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Theorem rnmptssd 39904
Description: The range of an operation given by the "maps to" notation as a subset. (Contributed by Glauco Siliprandi, 11-Oct-2020.)
Hypotheses
Ref Expression
rnmptssd.1 𝑥𝜑
rnmptssd.2 𝐹 = (𝑥𝐴𝐵)
rnmptssd.3 ((𝜑𝑥𝐴) → 𝐵𝐶)
Assertion
Ref Expression
rnmptssd (𝜑 → ran 𝐹𝐶)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐶
Allowed substitution hints:   𝜑(𝑥)   𝐵(𝑥)   𝐹(𝑥)

Proof of Theorem rnmptssd
StepHypRef Expression
1 rnmptssd.1 . . 3 𝑥𝜑
2 rnmptssd.3 . . . 4 ((𝜑𝑥𝐴) → 𝐵𝐶)
32ex 397 . . 3 (𝜑 → (𝑥𝐴𝐵𝐶))
41, 3ralrimi 3106 . 2 (𝜑 → ∀𝑥𝐴 𝐵𝐶)
5 rnmptssd.2 . . 3 𝐹 = (𝑥𝐴𝐵)
65rnmptss 6537 . 2 (∀𝑥𝐴 𝐵𝐶 → ran 𝐹𝐶)
74, 6syl 17 1 (𝜑 → ran 𝐹𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382   = wceq 1631  wnf 1856  wcel 2145  wral 3061  wss 3723  cmpt 4864  ran crn 5251
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 4916  ax-nul 4924  ax-pr 5035
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-ne 2944  df-ral 3066  df-rex 3067  df-rab 3070  df-v 3353  df-sbc 3588  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-uni 4576  df-br 4788  df-opab 4848  df-mpt 4865  df-id 5158  df-xp 5256  df-rel 5257  df-cnv 5258  df-co 5259  df-dm 5260  df-rn 5261  df-res 5262  df-ima 5263  df-iota 5993  df-fun 6032  df-fn 6033  df-f 6034  df-fv 6038
This theorem is referenced by:  infnsuprnmpt  39980  suprclrnmpt  39981  suprubrnmpt2  39982  suprubrnmpt  39983  fisupclrnmpt  40135  supxrleubrnmpt  40145  infxrlbrnmpt2  40150  supxrrernmpt  40161  suprleubrnmpt  40162  infrnmptle  40163  infxrunb3rnmpt  40168  supxrre3rnmpt  40169  supminfrnmpt  40185  infxrrnmptcl  40188  infxrgelbrnmpt  40196  infrpgernmpt  40208  supminfxrrnmpt  40214  liminfcl  40510  sge0xaddlem2  41165  sge0reuz  41178  sge0reuzb  41179  hoidmvlelem2  41327  iunhoiioolem  41406  vonioolem1  41411  smflimsuplem4  41546
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