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Theorem abfmpunirn 30419
Description: Membership in a union of a mapping function-defined family of sets. (Contributed by Thierry Arnoux, 28-Sep-2016.)
Hypotheses
Ref Expression
abfmpunirn.1 𝐹 = (𝑥𝑉 ↦ {𝑦𝜑})
abfmpunirn.2 {𝑦𝜑} ∈ V
abfmpunirn.3 (𝑦 = 𝐵 → (𝜑𝜓))
Assertion
Ref Expression
abfmpunirn (𝐵 ran 𝐹 ↔ (𝐵 ∈ V ∧ ∃𝑥𝑉 𝜓))
Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐹,𝑦   𝑥,𝑉,𝑦   𝜓,𝑦
Allowed substitution hints:   𝜑(𝑥,𝑦)   𝜓(𝑥)

Proof of Theorem abfmpunirn
StepHypRef Expression
1 elex 3462 . 2 (𝐵 ran 𝐹𝐵 ∈ V)
2 abfmpunirn.2 . . . . . 6 {𝑦𝜑} ∈ V
3 abfmpunirn.1 . . . . . 6 𝐹 = (𝑥𝑉 ↦ {𝑦𝜑})
42, 3fnmpti 6467 . . . . 5 𝐹 Fn 𝑉
5 fnunirn 6994 . . . . 5 (𝐹 Fn 𝑉 → (𝐵 ran 𝐹 ↔ ∃𝑥𝑉 𝐵 ∈ (𝐹𝑥)))
64, 5ax-mp 5 . . . 4 (𝐵 ran 𝐹 ↔ ∃𝑥𝑉 𝐵 ∈ (𝐹𝑥))
73fvmpt2 6760 . . . . . . 7 ((𝑥𝑉 ∧ {𝑦𝜑} ∈ V) → (𝐹𝑥) = {𝑦𝜑})
82, 7mpan2 690 . . . . . 6 (𝑥𝑉 → (𝐹𝑥) = {𝑦𝜑})
98eleq2d 2878 . . . . 5 (𝑥𝑉 → (𝐵 ∈ (𝐹𝑥) ↔ 𝐵 ∈ {𝑦𝜑}))
109rexbiia 3212 . . . 4 (∃𝑥𝑉 𝐵 ∈ (𝐹𝑥) ↔ ∃𝑥𝑉 𝐵 ∈ {𝑦𝜑})
116, 10bitri 278 . . 3 (𝐵 ran 𝐹 ↔ ∃𝑥𝑉 𝐵 ∈ {𝑦𝜑})
12 abfmpunirn.3 . . . . 5 (𝑦 = 𝐵 → (𝜑𝜓))
1312elabg 3617 . . . 4 (𝐵 ∈ V → (𝐵 ∈ {𝑦𝜑} ↔ 𝜓))
1413rexbidv 3259 . . 3 (𝐵 ∈ V → (∃𝑥𝑉 𝐵 ∈ {𝑦𝜑} ↔ ∃𝑥𝑉 𝜓))
1511, 14syl5bb 286 . 2 (𝐵 ∈ V → (𝐵 ran 𝐹 ↔ ∃𝑥𝑉 𝜓))
161, 15biadanii 821 1 (𝐵 ran 𝐹 ↔ (𝐵 ∈ V ∧ ∃𝑥𝑉 𝜓))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 209  wa 399   = wceq 1538  wcel 2112  {cab 2779  wrex 3110  Vcvv 3444   cuni 4803  cmpt 5113  ran crn 5524   Fn wfn 6323  cfv 6328
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2114  ax-9 2122  ax-10 2143  ax-11 2159  ax-12 2176  ax-ext 2773  ax-sep 5170  ax-nul 5177  ax-pow 5234  ax-pr 5298
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2601  df-eu 2632  df-clab 2780  df-cleq 2794  df-clel 2873  df-nfc 2941  df-ral 3114  df-rex 3115  df-rab 3118  df-v 3446  df-sbc 3724  df-csb 3832  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4247  df-if 4429  df-sn 4529  df-pr 4531  df-op 4535  df-uni 4804  df-br 5034  df-opab 5096  df-mpt 5114  df-id 5428  df-xp 5529  df-rel 5530  df-cnv 5531  df-co 5532  df-dm 5533  df-rn 5534  df-res 5535  df-ima 5536  df-iota 6287  df-fun 6330  df-fn 6331  df-fv 6336
This theorem is referenced by:  rabfmpunirn  30420  isrnsiga  31486  isrnmeas  31573
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