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Theorem rnmptssdf 39981
 Description: The range of an operation given by the "maps to" notation as a subset. (Contributed by Glauco Siliprandi, 23-Oct-2021.)
Hypotheses
Ref Expression
rnmptssdf.1 𝑥𝜑
rnmptssdf.2 𝑥𝐶
rnmptssdf.3 𝐹 = (𝑥𝐴𝐵)
rnmptssdf.4 ((𝜑𝑥𝐴) → 𝐵𝐶)
Assertion
Ref Expression
rnmptssdf (𝜑 → ran 𝐹𝐶)
Distinct variable group:   𝑥,𝐴
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)   𝐶(𝑥)   𝐹(𝑥)

Proof of Theorem rnmptssdf
StepHypRef Expression
1 rnmptssdf.1 . . 3 𝑥𝜑
2 rnmptssdf.4 . . . 4 ((𝜑𝑥𝐴) → 𝐵𝐶)
32ex 397 . . 3 (𝜑 → (𝑥𝐴𝐵𝐶))
41, 3ralrimi 3105 . 2 (𝜑 → ∀𝑥𝐴 𝐵𝐶)
5 rnmptssdf.2 . . 3 𝑥𝐶
6 rnmptssdf.3 . . 3 𝐹 = (𝑥𝐴𝐵)
75, 6rnmptssf 39974 . 2 (∀𝑥𝐴 𝐵𝐶 → ran 𝐹𝐶)
84, 7syl 17 1 (𝜑 → ran 𝐹𝐶)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 382   = wceq 1630  Ⅎwnf 1855   ∈ wcel 2144  Ⅎwnfc 2899  ∀wral 3060   ⊆ wss 3721   ↦ cmpt 4861  ran crn 5250 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750  ax-sep 4912  ax-nul 4920  ax-pr 5034 This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-eu 2621  df-mo 2622  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-ne 2943  df-ral 3065  df-rex 3066  df-rab 3069  df-v 3351  df-sbc 3586  df-csb 3681  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-uni 4573  df-br 4785  df-opab 4845  df-mpt 4862  df-id 5157  df-xp 5255  df-rel 5256  df-cnv 5257  df-co 5258  df-dm 5259  df-rn 5260  df-res 5261  df-ima 5262  df-iota 5994  df-fun 6033  df-fn 6034  df-f 6035  df-fv 6039 This theorem is referenced by:  rnmptss2  39984  supminfrnmpt  40182  supminfxrrnmpt  40211
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