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Theorem iunxdif2 3800
 Description: Indexed union with a class difference as its index. (Contributed by NM, 10-Dec-2004.)
Hypothesis
Ref Expression
iunxdif2.1 (𝑥 = 𝑦𝐶 = 𝐷)
Assertion
Ref Expression
iunxdif2 (∀𝑥𝐴𝑦 ∈ (𝐴𝐵)𝐶𝐷 𝑦 ∈ (𝐴𝐵)𝐷 = 𝑥𝐴 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
StepHypRef Expression
1 iunss2 3797 . . 3 (∀𝑥𝐴𝑦 ∈ (𝐴𝐵)𝐶𝐷 𝑥𝐴 𝐶 𝑦 ∈ (𝐴𝐵)𝐷)
2 difss 3141 . . . . 5 (𝐴𝐵) ⊆ 𝐴
3 iunss1 3763 . . . . 5 ((𝐴𝐵) ⊆ 𝐴 𝑦 ∈ (𝐴𝐵)𝐷 𝑦𝐴 𝐷)
42, 3ax-mp 7 . . . 4 𝑦 ∈ (𝐴𝐵)𝐷 𝑦𝐴 𝐷
5 iunxdif2.1 . . . . 5 (𝑥 = 𝑦𝐶 = 𝐷)
65cbviunv 3791 . . . 4 𝑥𝐴 𝐶 = 𝑦𝐴 𝐷
74, 6sseqtr4i 3074 . . 3 𝑦 ∈ (𝐴𝐵)𝐷 𝑥𝐴 𝐶
81, 7jctil 306 . 2 (∀𝑥𝐴𝑦 ∈ (𝐴𝐵)𝐶𝐷 → ( 𝑦 ∈ (𝐴𝐵)𝐷 𝑥𝐴 𝐶 𝑥𝐴 𝐶 𝑦 ∈ (𝐴𝐵)𝐷))
9 eqss 3054 . 2 ( 𝑦 ∈ (𝐴𝐵)𝐷 = 𝑥𝐴 𝐶 ↔ ( 𝑦 ∈ (𝐴𝐵)𝐷 𝑥𝐴 𝐶 𝑥𝐴 𝐶 𝑦 ∈ (𝐴𝐵)𝐷))
108, 9sylibr 133 1 (∀𝑥𝐴𝑦 ∈ (𝐴𝐵)𝐶𝐷 𝑦 ∈ (𝐴𝐵)𝐷 = 𝑥𝐴 𝐶)
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   = wceq 1296  ∀wral 2370  ∃wrex 2371   ∖ cdif 3010   ⊆ wss 3013  ∪ ciun 3752 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077 This theorem depends on definitions:  df-bi 116  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-v 2635  df-dif 3015  df-in 3019  df-ss 3026  df-iun 3754 This theorem is referenced by: (None)
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