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Theorem elunirn2 29781
 Description: Condition for the membership in the union of the range of a function. (Contributed by Thierry Arnoux, 13-Nov-2016.)
Assertion
Ref Expression
elunirn2 ((Fun 𝐹𝐵 ∈ (𝐹𝐴)) → 𝐵 ran 𝐹)

Proof of Theorem elunirn2
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 elfvdm 6382 . . . 4 (𝐵 ∈ (𝐹𝐴) → 𝐴 ∈ dom 𝐹)
2 fveq2 6353 . . . . . 6 (𝑥 = 𝐴 → (𝐹𝑥) = (𝐹𝐴))
32eleq2d 2825 . . . . 5 (𝑥 = 𝐴 → (𝐵 ∈ (𝐹𝑥) ↔ 𝐵 ∈ (𝐹𝐴)))
43rspcev 3449 . . . 4 ((𝐴 ∈ dom 𝐹𝐵 ∈ (𝐹𝐴)) → ∃𝑥 ∈ dom 𝐹 𝐵 ∈ (𝐹𝑥))
51, 4mpancom 706 . . 3 (𝐵 ∈ (𝐹𝐴) → ∃𝑥 ∈ dom 𝐹 𝐵 ∈ (𝐹𝑥))
65adantl 473 . 2 ((Fun 𝐹𝐵 ∈ (𝐹𝐴)) → ∃𝑥 ∈ dom 𝐹 𝐵 ∈ (𝐹𝑥))
7 elunirn 6673 . . 3 (Fun 𝐹 → (𝐵 ran 𝐹 ↔ ∃𝑥 ∈ dom 𝐹 𝐵 ∈ (𝐹𝑥)))
87adantr 472 . 2 ((Fun 𝐹𝐵 ∈ (𝐹𝐴)) → (𝐵 ran 𝐹 ↔ ∃𝑥 ∈ dom 𝐹 𝐵 ∈ (𝐹𝑥)))
96, 8mpbird 247 1 ((Fun 𝐹𝐵 ∈ (𝐹𝐴)) → 𝐵 ran 𝐹)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 196   ∧ wa 383   = wceq 1632   ∈ wcel 2139  ∃wrex 3051  ∪ cuni 4588  dom cdm 5266  ran crn 5267  Fun wfun 6043  ‘cfv 6049 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-mpt 4882  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-iota 6012  df-fun 6051  df-fn 6052  df-fv 6057 This theorem is referenced by:  measbasedom  30595  sxbrsigalem0  30663
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