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Theorem iunssd 38756
Description: Subset theorem for an indexed union. (Contributed by Glauco Siliprandi, 8-Apr-2021.)
Hypothesis
Ref Expression
iunssd.1 ((𝜑𝑥𝐴) → 𝐵𝐶)
Assertion
Ref Expression
iunssd (𝜑 𝑥𝐴 𝐵𝐶)
Distinct variable groups:   𝑥,𝐶   𝜑,𝑥
Allowed substitution hints:   𝐴(𝑥)   𝐵(𝑥)

Proof of Theorem iunssd
StepHypRef Expression
1 iunssd.1 . . 3 ((𝜑𝑥𝐴) → 𝐵𝐶)
21ralrimiva 2960 . 2 (𝜑 → ∀𝑥𝐴 𝐵𝐶)
3 iunss 4527 . 2 ( 𝑥𝐴 𝐵𝐶 ↔ ∀𝑥𝐴 𝐵𝐶)
42, 3sylibr 224 1 (𝜑 𝑥𝐴 𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wcel 1987  wral 2907  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:  meaiininclem  40007  smflim  40292  smfresal  40302  smfmullem4  40308
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