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Theorem elpreimad 39971
Description: Membership in the preimage of a set under a function. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypotheses
Ref Expression
elpreimad.f (𝜑𝐹 Fn 𝐴)
elpreimad.b (𝜑𝐵𝐴)
elpreimad.c (𝜑 → (𝐹𝐵) ∈ 𝐶)
Assertion
Ref Expression
elpreimad (𝜑𝐵 ∈ (𝐹𝐶))

Proof of Theorem elpreimad
StepHypRef Expression
1 elpreimad.b . . 3 (𝜑𝐵𝐴)
2 elpreimad.c . . 3 (𝜑 → (𝐹𝐵) ∈ 𝐶)
31, 2jca 555 . 2 (𝜑 → (𝐵𝐴 ∧ (𝐹𝐵) ∈ 𝐶))
4 elpreimad.f . . 3 (𝜑𝐹 Fn 𝐴)
5 elpreima 6501 . . 3 (𝐹 Fn 𝐴 → (𝐵 ∈ (𝐹𝐶) ↔ (𝐵𝐴 ∧ (𝐹𝐵) ∈ 𝐶)))
64, 5syl 17 . 2 (𝜑 → (𝐵 ∈ (𝐹𝐶) ↔ (𝐵𝐴 ∧ (𝐹𝐵) ∈ 𝐶)))
73, 6mpbird 247 1 (𝜑𝐵 ∈ (𝐹𝐶))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 196  wa 383  wcel 2139  ccnv 5265  cima 5269   Fn wfn 6044  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-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  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-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-iota 6012  df-fun 6051  df-fn 6052  df-fv 6057
This theorem is referenced by:  smfsuplem1  41541
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