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Theorem elimampt 30385
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 5570 . . 3 (𝐹𝐷) = ran (𝐹𝐷)
21eleq2i 2906 . 2 (𝐶 ∈ (𝐹𝐷) ↔ 𝐶 ∈ ran (𝐹𝐷))
3 elimampt.d . . . 4 (𝜑𝐷𝐴)
4 elimampt.f . . . . . . . 8 𝐹 = (𝑥𝐴𝐵)
54reseq1i 5851 . . . . . . 7 (𝐹𝐷) = ((𝑥𝐴𝐵) ↾ 𝐷)
6 resmpt 5907 . . . . . . 7 (𝐷𝐴 → ((𝑥𝐴𝐵) ↾ 𝐷) = (𝑥𝐷𝐵))
75, 6syl5eq 2870 . . . . . 6 (𝐷𝐴 → (𝐹𝐷) = (𝑥𝐷𝐵))
87rneqd 5810 . . . . 5 (𝐷𝐴 → ran (𝐹𝐷) = ran (𝑥𝐷𝐵))
98eleq2d 2900 . . . 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 2823 . . . . 5 (𝑥𝐷𝐵) = (𝑥𝐷𝐵)
1312elrnmpt 5830 . . . 4 (𝐶𝑊 → (𝐶 ∈ ran (𝑥𝐷𝐵) ↔ ∃𝑥𝐷 𝐶 = 𝐵))
1411, 13syl 17 . . 3 (𝜑 → (𝐶 ∈ ran (𝑥𝐷𝐵) ↔ ∃𝑥𝐷 𝐶 = 𝐵))
1510, 14bitrd 281 . 2 (𝜑 → (𝐶 ∈ ran (𝐹𝐷) ↔ ∃𝑥𝐷 𝐶 = 𝐵))
162, 15syl5bb 285 1 (𝜑 → (𝐶 ∈ (𝐹𝐷) ↔ ∃𝑥𝐷 𝐶 = 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 208   = wceq 1537  wcel 2114  wrex 3141  wss 3938  cmpt 5148  ran crn 5558  cres 5559  cima 5560
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795  ax-sep 5205  ax-nul 5212  ax-pr 5332
This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2802  df-cleq 2816  df-clel 2895  df-nfc 2965  df-rex 3146  df-rab 3149  df-v 3498  df-dif 3941  df-un 3943  df-in 3945  df-ss 3954  df-nul 4294  df-if 4470  df-sn 4570  df-pr 4572  df-op 4576  df-br 5069  df-opab 5131  df-mpt 5149  df-xp 5563  df-rel 5564  df-cnv 5565  df-dm 5567  df-rn 5568  df-res 5569  df-ima 5570
This theorem is referenced by:  reprpmtf1o  31899
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