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Theorem elrnmpt1sf 41755
Description: Elementhood in an image set. Same as elrnmpt1s 5806, 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 2822 . . 3 𝐶 = 𝐶
2 elrnmpt1sf.1 . . . . 5 𝑥𝐶
32, 2nfeq 2992 . . . 4 𝑥 𝐶 = 𝐶
4 elrnmpt1sf.3 . . . . 5 (𝑥 = 𝐷𝐵 = 𝐶)
54eqeq2d 2833 . . . 4 (𝑥 = 𝐷 → (𝐶 = 𝐵𝐶 = 𝐶))
63, 5rspce 3587 . . 3 ((𝐷𝐴𝐶 = 𝐶) → ∃𝑥𝐴 𝐶 = 𝐵)
71, 6mpan2 690 . 2 (𝐷𝐴 → ∃𝑥𝐴 𝐶 = 𝐵)
8 elrnmpt1sf.2 . . . 4 𝐹 = (𝑥𝐴𝐵)
92, 8elrnmptf 41746 . . 3 (𝐶𝑉 → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥𝐴 𝐶 = 𝐵))
109biimparc 483 . 2 ((∃𝑥𝐴 𝐶 = 𝐵𝐶𝑉) → 𝐶 ∈ ran 𝐹)
117, 10sylan 583 1 ((𝐷𝐴𝐶𝑉) → 𝐶 ∈ ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 399   = wceq 1538  wcel 2114  wnfc 2960  wrex 3131  cmpt 5122  ran crn 5533
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2178  ax-ext 2794  ax-sep 5179  ax-nul 5186  ax-pr 5307
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2622  df-eu 2653  df-clab 2801  df-cleq 2815  df-clel 2894  df-nfc 2962  df-rex 3136  df-v 3471  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4266  df-if 4440  df-sn 4540  df-pr 4542  df-op 4546  df-br 5043  df-opab 5105  df-mpt 5123  df-cnv 5540  df-dm 5542  df-rn 5543
This theorem is referenced by:  sge0f1o  42961
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