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Theorem ssiun2sf 29223
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 2773 . . . 4 𝑥 𝐶𝐴
4 ssiun2sf.3 . . . . 5 𝑥𝐷
5 nfiu1 4516 . . . . 5 𝑥 𝑥𝐴 𝐵
64, 5nfss 3576 . . . 4 𝑥 𝐷 𝑥𝐴 𝐵
73, 6nfim 1822 . . 3 𝑥(𝐶𝐴𝐷 𝑥𝐴 𝐵)
8 eleq1 2686 . . . 4 (𝑥 = 𝐶 → (𝑥𝐴𝐶𝐴))
9 ssiun2sf.4 . . . . 5 (𝑥 = 𝐶𝐵 = 𝐷)
109sseq1d 3611 . . . 4 (𝑥 = 𝐶 → (𝐵 𝑥𝐴 𝐵𝐷 𝑥𝐴 𝐵))
118, 10imbi12d 334 . . 3 (𝑥 = 𝐶 → ((𝑥𝐴𝐵 𝑥𝐴 𝐵) ↔ (𝐶𝐴𝐷 𝑥𝐴 𝐵)))
12 ssiun2 4529 . . 3 (𝑥𝐴𝐵 𝑥𝐴 𝐵)
131, 7, 11, 12vtoclgf 3250 . 2 (𝐶𝐴 → (𝐶𝐴𝐷 𝑥𝐴 𝐵))
1413pm2.43i 52 1 (𝐶𝐴𝐷 𝑥𝐴 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1987  wnfc 2748  wss 3555   ciun 4485
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-v 3188  df-in 3562  df-ss 3569  df-iun 4487
This theorem is referenced by:  iundisj2f  29248  esum2dlem  29935  voliune  30073  volfiniune  30074
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