Theorem unixpm 4853

Description: The double class union of an inhabited cross product is the union of its members. (Contributed by Jim Kingdon, 18-Dec-2018.)

Assertion

Ref	Expression
unixpm	⊢ (∃𝑥 𝑥 ∈ (𝐴 × 𝐵) → ∪ ∪ (𝐴 × 𝐵) = (𝐴 ∪ 𝐵))

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem unixpm

Dummy variables 𝑎 𝑏 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		relxp 4447	. . 3 ⊢ Rel (𝐴 × 𝐵)
2		relfld 4846	. . 3 ⊢ (Rel (𝐴 × 𝐵) → ∪ ∪ (𝐴 × 𝐵) = (dom (𝐴 × 𝐵) ∪ ran (𝐴 × 𝐵)))
3	1, 2	ax-mp 7	. 2 ⊢ ∪ ∪ (𝐴 × 𝐵) = (dom (𝐴 × 𝐵) ∪ ran (𝐴 × 𝐵))
4		ancom 253	. . . 4 ⊢ ((∃𝑏 𝑏 ∈ 𝐵 ∧ ∃𝑎 𝑎 ∈ 𝐴) ↔ (∃𝑎 𝑎 ∈ 𝐴 ∧ ∃𝑏 𝑏 ∈ 𝐵))
5		xpm 4745	. . . 4 ⊢ ((∃𝑎 𝑎 ∈ 𝐴 ∧ ∃𝑏 𝑏 ∈ 𝐵) ↔ ∃𝑥 𝑥 ∈ (𝐴 × 𝐵))
6	4, 5	bitri 173	. . 3 ⊢ ((∃𝑏 𝑏 ∈ 𝐵 ∧ ∃𝑎 𝑎 ∈ 𝐴) ↔ ∃𝑥 𝑥 ∈ (𝐴 × 𝐵))
7		dmxpm 4555	. . . 4 ⊢ (∃𝑏 𝑏 ∈ 𝐵 → dom (𝐴 × 𝐵) = 𝐴)
8		rnxpm 4752	. . . 4 ⊢ (∃𝑎 𝑎 ∈ 𝐴 → ran (𝐴 × 𝐵) = 𝐵)
9		uneq12 3092	. . . 4 ⊢ ((dom (𝐴 × 𝐵) = 𝐴 ∧ ran (𝐴 × 𝐵) = 𝐵) → (dom (𝐴 × 𝐵) ∪ ran (𝐴 × 𝐵)) = (𝐴 ∪ 𝐵))
10	7, 8, 9	syl2an 273	. . 3 ⊢ ((∃𝑏 𝑏 ∈ 𝐵 ∧ ∃𝑎 𝑎 ∈ 𝐴) → (dom (𝐴 × 𝐵) ∪ ran (𝐴 × 𝐵)) = (𝐴 ∪ 𝐵))
11	6, 10	sylbir 125	. 2 ⊢ (∃𝑥 𝑥 ∈ (𝐴 × 𝐵) → (dom (𝐴 × 𝐵) ∪ ran (𝐴 × 𝐵)) = (𝐴 ∪ 𝐵))
12	3, 11	syl5eq 2084	1 ⊢ (∃𝑥 𝑥 ∈ (𝐴 × 𝐵) → ∪ ∪ (𝐴 × 𝐵) = (𝐴 ∪ 𝐵))

Syntax hints:   → wi 4   ∧ wa 97   = wceq 1243  ∃wex 1381   ∈ wcel 1393   ∪ cun 2915  ∪ cuni 3580   × cxp 4343  dom cdm 4345  ran crn 4346  Rel wrel 4350

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944

This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-xp 4351  df-rel 4352  df-cnv 4353  df-dm 4355  df-rn 4356

This theorem is referenced by: (None)