Theorem iunxdif2 3965

Description: Indexed union with a class difference as its index. (Contributed by NM, 10-Dec-2004.)

Hypothesis

Ref	Expression
iunxdif2.1	⊢ (𝑥 = 𝑦 → 𝐶 = 𝐷)

Assertion

Ref	Expression
iunxdif2	⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ (𝐴 ∖ 𝐵)𝐶 ⊆ 𝐷 → ∪ 𝑦 ∈ (𝐴 ∖ 𝐵)𝐷 = ∪ 𝑥 ∈ 𝐴 𝐶)

Distinct variable groups:   𝑥,𝑦,𝐴   𝑥,𝐵,𝑦   𝑦,𝐶   𝑥,𝐷

Allowed substitution hints:   𝐶(𝑥)   𝐷(𝑦)

Proof of Theorem iunxdif2

Step	Hyp	Ref	Expression
1		iunss2 3961	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ (𝐴 ∖ 𝐵)𝐶 ⊆ 𝐷 → ∪ 𝑥 ∈ 𝐴 𝐶 ⊆ ∪ 𝑦 ∈ (𝐴 ∖ 𝐵)𝐷)
2		difss 3289	. . . . 5 ⊢ (𝐴 ∖ 𝐵) ⊆ 𝐴
3		iunss1 3927	. . . . 5 ⊢ ((𝐴 ∖ 𝐵) ⊆ 𝐴 → ∪ 𝑦 ∈ (𝐴 ∖ 𝐵)𝐷 ⊆ ∪ 𝑦 ∈ 𝐴 𝐷)
4	2, 3	ax-mp 5	. . . 4 ⊢ ∪ 𝑦 ∈ (𝐴 ∖ 𝐵)𝐷 ⊆ ∪ 𝑦 ∈ 𝐴 𝐷
5		iunxdif2.1	. . . . 5 ⊢ (𝑥 = 𝑦 → 𝐶 = 𝐷)
6	5	cbviunv 3955	. . . 4 ⊢ ∪ 𝑥 ∈ 𝐴 𝐶 = ∪ 𝑦 ∈ 𝐴 𝐷
7	4, 6	sseqtrri 3218	. . 3 ⊢ ∪ 𝑦 ∈ (𝐴 ∖ 𝐵)𝐷 ⊆ ∪ 𝑥 ∈ 𝐴 𝐶
8	1, 7	jctil 312	. 2 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ (𝐴 ∖ 𝐵)𝐶 ⊆ 𝐷 → (∪ 𝑦 ∈ (𝐴 ∖ 𝐵)𝐷 ⊆ ∪ 𝑥 ∈ 𝐴 𝐶 ∧ ∪ 𝑥 ∈ 𝐴 𝐶 ⊆ ∪ 𝑦 ∈ (𝐴 ∖ 𝐵)𝐷))
9		eqss 3198	. 2 ⊢ (∪ 𝑦 ∈ (𝐴 ∖ 𝐵)𝐷 = ∪ 𝑥 ∈ 𝐴 𝐶 ↔ (∪ 𝑦 ∈ (𝐴 ∖ 𝐵)𝐷 ⊆ ∪ 𝑥 ∈ 𝐴 𝐶 ∧ ∪ 𝑥 ∈ 𝐴 𝐶 ⊆ ∪ 𝑦 ∈ (𝐴 ∖ 𝐵)𝐷))
10	8, 9	sylibr 134	1 ⊢ (∀𝑥 ∈ 𝐴 ∃𝑦 ∈ (𝐴 ∖ 𝐵)𝐶 ⊆ 𝐷 → ∪ 𝑦 ∈ (𝐴 ∖ 𝐵)𝐷 = ∪ 𝑥 ∈ 𝐴 𝐶)

Syntax hints:   → wi 4   ∧ wa 104   = wceq 1364  ∀wral 2475  ∃wrex 2476   ∖ cdif 3154   ⊆ wss 3157  ∪ ciun 3916

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-in1 615  ax-in2 616  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-ral 2480  df-rex 2481  df-v 2765  df-dif 3159  df-in 3163  df-ss 3170  df-iun 3918

This theorem is referenced by: (None)