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Theorem rnmptssd 38877
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 450 . . 3 (𝜑 → (𝑥𝐴𝐵𝐶))
41, 3ralrimi 2951 . 2 (𝜑 → ∀𝑥𝐴 𝐵𝐶)
5 rnmptssd.2 . . 3 𝐹 = (𝑥𝐴𝐵)
65rnmptss 6350 . 2 (∀𝑥𝐴 𝐵𝐶 → ran 𝐹𝐶)
74, 6syl 17 1 (𝜑 → ran 𝐹𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384   = wceq 1480  wnf 1705  wcel 1987  wral 2907  wss 3556  cmpt 4675  ran crn 5077
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4743  ax-nul 4751  ax-pr 4869
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ne 2791  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-sbc 3419  df-dif 3559  df-un 3561  df-in 3563  df-ss 3570  df-nul 3894  df-if 4061  df-sn 4151  df-pr 4153  df-op 4157  df-uni 4405  df-br 4616  df-opab 4676  df-mpt 4677  df-id 4991  df-xp 5082  df-rel 5083  df-cnv 5084  df-co 5085  df-dm 5086  df-rn 5087  df-res 5088  df-ima 5089  df-iota 5812  df-fun 5851  df-fn 5852  df-f 5853  df-fv 5857
This theorem is referenced by:  infnsuprnmpt  38959  suprclrnmpt  38960  suprubrnmpt2  38961  suprubrnmpt  38962  fisupclrnmpt  39104  supxrleubrnmpt  39114  infxrlbrnmpt2  39119  supxrrernmpt  39130  suprleubrnmpt  39131  infrnmptle  39132  infxrunb3rnmpt  39137  supxrre3rnmpt  39138  sge0xaddlem2  39974  sge0reuz  39987  sge0reuzb  39988  hoidmvlelem2  40133  iunhoiioolem  40212  vonioolem1  40217  smflimsuplem4  40352
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