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Theorem elrnmpt1sf 43887
Description: Elementhood in an image set. Same as elrnmpt1s 5957, 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 2733 . . 3 𝐶 = 𝐶
2 elrnmpt1sf.1 . . . . 5 𝑥𝐶
32, 2nfeq 2917 . . . 4 𝑥 𝐶 = 𝐶
4 elrnmpt1sf.3 . . . . 5 (𝑥 = 𝐷𝐵 = 𝐶)
54eqeq2d 2744 . . . 4 (𝑥 = 𝐷 → (𝐶 = 𝐵𝐶 = 𝐶))
63, 5rspce 3602 . . 3 ((𝐷𝐴𝐶 = 𝐶) → ∃𝑥𝐴 𝐶 = 𝐵)
71, 6mpan2 690 . 2 (𝐷𝐴 → ∃𝑥𝐴 𝐶 = 𝐵)
8 elrnmpt1sf.2 . . . 4 𝐹 = (𝑥𝐴𝐵)
92, 8elrnmptf 43878 . . 3 (𝐶𝑉 → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥𝐴 𝐶 = 𝐵))
109biimparc 481 . 2 ((∃𝑥𝐴 𝐶 = 𝐵𝐶𝑉) → 𝐶 ∈ ran 𝐹)
117, 10sylan 581 1 ((𝐷𝐴𝐶𝑉) → 𝐶 ∈ ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 397   = wceq 1542  wcel 2107  wnfc 2884  wrex 3071  cmpt 5232  ran crn 5678
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-mpt 5233  df-cnv 5685  df-dm 5687  df-rn 5688
This theorem is referenced by:  sge0f1o  45098
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