dffn5im 5700	. . . . . . . 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 5580	. . . . . . 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 5593	. . . . . . . 8 ⊢ (𝑔:𝐶–1-1-onto→𝐷 → 𝑔 Fn 𝐶)
22		dffn5im 5700	. . . . . . . 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 5580	. . . . . . 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 7073	. . . 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 6973	. . . 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 1947	. 2 ⊢ (((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) ∧ 𝑓:𝐴–1-1-onto→𝐵) → (𝐴 × 𝐶) ≈ (𝐵 × 𝐷))
31	3, 30	exlimddv 1947	1 ⊢ ((𝐴 ≈ 𝐵 ∧ 𝐶 ≈ 𝐷) → (𝐴 × 𝐶) ≈ (𝐵 × 𝐷))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398  ∃wex 1541   ∈ wcel 2202  Vcvv 2803  ⟨cop 3676   class class class wbr 4093   ↦ cmpt 4155   × cxp 4729   Fn wfn 5328  –1-1-onto→wf1o 5332  ‘cfv 5333   ∈ cmpo 6030   ≈ cen 6950

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2204  ax-14 2205  ax-ext 2213  ax-coll 4209  ax-sep 4212  ax-pow 4270  ax-pr 4305  ax-un 4536

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2364  df-ral 2516  df-rex 2517  df-reu 2518  df-rab 2520  df-v 2805  df-sbc 3033  df-csb 3129  df-un 3205  df-in 3207  df-ss 3214  df-pw 3658  df-sn 3679  df-pr 3680  df-op 3682  df-uni 3899  df-iun 3977  df-br 4094  df-opab 4156  df-mpt 4157  df-id 4396  df-xp 4737  df-rel 4738  df-cnv 4739  df-co 4740  df-dm 4741  df-rn 4742  df-res 4743  df-ima 4744  df-iota 5293  df-fun 5335  df-fn 5336  df-f 5337  df-f1 5338  df-fo 5339  df-f1o 5340  df-fv 5341  df-oprab 6032  df-mpo 6033  df-1st 6312  df-2nd 6313  df-en 6953

This theorem is referenced by:  xpdjuen  7476  xpnnen  13078  xpomen  13079  qnnen  13115