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Theorem ssiun2sf 29504
Description: Subset relationship for an indexed union. (Contributed by Thierry Arnoux, 31-Dec-2016.)
Hypotheses
Ref Expression
ssiun2sf.1 𝑥𝐴
ssiun2sf.2 𝑥𝐶
ssiun2sf.3 𝑥𝐷
ssiun2sf.4 (𝑥 = 𝐶𝐵 = 𝐷)
Assertion
Ref Expression
ssiun2sf (𝐶𝐴𝐷 𝑥𝐴 𝐵)

Proof of Theorem ssiun2sf
StepHypRef Expression
1 ssiun2sf.2 . . 3 𝑥𝐶
2 ssiun2sf.1 . . . . 5 𝑥𝐴
31, 2nfel 2806 . . . 4 𝑥 𝐶𝐴
4 ssiun2sf.3 . . . . 5 𝑥𝐷
5 nfiu1 4582 . . . . 5 𝑥 𝑥𝐴 𝐵
64, 5nfss 3629 . . . 4 𝑥 𝐷 𝑥𝐴 𝐵
73, 6nfim 1865 . . 3 𝑥(𝐶𝐴𝐷 𝑥𝐴 𝐵)
8 eleq1 2718 . . . 4 (𝑥 = 𝐶 → (𝑥𝐴𝐶𝐴))
9 ssiun2sf.4 . . . . 5 (𝑥 = 𝐶𝐵 = 𝐷)
109sseq1d 3665 . . . 4 (𝑥 = 𝐶 → (𝐵 𝑥𝐴 𝐵𝐷 𝑥𝐴 𝐵))
118, 10imbi12d 333 . . 3 (𝑥 = 𝐶 → ((𝑥𝐴𝐵 𝑥𝐴 𝐵) ↔ (𝐶𝐴𝐷 𝑥𝐴 𝐵)))
12 ssiun2 4595 . . 3 (𝑥𝐴𝐵 𝑥𝐴 𝐵)
131, 7, 11, 12vtoclgf 3295 . 2 (𝐶𝐴 → (𝐶𝐴𝐷 𝑥𝐴 𝐵))
1413pm2.43i 52 1 (𝐶𝐴𝐷 𝑥𝐴 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1523  wcel 2030  wnfc 2780  wss 3607   ciun 4552
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-v 3233  df-in 3614  df-ss 3621  df-iun 4554
This theorem is referenced by:  iundisj2f  29529  esum2dlem  30282  voliune  30420  volfiniune  30421
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