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Theorem elimampt 6063
Description: Membership in the image of a mapping. (Contributed by Thierry Arnoux, 3-Jan-2022.)
Hypotheses
Ref Expression
elimampt.f 𝐹 = (𝑥𝐴𝐵)
elimampt.c (𝜑𝐶𝑊)
elimampt.d (𝜑𝐷𝐴)
Assertion
Ref Expression
elimampt (𝜑 → (𝐶 ∈ (𝐹𝐷) ↔ ∃𝑥𝐷 𝐶 = 𝐵))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶   𝑥,𝐷
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)   𝐹(𝑥)   𝑊(𝑥)

Proof of Theorem elimampt
StepHypRef Expression
1 df-ima 5702 . . 3 (𝐹𝐷) = ran (𝐹𝐷)
21eleq2i 2831 . 2 (𝐶 ∈ (𝐹𝐷) ↔ 𝐶 ∈ ran (𝐹𝐷))
3 elimampt.d . . . 4 (𝜑𝐷𝐴)
4 elimampt.f . . . . . . . 8 𝐹 = (𝑥𝐴𝐵)
54reseq1i 5996 . . . . . . 7 (𝐹𝐷) = ((𝑥𝐴𝐵) ↾ 𝐷)
6 resmpt 6057 . . . . . . 7 (𝐷𝐴 → ((𝑥𝐴𝐵) ↾ 𝐷) = (𝑥𝐷𝐵))
75, 6eqtrid 2787 . . . . . 6 (𝐷𝐴 → (𝐹𝐷) = (𝑥𝐷𝐵))
87rneqd 5952 . . . . 5 (𝐷𝐴 → ran (𝐹𝐷) = ran (𝑥𝐷𝐵))
98eleq2d 2825 . . . 4 (𝐷𝐴 → (𝐶 ∈ ran (𝐹𝐷) ↔ 𝐶 ∈ ran (𝑥𝐷𝐵)))
103, 9syl 17 . . 3 (𝜑 → (𝐶 ∈ ran (𝐹𝐷) ↔ 𝐶 ∈ ran (𝑥𝐷𝐵)))
11 elimampt.c . . . 4 (𝜑𝐶𝑊)
12 eqid 2735 . . . . 5 (𝑥𝐷𝐵) = (𝑥𝐷𝐵)
1312elrnmpt 5972 . . . 4 (𝐶𝑊 → (𝐶 ∈ ran (𝑥𝐷𝐵) ↔ ∃𝑥𝐷 𝐶 = 𝐵))
1411, 13syl 17 . . 3 (𝜑 → (𝐶 ∈ ran (𝑥𝐷𝐵) ↔ ∃𝑥𝐷 𝐶 = 𝐵))
1510, 14bitrd 279 . 2 (𝜑 → (𝐶 ∈ ran (𝐹𝐷) ↔ ∃𝑥𝐷 𝐶 = 𝐵))
162, 15bitrid 283 1 (𝜑 → (𝐶 ∈ (𝐹𝐷) ↔ ∃𝑥𝐷 𝐶 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206   = wceq 1537  wcel 2106  wrex 3068  wss 3963  cmpt 5231  ran crn 5690  cres 5691  cima 5692
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-mpt 5232  df-xp 5695  df-rel 5696  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702
This theorem is referenced by:  reprpmtf1o  34620  ellcsrspsn  35626
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