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Theorem elrnmpt1sf 38850
 Description: Elementhood in an image set. Same as elrnmpt1s 5333, 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 2621 . . 3 𝐶 = 𝐶
21nfth 1724 . . . 4 𝑥 𝐶 = 𝐶
3 elrnmpt1sf.3 . . . . 5 (𝑥 = 𝐷𝐵 = 𝐶)
43eqeq2d 2631 . . . 4 (𝑥 = 𝐷 → (𝐶 = 𝐵𝐶 = 𝐶))
52, 4rspce 3290 . . 3 ((𝐷𝐴𝐶 = 𝐶) → ∃𝑥𝐴 𝐶 = 𝐵)
61, 5mpan2 706 . 2 (𝐷𝐴 → ∃𝑥𝐴 𝐶 = 𝐵)
7 elrnmpt1sf.1 . . . 4 𝑥𝐶
8 elrnmpt1sf.2 . . . 4 𝐹 = (𝑥𝐴𝐵)
97, 8elrnmptf 38841 . . 3 (𝐶𝑉 → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥𝐴 𝐶 = 𝐵))
109biimparc 504 . 2 ((∃𝑥𝐴 𝐶 = 𝐵𝐶𝑉) → 𝐶 ∈ ran 𝐹)
116, 10sylan 488 1 ((𝐷𝐴𝐶𝑉) → 𝐶 ∈ ran 𝐹)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 384   = wceq 1480   ∈ wcel 1987  Ⅎwnfc 2748  ∃wrex 2908   ↦ cmpt 4673  ran crn 5075 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867 This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-br 4614  df-opab 4674  df-mpt 4675  df-cnv 5082  df-dm 5084  df-rn 5085 This theorem is referenced by:  sge0f1o  39906
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