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Theorem iunxdif2 3705
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 3702 . . 3 (∀𝑥𝐴𝑦 ∈ (𝐴𝐵)𝐶𝐷 𝑥𝐴 𝐶 𝑦 ∈ (𝐴𝐵)𝐷)
2 difss 3070 . . . . 5 (𝐴𝐵) ⊆ 𝐴
3 iunss1 3668 . . . . 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 3696 . . . 4 𝑥𝐴 𝐶 = 𝑦𝐴 𝐷
74, 6sseqtr4i 2978 . . 3 𝑦 ∈ (𝐴𝐵)𝐷 𝑥𝐴 𝐶
81, 7jctil 295 . 2 (∀𝑥𝐴𝑦 ∈ (𝐴𝐵)𝐶𝐷 → ( 𝑦 ∈ (𝐴𝐵)𝐷 𝑥𝐴 𝐶 𝑥𝐴 𝐶 𝑦 ∈ (𝐴𝐵)𝐷))
9 eqss 2960 . 2 ( 𝑦 ∈ (𝐴𝐵)𝐷 = 𝑥𝐴 𝐶 ↔ ( 𝑦 ∈ (𝐴𝐵)𝐷 𝑥𝐴 𝐶 𝑥𝐴 𝐶 𝑦 ∈ (𝐴𝐵)𝐷))
108, 9sylibr 137 1 (∀𝑥𝐴𝑦 ∈ (𝐴𝐵)𝐶𝐷 𝑦 ∈ (𝐴𝐵)𝐷 = 𝑥𝐴 𝐶)
Colors of variables: wff set class
Syntax hints:  wi 4  wa 97   = wceq 1243  wral 2306  wrex 2307  cdif 2914  wss 2917   ciun 3657
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-dif 2920  df-in 2924  df-ss 2931  df-iun 3659
This theorem is referenced by: (None)
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