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Theorem iunxdif2 3734
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 3731 . . 3 (∀𝑥𝐴𝑦 ∈ (𝐴𝐵)𝐶𝐷 𝑥𝐴 𝐶 𝑦 ∈ (𝐴𝐵)𝐷)
2 difss 3099 . . . . 5 (𝐴𝐵) ⊆ 𝐴
3 iunss1 3697 . . . . 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 3725 . . . 4 𝑥𝐴 𝐶 = 𝑦𝐴 𝐷
74, 6sseqtr4i 3033 . . 3 𝑦 ∈ (𝐴𝐵)𝐷 𝑥𝐴 𝐶
81, 7jctil 305 . 2 (∀𝑥𝐴𝑦 ∈ (𝐴𝐵)𝐶𝐷 → ( 𝑦 ∈ (𝐴𝐵)𝐷 𝑥𝐴 𝐶 𝑥𝐴 𝐶 𝑦 ∈ (𝐴𝐵)𝐷))
9 eqss 3015 . 2 ( 𝑦 ∈ (𝐴𝐵)𝐷 = 𝑥𝐴 𝐶 ↔ ( 𝑦 ∈ (𝐴𝐵)𝐷 𝑥𝐴 𝐶 𝑥𝐴 𝐶 𝑦 ∈ (𝐴𝐵)𝐷))
108, 9sylibr 132 1 (∀𝑥𝐴𝑦 ∈ (𝐴𝐵)𝐶𝐷 𝑦 ∈ (𝐴𝐵)𝐷 = 𝑥𝐴 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 102   = wceq 1285  wral 2349  wrex 2350  cdif 2971  wss 2974   ciun 3686
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064
This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-rex 2355  df-v 2604  df-dif 2976  df-in 2980  df-ss 2987  df-iun 3688
This theorem is referenced by: (None)
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