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Theorem afvelrnb0 44219
Description: A member of a function's range is a value of the function, only one direction of implication of fvelrnb 6733. (Contributed by Alexander van der Vekens, 1-Jun-2017.)
Assertion
Ref Expression
afvelrnb0 (𝐹 Fn 𝐴 → (𝐵 ∈ ran 𝐹 → ∃𝑥𝐴 (𝐹'''𝑥) = 𝐵))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐹

Proof of Theorem afvelrnb0
Dummy variable 𝑦 is distinct from all other variables.
StepHypRef Expression
1 fnrnafv 44217 . . 3 (𝐹 Fn 𝐴 → ran 𝐹 = {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹'''𝑥)})
21eleq2d 2819 . 2 (𝐹 Fn 𝐴 → (𝐵 ∈ ran 𝐹𝐵 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹'''𝑥)}))
3 eqeq1 2743 . . . . . 6 (𝑦 = 𝐵 → (𝑦 = (𝐹'''𝑥) ↔ 𝐵 = (𝐹'''𝑥)))
4 eqcom 2746 . . . . . 6 (𝐵 = (𝐹'''𝑥) ↔ (𝐹'''𝑥) = 𝐵)
53, 4bitrdi 290 . . . . 5 (𝑦 = 𝐵 → (𝑦 = (𝐹'''𝑥) ↔ (𝐹'''𝑥) = 𝐵))
65rexbidv 3208 . . . 4 (𝑦 = 𝐵 → (∃𝑥𝐴 𝑦 = (𝐹'''𝑥) ↔ ∃𝑥𝐴 (𝐹'''𝑥) = 𝐵))
76elabg 3572 . . 3 (𝐵 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹'''𝑥)} → (𝐵 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹'''𝑥)} ↔ ∃𝑥𝐴 (𝐹'''𝑥) = 𝐵))
87ibi 270 . 2 (𝐵 ∈ {𝑦 ∣ ∃𝑥𝐴 𝑦 = (𝐹'''𝑥)} → ∃𝑥𝐴 (𝐹'''𝑥) = 𝐵)
92, 8syl6bi 256 1 (𝐹 Fn 𝐴 → (𝐵 ∈ ran 𝐹 → ∃𝑥𝐴 (𝐹'''𝑥) = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wcel 2114  {cab 2717  wrex 3055  ran crn 5527   Fn wfn 6335  '''cafv 44172
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2020  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2162  ax-12 2179  ax-ext 2711  ax-sep 5168  ax-nul 5175  ax-pow 5233  ax-pr 5297
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2075  df-mo 2541  df-eu 2571  df-clab 2718  df-cleq 2731  df-clel 2812  df-nfc 2882  df-ne 2936  df-ral 3059  df-rex 3060  df-rab 3063  df-v 3401  df-sbc 3682  df-csb 3792  df-dif 3847  df-un 3849  df-in 3851  df-ss 3861  df-nul 4213  df-if 4416  df-sn 4518  df-pr 4520  df-op 4524  df-uni 4798  df-int 4838  df-br 5032  df-opab 5094  df-mpt 5112  df-id 5430  df-xp 5532  df-rel 5533  df-cnv 5534  df-co 5535  df-dm 5536  df-rn 5537  df-res 5538  df-iota 6298  df-fun 6342  df-fn 6343  df-fv 6348  df-aiota 44139  df-dfat 44174  df-afv 44175
This theorem is referenced by:  ffnafv  44226
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