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Theorem mptex2 6382
Description: If a class given as a map-to notation is a set, it's image values are set. (Contributed by Glauco Siliprandi, 11-Dec-2019.)
Hypothesis
Ref Expression
mptex2.1 (𝜑 → (𝑡𝐴𝐵):𝐴𝐶)
Assertion
Ref Expression
mptex2 ((𝜑𝑡𝐴) → 𝐵𝐶)
Distinct variable groups:   𝑡,𝐴   𝑡,𝐶
Allowed substitution hints:   𝜑(𝑡)   𝐵(𝑡)

Proof of Theorem mptex2
StepHypRef Expression
1 mptex2.1 . . 3 (𝜑 → (𝑡𝐴𝐵):𝐴𝐶)
2 eqid 2621 . . . 4 (𝑡𝐴𝐵) = (𝑡𝐴𝐵)
32fmpt 6379 . . 3 (∀𝑡𝐴 𝐵𝐶 ↔ (𝑡𝐴𝐵):𝐴𝐶)
41, 3sylibr 224 . 2 (𝜑 → ∀𝑡𝐴 𝐵𝐶)
54r19.21bi 2931 1 ((𝜑𝑡𝐴) → 𝐵𝐶)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 384  wcel 1989  wral 2911  cmpt 4727  wf 5882
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pr 4904
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-sbc 3434  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-uni 4435  df-br 4652  df-opab 4711  df-mpt 4728  df-id 5022  df-xp 5118  df-rel 5119  df-cnv 5120  df-co 5121  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125  df-iota 5849  df-fun 5888  df-fn 5889  df-f 5890  df-fv 5894
This theorem is referenced by:  divcncf  23210  cncfcompt  39865  cncficcgt0  39870  cncfcompt2  39881  itgsubsticclem  39960  sge0iunmptlemre  40401  hoicvrrex  40539  smfadd  40742
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