Theorem xp11m 5221

Description: The cross product of inhabited classes is one-to-one. (Contributed by Jim Kingdon, 13-Dec-2018.)

Assertion

Ref	Expression
xp11m	⊢ ((∃𝑥 𝑥 ∈ 𝐴 ∧ ∃𝑦 𝑦 ∈ 𝐵) → ((𝐴 × 𝐵) = (𝐶 × 𝐷) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))

Distinct variable groups:   𝑥,𝐴   𝑦,𝐵

Allowed substitution hints:   𝐴(𝑦)   𝐵(𝑥)   𝐶(𝑥,𝑦)   𝐷(𝑥,𝑦)

Proof of Theorem xp11m

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		xpm 5204	. . 3 ⊢ ((∃𝑥 𝑥 ∈ 𝐴 ∧ ∃𝑦 𝑦 ∈ 𝐵) ↔ ∃𝑧 𝑧 ∈ (𝐴 × 𝐵))
2		anidm 400	. . . . . 6 ⊢ ((∃𝑧 𝑧 ∈ (𝐴 × 𝐵) ∧ ∃𝑧 𝑧 ∈ (𝐴 × 𝐵)) ↔ ∃𝑧 𝑧 ∈ (𝐴 × 𝐵))
3		eleq2 2302	. . . . . . . 8 ⊢ ((𝐴 × 𝐵) = (𝐶 × 𝐷) → (𝑧 ∈ (𝐴 × 𝐵) ↔ 𝑧 ∈ (𝐶 × 𝐷)))
4	3	exbidv 1878	. . . . . . 7 ⊢ ((𝐴 × 𝐵) = (𝐶 × 𝐷) → (∃𝑧 𝑧 ∈ (𝐴 × 𝐵) ↔ ∃𝑧 𝑧 ∈ (𝐶 × 𝐷)))
5	4	anbi2d 468	. . . . . 6 ⊢ ((𝐴 × 𝐵) = (𝐶 × 𝐷) → ((∃𝑧 𝑧 ∈ (𝐴 × 𝐵) ∧ ∃𝑧 𝑧 ∈ (𝐴 × 𝐵)) ↔ (∃𝑧 𝑧 ∈ (𝐴 × 𝐵) ∧ ∃𝑧 𝑧 ∈ (𝐶 × 𝐷))))
6	2, 5	bitr3id 194	. . . . 5 ⊢ ((𝐴 × 𝐵) = (𝐶 × 𝐷) → (∃𝑧 𝑧 ∈ (𝐴 × 𝐵) ↔ (∃𝑧 𝑧 ∈ (𝐴 × 𝐵) ∧ ∃𝑧 𝑧 ∈ (𝐶 × 𝐷))))
7		eqimss 3302	. . . . . . . 8 ⊢ ((𝐴 × 𝐵) = (𝐶 × 𝐷) → (𝐴 × 𝐵) ⊆ (𝐶 × 𝐷))
8		ssxpbm 5218	. . . . . . . 8 ⊢ (∃𝑧 𝑧 ∈ (𝐴 × 𝐵) → ((𝐴 × 𝐵) ⊆ (𝐶 × 𝐷) ↔ (𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐷)))
9	7, 8	syl5ibcom 155	. . . . . . 7 ⊢ ((𝐴 × 𝐵) = (𝐶 × 𝐷) → (∃𝑧 𝑧 ∈ (𝐴 × 𝐵) → (𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐷)))
10		eqimss2 3303	. . . . . . . 8 ⊢ ((𝐴 × 𝐵) = (𝐶 × 𝐷) → (𝐶 × 𝐷) ⊆ (𝐴 × 𝐵))
11		ssxpbm 5218	. . . . . . . 8 ⊢ (∃𝑧 𝑧 ∈ (𝐶 × 𝐷) → ((𝐶 × 𝐷) ⊆ (𝐴 × 𝐵) ↔ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐵)))
12	10, 11	syl5ibcom 155	. . . . . . 7 ⊢ ((𝐴 × 𝐵) = (𝐶 × 𝐷) → (∃𝑧 𝑧 ∈ (𝐶 × 𝐷) → (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐵)))
13	9, 12	anim12d 335	. . . . . 6 ⊢ ((𝐴 × 𝐵) = (𝐶 × 𝐷) → ((∃𝑧 𝑧 ∈ (𝐴 × 𝐵) ∧ ∃𝑧 𝑧 ∈ (𝐶 × 𝐷)) → ((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐷) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐵))))
14		an4 592	. . . . . . 7 ⊢ (((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐷) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐵)) ↔ ((𝐴 ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐴) ∧ (𝐵 ⊆ 𝐷 ∧ 𝐷 ⊆ 𝐵)))
15		eqss 3263	. . . . . . . 8 ⊢ (𝐴 = 𝐶 ↔ (𝐴 ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐴))
16		eqss 3263	. . . . . . . 8 ⊢ (𝐵 = 𝐷 ↔ (𝐵 ⊆ 𝐷 ∧ 𝐷 ⊆ 𝐵))
17	15, 16	anbi12i 464	. . . . . . 7 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) ↔ ((𝐴 ⊆ 𝐶 ∧ 𝐶 ⊆ 𝐴) ∧ (𝐵 ⊆ 𝐷 ∧ 𝐷 ⊆ 𝐵)))
18	14, 17	bitr4i 187	. . . . . 6 ⊢ (((𝐴 ⊆ 𝐶 ∧ 𝐵 ⊆ 𝐷) ∧ (𝐶 ⊆ 𝐴 ∧ 𝐷 ⊆ 𝐵)) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷))
19	13, 18	imbitrdi 161	. . . . 5 ⊢ ((𝐴 × 𝐵) = (𝐶 × 𝐷) → ((∃𝑧 𝑧 ∈ (𝐴 × 𝐵) ∧ ∃𝑧 𝑧 ∈ (𝐶 × 𝐷)) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))
20	6, 19	sylbid 150	. . . 4 ⊢ ((𝐴 × 𝐵) = (𝐶 × 𝐷) → (∃𝑧 𝑧 ∈ (𝐴 × 𝐵) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))
21	20	com12 30	. . 3 ⊢ (∃𝑧 𝑧 ∈ (𝐴 × 𝐵) → ((𝐴 × 𝐵) = (𝐶 × 𝐷) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))
22	1, 21	sylbi 121	. 2 ⊢ ((∃𝑥 𝑥 ∈ 𝐴 ∧ ∃𝑦 𝑦 ∈ 𝐵) → ((𝐴 × 𝐵) = (𝐶 × 𝐷) → (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))
23		xpeq12 4788	. 2 ⊢ ((𝐴 = 𝐶 ∧ 𝐵 = 𝐷) → (𝐴 × 𝐵) = (𝐶 × 𝐷))
24	22, 23	impbid1 142	1 ⊢ ((∃𝑥 𝑥 ∈ 𝐴 ∧ ∃𝑦 𝑦 ∈ 𝐵) → ((𝐴 × 𝐵) = (𝐶 × 𝐷) ↔ (𝐴 = 𝐶 ∧ 𝐵 = 𝐷)))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1402  ∃wex 1545   ∈ wcel 2209   ⊆ wss 3220   × cxp 4767

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-sep 4244  ax-pow 4306  ax-pr 4341

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-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-op 3714  df-br 4126  df-opab 4188  df-xp 4775  df-rel 4776  df-cnv 4777  df-dm 4779  df-rn 4780

This theorem is referenced by:  cc2lem  7622  lmodfopnelem1  14633