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Theorem elmapdd 8831
Description: Deduction associated with elmapd 8830. (Contributed by SN, 29-Jul-2024.)
Hypotheses
Ref Expression
elmapdd.a (𝜑𝐴𝑉)
elmapdd.b (𝜑𝐵𝑊)
elmapdd.c (𝜑𝐶:𝐵𝐴)
Assertion
Ref Expression
elmapdd (𝜑𝐶 ∈ (𝐴m 𝐵))

Proof of Theorem elmapdd
StepHypRef Expression
1 elmapdd.c . 2 (𝜑𝐶:𝐵𝐴)
2 elmapdd.a . . 3 (𝜑𝐴𝑉)
3 elmapdd.b . . 3 (𝜑𝐵𝑊)
42, 3elmapd 8830 . 2 (𝜑 → (𝐶 ∈ (𝐴m 𝐵) ↔ 𝐶:𝐵𝐴))
51, 4mpbird 257 1 (𝜑𝐶 ∈ (𝐴m 𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  wf 6536  (class class class)co 7404  m cmap 8816
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7720
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-mo 2535  df-eu 2564  df-clab 2711  df-cleq 2725  df-clel 2811  df-nfc 2886  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-sbc 3777  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-fv 6548  df-ov 7407  df-oprab 7408  df-mpo 7409  df-map 8818
This theorem is referenced by:  elrspunsn  32505  mapcod  41016  mhmcompl  41070  mhmcoaddmpl  41073  evlsbagval  41088  selvvvval  41107  evlselv  41109  mhphf  41119
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