Theorem iunxpf 4880

Description: Indexed union on a cross product is equals a double indexed union. The hypothesis specifies an implicit substitution. (Contributed by NM, 19-Dec-2008.)

Hypotheses

Ref	Expression
iunxpf.1	⊢ Ⅎ𝑦𝐶
iunxpf.2	⊢ Ⅎ𝑧𝐶
iunxpf.3	⊢ Ⅎ𝑥𝐷
iunxpf.4	⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → 𝐶 = 𝐷)

Assertion

Ref	Expression
iunxpf	⊢ ∪ 𝑥 ∈ (𝐴 × 𝐵)𝐶 = ∪ 𝑦 ∈ 𝐴 ∪ 𝑧 ∈ 𝐵 𝐷

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝑧,𝐵,𝑦

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

Proof of Theorem iunxpf

Dummy variable 𝑤 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		iunxpf.1	. . . . 5 ⊢ Ⅎ𝑦𝐶
2	1	nfcri 2367	. . . 4 ⊢ Ⅎ𝑦 𝑤 ∈ 𝐶
3		iunxpf.2	. . . . 5 ⊢ Ⅎ𝑧𝐶
4	3	nfcri 2367	. . . 4 ⊢ Ⅎ𝑧 𝑤 ∈ 𝐶
5		iunxpf.3	. . . . 5 ⊢ Ⅎ𝑥𝐷
6	5	nfcri 2367	. . . 4 ⊢ Ⅎ𝑥 𝑤 ∈ 𝐷
7		iunxpf.4	. . . . 5 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → 𝐶 = 𝐷)
8	7	eleq2d 2300	. . . 4 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑤 ∈ 𝐶 ↔ 𝑤 ∈ 𝐷))
9	2, 4, 6, 8	rexxpf 4879	. . 3 ⊢ (∃𝑥 ∈ (𝐴 × 𝐵)𝑤 ∈ 𝐶 ↔ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑤 ∈ 𝐷)
10		eliun 3975	. . 3 ⊢ (𝑤 ∈ ∪ 𝑥 ∈ (𝐴 × 𝐵)𝐶 ↔ ∃𝑥 ∈ (𝐴 × 𝐵)𝑤 ∈ 𝐶)
11		eliun 3975	. . . 4 ⊢ (𝑤 ∈ ∪ 𝑦 ∈ 𝐴 ∪ 𝑧 ∈ 𝐵 𝐷 ↔ ∃𝑦 ∈ 𝐴 𝑤 ∈ ∪ 𝑧 ∈ 𝐵 𝐷)
12		eliun 3975	. . . . 5 ⊢ (𝑤 ∈ ∪ 𝑧 ∈ 𝐵 𝐷 ↔ ∃𝑧 ∈ 𝐵 𝑤 ∈ 𝐷)
13	12	rexbii 2538	. . . 4 ⊢ (∃𝑦 ∈ 𝐴 𝑤 ∈ ∪ 𝑧 ∈ 𝐵 𝐷 ↔ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑤 ∈ 𝐷)
14	11, 13	bitri 184	. . 3 ⊢ (𝑤 ∈ ∪ 𝑦 ∈ 𝐴 ∪ 𝑧 ∈ 𝐵 𝐷 ↔ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑤 ∈ 𝐷)
15	9, 10, 14	3bitr4i 212	. 2 ⊢ (𝑤 ∈ ∪ 𝑥 ∈ (𝐴 × 𝐵)𝐶 ↔ 𝑤 ∈ ∪ 𝑦 ∈ 𝐴 ∪ 𝑧 ∈ 𝐵 𝐷)
16	15	eqriv 2227	1 ⊢ ∪ 𝑥 ∈ (𝐴 × 𝐵)𝐶 = ∪ 𝑦 ∈ 𝐴 ∪ 𝑧 ∈ 𝐵 𝐷

Syntax hints:   → wi 4   = wceq 1397   ∈ wcel 2201  Ⅎwnfc 2360  ∃wrex 2510  ⟨cop 3673  ∪ ciun 3971   × cxp 4725

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 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-14 2204  ax-ext 2212  ax-sep 4208  ax-pow 4266  ax-pr 4301

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1810  df-clab 2217  df-cleq 2223  df-clel 2226  df-nfc 2362  df-ral 2514  df-rex 2515  df-v 2803  df-sbc 3031  df-csb 3127  df-un 3203  df-in 3205  df-ss 3212  df-pw 3655  df-sn 3676  df-pr 3677  df-op 3679  df-iun 3973  df-opab 4152  df-xp 4733  df-rel 4734

This theorem is referenced by:  dfmpo  6393