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Theorem elrnmpt1sf 39889
Description: Elementhood in an image set. Same as elrnmpt1s 5509, 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 2771 . . 3 𝐶 = 𝐶
21nfth 1875 . . . 4 𝑥 𝐶 = 𝐶
3 elrnmpt1sf.3 . . . . 5 (𝑥 = 𝐷𝐵 = 𝐶)
43eqeq2d 2781 . . . 4 (𝑥 = 𝐷 → (𝐶 = 𝐵𝐶 = 𝐶))
52, 4rspce 3455 . . 3 ((𝐷𝐴𝐶 = 𝐶) → ∃𝑥𝐴 𝐶 = 𝐵)
61, 5mpan2 671 . 2 (𝐷𝐴 → ∃𝑥𝐴 𝐶 = 𝐵)
7 elrnmpt1sf.1 . . . 4 𝑥𝐶
8 elrnmpt1sf.2 . . . 4 𝐹 = (𝑥𝐴𝐵)
97, 8elrnmptf 39880 . . 3 (𝐶𝑉 → (𝐶 ∈ ran 𝐹 ↔ ∃𝑥𝐴 𝐶 = 𝐵))
109biimparc 465 . 2 ((∃𝑥𝐴 𝐶 = 𝐵𝐶𝑉) → 𝐶 ∈ ran 𝐹)
116, 10sylan 569 1 ((𝐷𝐴𝐶𝑉) → 𝐶 ∈ ran 𝐹)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wcel 2145  wnfc 2900  wrex 3062  cmpt 4863  ran crn 5250
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751  ax-sep 4915  ax-nul 4923  ax-pr 5034
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-eu 2622  df-mo 2623  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-rex 3067  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4226  df-sn 4317  df-pr 4319  df-op 4323  df-br 4787  df-opab 4847  df-mpt 4864  df-cnv 5257  df-dm 5259  df-rn 5260
This theorem is referenced by:  sge0f1o  41109
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