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Theorem elrnmpt1sf 40126
Description: Elementhood in an image set. Same as elrnmpt1s 5577, but using bound-variable hypotheses instead of distinct variable conditions. (Contributed by Glauco Siliprandi, 17-Aug-2020.)
Hypotheses
Ref Expression
elrnmpt1sf.1 𝑥𝐶
elrnmpt1sf.2 𝐹 = (𝑥𝐴𝐵)
elrnmpt1sf.3 (𝑥 = 𝐷𝐵 = 𝐶)
Assertion
Ref Expression
elrnmpt1sf ((𝐷𝐴𝐶𝑉) → 𝐶 ∈ ran 𝐹)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐷
Allowed substitution hints:   𝐵(𝑥)   𝐶(𝑥)   𝐹(𝑥)   𝑉(𝑥)

Proof of Theorem elrnmpt1sf
StepHypRef Expression
1 eqid 2799 . . 3 𝐶 = 𝐶
21nfth 1897 . . . 4 𝑥 𝐶 = 𝐶
3 elrnmpt1sf.3 . . . . 5 (𝑥 = 𝐷𝐵 = 𝐶)
43eqeq2d 2809 . . . 4 (𝑥 = 𝐷 → (𝐶 = 𝐵𝐶 = 𝐶))
52, 4rspce 3492 . . 3 ((𝐷𝐴𝐶 = 𝐶) → ∃𝑥𝐴 𝐶 = 𝐵)
61, 5mpan2 683 . 2 (𝐷𝐴 → ∃𝑥𝐴 𝐶 = 𝐵)
7 elrnmpt1sf.1 . . . 4 𝑥𝐶
8 elrnmpt1sf.2 . . . 4 𝐹 = (𝑥𝐴𝐵)
97, 8elrnmptf 40117 . . 3 (𝐶𝑉 → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥𝐴 𝐶 = 𝐵))
109biimparc 472 . 2 ((∃𝑥𝐴 𝐶 = 𝐵𝐶𝑉) → 𝐶 ∈ ran 𝐹)
116, 10sylan 576 1 ((𝐷𝐴𝐶𝑉) → 𝐶 ∈ ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 385   = wceq 1653  wcel 2157  wnfc 2928  wrex 3090  cmpt 4922  ran crn 5313
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-rex 3095  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-br 4844  df-opab 4906  df-mpt 4923  df-cnv 5320  df-dm 5322  df-rn 5323
This theorem is referenced by:  sge0f1o  41338
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