Theorem iunxpf 4746

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 2300	. . . 4 ⊢ Ⅎ𝑦 𝑤 ∈ 𝐶
3		iunxpf.2	. . . . 5 ⊢ Ⅎ𝑧𝐶
4	3	nfcri 2300	. . . 4 ⊢ Ⅎ𝑧 𝑤 ∈ 𝐶
5		iunxpf.3	. . . . 5 ⊢ Ⅎ𝑥𝐷
6	5	nfcri 2300	. . . 4 ⊢ Ⅎ𝑥 𝑤 ∈ 𝐷
7		iunxpf.4	. . . . 5 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → 𝐶 = 𝐷)
8	7	eleq2d 2234	. . . 4 ⊢ (𝑥 = ⟨𝑦, 𝑧⟩ → (𝑤 ∈ 𝐶 ↔ 𝑤 ∈ 𝐷))
9	2, 4, 6, 8	rexxpf 4745	. . 3 ⊢ (∃𝑥 ∈ (𝐴 × 𝐵)𝑤 ∈ 𝐶 ↔ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑤 ∈ 𝐷)
10		eliun 3864	. . 3 ⊢ (𝑤 ∈ ∪ 𝑥 ∈ (𝐴 × 𝐵)𝐶 ↔ ∃𝑥 ∈ (𝐴 × 𝐵)𝑤 ∈ 𝐶)
11		eliun 3864	. . . 4 ⊢ (𝑤 ∈ ∪ 𝑦 ∈ 𝐴 ∪ 𝑧 ∈ 𝐵 𝐷 ↔ ∃𝑦 ∈ 𝐴 𝑤 ∈ ∪ 𝑧 ∈ 𝐵 𝐷)
12		eliun 3864	. . . . 5 ⊢ (𝑤 ∈ ∪ 𝑧 ∈ 𝐵 𝐷 ↔ ∃𝑧 ∈ 𝐵 𝑤 ∈ 𝐷)
13	12	rexbii 2471	. . . 4 ⊢ (∃𝑦 ∈ 𝐴 𝑤 ∈ ∪ 𝑧 ∈ 𝐵 𝐷 ↔ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑤 ∈ 𝐷)
14	11, 13	bitri 183	. . 3 ⊢ (𝑤 ∈ ∪ 𝑦 ∈ 𝐴 ∪ 𝑧 ∈ 𝐵 𝐷 ↔ ∃𝑦 ∈ 𝐴 ∃𝑧 ∈ 𝐵 𝑤 ∈ 𝐷)
15	9, 10, 14	3bitr4i 211	. 2 ⊢ (𝑤 ∈ ∪ 𝑥 ∈ (𝐴 × 𝐵)𝐶 ↔ 𝑤 ∈ ∪ 𝑦 ∈ 𝐴 ∪ 𝑧 ∈ 𝐵 𝐷)
16	15	eqriv 2161	1 ⊢ ∪ 𝑥 ∈ (𝐴 × 𝐵)𝐶 = ∪ 𝑦 ∈ 𝐴 ∪ 𝑧 ∈ 𝐵 𝐷

Syntax hints:   → wi 4   = wceq 1342   ∈ wcel 2135  Ⅎwnfc 2293  ∃wrex 2443  ⟨cop 3573  ∪ ciun 3860   × cxp 4596

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-14 2138  ax-ext 2146  ax-sep 4094  ax-pow 4147  ax-pr 4181

This theorem depends on definitions:  df-bi 116  df-3an 969  df-tru 1345  df-nf 1448  df-sb 1750  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ral 2447  df-rex 2448  df-v 2723  df-sbc 2947  df-csb 3041  df-un 3115  df-in 3117  df-ss 3124  df-pw 3555  df-sn 3576  df-pr 3577  df-op 3579  df-iun 3862  df-opab 4038  df-xp 4604  df-rel 4605

This theorem is referenced by:  dfmpo  6182